diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzadgd" "b/data_all_eng_slimpj/shuffled/split2/finalzzadgd"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzadgd"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nThe edge computing, offers a decentralized solution to cloud-based datacenters \\cite{2018edge} and intelligence-at-the-edge of mobile networks. However, training on the edge is a challenge for many deep neural networks (DNNs). This arises due to the significant cost of multiply-and-accumulate (MAC) units, an ubiquitous operation in all DNNs. In a 45~nm CMOS process, energy consumption doubles from 16-bit floats to 32-bit floats for addition and by ${\\sim}4\\times$ for multiplication \\cite{horowitz20141}. Memory access cost increases by ${\\sim}10\\times$ from 8~kB to 1~MB memory with a 64-bit cache \\cite{horowitz20141}. In general, there is a gap between memory storage, bandwidth, compute requirements, and energy consumption of modern DNNs and hardware resources available on edge devices \\cite{facebookEdge}.\n\nAn apparent solution to address this gap is to compress such networks, thus reducing the compute requirements to match putative edge resources. Several groups have proposed compressed new compute- and memory-efficient DNN architectures \\cite{MobileNet,chen2019drop,MEC} and parameter-efficient neural networks, using methods such as DNN pruning \\cite{ren2018sbnet}, distillation \\cite{zhou2018revisiting}, and low-precision arithmetic \\cite{Jacob_2018_CVPR,IBM8}. Among these approaches, low-precision arithmetic is noted for its ability to reduce memory capacity, bandwidth, latency, and energy consumption associated with MAC units in DNNs and an increase in the level of data parallelism \\cite{Jacob_2018_CVPR, hashemi2017, gysel2018}.\n\n\n\n\n\nThe ultimate goal of low-precision DNN design is to reduce the original hardware complexity of the high-precision DNN model to a level suitable for edge devices without significantly degrading performance.\n\nTo address the gaps in previous studies, we are motivated to study low-precision posit for DNN training on the edge.\n\n\n\\section{Posit Numerical Format}\n\n\nAn alternative to IEEE-754 floating point numbers, posits were recently introduced and exhibit a tapered-precision characteristic \\cite{gustafson2017beating}. Posits, a Type III unum, offer an elegant resolution to many of the shortcomings of IEEE-754 floating format and address limitations of both Type I and Type II unums \\cite{tichy2016unums}. Moreover, posits provide better accuracy, dynamic range, and program reproducibility than IEEE floating point. The essential advantage of posits is their capability to represent nonlinearly distributed numbers in a specific dynamic range about unity ({$\\pm$}1.0) with high accuracy.\nThe value of a posit bit-string is governed by \\eqref{equ:equ00}, where $s$ represents the sign, $es$ the maximal exponential bits, $fs$ the maximal fractional bits, $e$ the exponent value, $f$ the fraction value, and $k$ the regime value (as given by \\eqref{equ:equ01}).\n\\begin{equation}\n    x= \n\\begin{cases}\n    0, & \\text{if } ({\\tt 00...0})   \\\\\n    NaR, & \\text{if }  ({\\tt 10...0}) \\\\\n    (-1)^{s}\\times 2^{2^{es} \\times k} \\times 2^e \\times \\left(1+\\frac{f}{2^{fs}} \\right), & \\text{otherwise}\n\\end{cases}\n\\label{equ:equ00}\n\\end{equation}\nThe regime is encoded based on the \\emph{runlength} $m$ of identical bits $(r...r)$ terminated by either a \\emph{regime terminating bit}~$\\overline{r}$ or the end of the bit-string of size $n$. Note that there is no requirement to distinguish between negative and positive zero since only a single bit pattern ${\\tt (00...0)}$ represents zero. Furthermore, instead of defining a ``Not-a-Number'' ($NaN$) for exceptional values and infinity by various bit patterns, a single bit pattern $({\\tt 10...0})$, ``Not-a-Real'' ($NaR$), represents all such values. Furthermore, $NaR$ never arises due to overflow or underflow. More details about the posit numerical format can be found in \\cite{gustafson2017beating}.\n\\begin{equation}\n    k= \n\\begin{cases}\n    -m , & \\text{if } r= 0\\\\\n     m-1, & \\text{if } r= 1\n\\end{cases}\n\\label{equ:equ01}\n\\end{equation}\n\n\\section{Related Work}\n\n\nAs early as the 1980s, low-precision arithmetic has been explored in shallow neural networks to decrease both compute and memory complexity for training and inference without deteriorating performance \\cite{graf1988vlsi,iwata1989artificial,hammerstrom1990vlsi, asanovic1991experimental}. In some scenarios, this bit-precision constraint also improves DNN performance due to the quantization noise acting as a regularization method \\cite{asanovic1991experimental,bishop1995training}.\nThe outcome of these studies indicate that 16- and 8-bit precision DNN parameters are capable of satisfactorily maintaining performance for both training and inference in shallow networks \\cite{iwata1989artificial,hammerstrom1990vlsi,asanovic1991experimental}. The capability of low-precision arithmetic is reevaluated in the deep learning era to reduce memory footprint and energy consumption during training and inference \\cite{Courbariaux14,Gupta,micikevicius2017mixed,flexpoint2017,IBM8,mellempudi2019mixed,kalamkar2019study,gysel2018,hashemi2017,Microsoft2018,carmichael2019positron,carmichael2019performance,Hamed2018,johnson2018rethinking}. \n\nIn performing DNN training, several previous studies utilize either variants of low-precision block floating point (BFP) (blocks of floating point DNN parameters that share an exponent \\cite{wilkinson1965rounding})\nor mixed-precision floating point.\nThese methods are sufficient to maintain similar performance as 32-bit high-precision floating point. For instance, Courbariaux \\textit{et al.} trained a low-precision DNN on the MNIST, CIFAR-10, and SVHN datasets with the floating point, fixed-point, and BFP numerical formats \\cite{Courbariaux14} and show that BFP achieves the best performance due to variability between the dynamic range and precision of DNN parameters \\cite{Courbariaux14}. Following, Koster \\textit{et al.} proposed the Flexpoint numerical format and a new algorithm, Autoflex, which analyzes the statistics of the history of DNN parameters, to optimally select the shared exponents for DNN parameters iteration-wise during gradient descent \\cite{flexpoint2017}.\n\nAside from\nthe BFP numerical format, Narang \\textit{et al.} explored mixed-precision floating point \\cite{micikevicius2017mixed} using 16-bit floating point weights, activations, and gradients during both the forward and backward passes. To prevent accuracy loss caused by underflow\nin 16-bit floating point, the weights are updated with 32-bit floating point. Additionally, to prevent gradients with very small magnitude from becoming zero when represented by 16-bit floating point, a new loss scaling approach is proposed.\n\nRecently, Wang \\textit{et al.} and Mellempudi \\textit{et al.} propose a method to reduce the bit-precision of weights, activations, and gradients to 8 bits by exhaustively analyzing DNN parameters during training \\cite{IBM8,mellempudi2019mixed}. In \\cite{mellempudi2019mixed}, a new chunk-based addition is presented to solve the truncation issue caused by the addition of large- and small-magnitude numbers, thus successfully reducing the number of bits for the accumulator and weight updates to 16 bits. To mitigate requiring loss scaling in mixed-precision floating point training, Kalamkar \\textit{et al.} \\cite{kalamkar2019study} proposed the brain floating point (BFLOAT-16) half-precision format with a reduced 8-bit fractional precision\nand similar dynamic range (7-bit exponent)\nto 32-bit floating point. A side effect of this representation is that the conversion complexity between these BFLOAT-16 and IEEE floating point is reduced during training. In training a ResNet model on the ImageNet dataset, BFLOAT-16s achieve the same performance as 32-bit floating point.\n\nThis research builds on earlier studies \\cite{carmichael2019positron,carmichael2019performance,Hamed2018,johnson2018rethinking,montero2019template} and for the first time studies feedforward neural network training with posits on MNIST and Fashion MNIST datasets .\n\n\n\\section{Proposed Work, Results \\& Analysis}\nTo perform DNN training in feedforward neural networks, we study two numerical formats (floating point and posit) with 32-bit and 16-bit precision. For simplicity, the\narchitecture\nexplained here is based on three hidden layers feedforward neural network training with the posit numerical format as shown in Fig. \\ref{fig:Software-Cheetah}.\n\n\\begin{figure*\n\\centering\n\\includegraphics[width=0.6\\linewidth]{Figures\/ShortPaperPage2.pdf}\n\\caption{The software framework for training the feedforward neural networks with three hidden layers using posit numerical format. The framework scales to any DNN architecture. P16: Posit16; $\\beta_1 \\& \\beta_2$: Exponential decay rates. }\n\\label{fig:Software-Cheetah}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\noindent\n\n\n\nThe networks are implemented in the Keras \\cite{keras} and TensorFlow \\cite{tensorflow2015} frameworks. \\{16, 32\\}-bit floating point and posit numbers for DNN training are extended to these frameworks via software emulation.\n\nTo compare the posit and floating point numerical formats, a four-layer feedforward neural network is trained with each of the number system on the MNIST and Fashion-MNIST datasets. The results indicate that posits have improved accuracy in comparison to floating point at both 16- and 32-bit precision, as shown in Table \\ref{tab:train_acc}.\nAlthough the evaluation is demonstrated on small datasets, there are two advantages compared to \\cite{IBM8,mellempudi2019mixed}. Mellempudi \\textit{et al.} \\cite{mellempudi2019mixed} use 32-bit numbers for accumulation to reduce the hardware cost of stochastic rounding. Wang \\textit{et al.} \\cite{IBM8} reduce the accumulation bit-precision to 16 by using stochastic rounding. However, in this paper, we show the potential of using 16-bit posits for \\textit{all DNN parameters} with a simple and hardware-friendly round-to-nearest algorithm and show less than 2\\% accuracy degradation without exhaustively analyzing the network training parameters.\n\n\\begin{table}[H]\n\\caption{Average accuracy over 10 independent runs {on the test set of the respective dataset. Networks are trained using only the specified numerical format}.}\\label{tab:train_acc}\n\\centering\n\\ra{1.2}\n\\begin{threeparttable}\n\\begin{tabular}{@{}lcc@{}}\n    \\toprule\n    Task          & Format   & Accuracy \\\\\n    \\midrule\n    \\multirow{4}{*}{MNIST}         & Posit-32 & \\textbf{98.131\\%} \\\\\n                                   & Float-32 &         98.087\\%  \\\\\n                                   & Posit-16 &         96.535\\%  \\\\\n                                   & Float-16 &         90.646\\%  \\\\[+1ex]\n    \\multirow{4}{*}{Fashion MNIST} & Posit-32 & \\textbf{89.263\\%} \\\\\n                                   & Float-32 &         89.105\\%  \\\\\n                                   & Posit-16 &         87.400\\%  \\\\\n                                   & Float-16 &         81.725\\%  \\\\\n    \\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{table}\n\n\nA summary of recent studies that propose low-precision training frameworks are shown in Table \\ref{tab:compare}. Several research groups have explored the efficacy of floats and BFP on the performance of DNNs with multiple image classification tasks \\cite{micikevicius2017mixed,flexpoint2017,mellempudi2019mixed,IBM8,kalamkar2019study,montero2019template}. However, the majority of these works analyze the appropriateness of the posit numerical format for DNN training.\n\n\n\\begin{table*}\n   \\caption{High-level summary of proposed work and other low-precision training frameworks. All datasets are image classification tasks. FMNIST: Fashion MNIST; FP: floating point; FX: fixed-point; PS: posit.}\n   \\label{tab:compare}\n   \\ra{1.3}\n   \\centering\n   \\resizebox{\\linewidth}{!}{%\n   \\begin{tabular}{@{}cccccccc@{}}\n     \\toprule\n     & Montero \\textit{et al.} \\cite{montero2019template}& Narang \\textit{et al.} \\cite{micikevicius2017mixed} & Koster \\textit{et al.} \\cite{flexpoint2017} & Mellempudi \\textit{et al.} \\cite{mellempudi2019mixed} & Wang  \\textit{et al.} \\cite{IBM8} & Kalamkar \\textit{et al.} \\cite{kalamkar2019study} & This Work \\\\\n     \\midrule\n   \n   \n   \n     \\multirow{2}{*}{Dataset} & \\multirow{2}{*}{Synthetic} & \\multirow{2}{*}{ImageNet} & \\multirow{2}{*}{{CIFAR}-10} & \\multirow{2}{*}{ImageNet} & \\multirow{2}{*}{ImageNet} & \\multirow{2}{*}{ImageNet} & \\multirow{2}{*}{MNIST, FMNIST} \\\\[-1ex]\n     &  && & &&&  \\\\\n     \\multirow{2}{*}{Numerical Format} & \\multirow{2}{*}{PS} & \\multirow{2}{*}{FP} & \\multirow{2}{*}{BFP} & \\multirow{2}{*}{FP} & \\multirow{2}{*}{FP} & \\multirow{2}{*}{BFLOAT} & \\multirow{2}{*} {PS, FP} \\\\[-1ex]\n     &  &  & & & & &  \\\\\n   \n     Bit-precision & [8..16] & 16\/32 & 16+5 & 8\/32 & 8\/16 & 16 & 16 \\\\ \n   \n     \\multirow{2}{*} {DNN library} & \\multirow{2}{*} {PySigmoid} & \\multirow{2}{*} {Caffe\/PyTorch} & \\multirow{2}{*} {Neon} & \\multirow{2}{*} {TensorFlow} & \\multirow{2}{*} {Home {S}uite} & \\multirow{1}{*} {IntelCaffe\/Caffe2} & \\multirow{2}{*} {Keras\/TensorFlow} \\\\[-1ex]\n     & & & & & & Neon\/Tensorflow& \\\\\n     \\bottomrule\n   \\end{tabular}\n   }\n   \\label{tab:OthersWork}\n \\end{table*}\n \n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\nThis work presented low-precision posit designs for both training and inference on edge devices.\nWe show that the novel posit numerical format has high efficacy for DNN training at \\{16, 32\\}-bit precision training, surpassing the equal-bandwidth fixed-point and floating point counterparts.\nThe success of posits in these experiments, needs further exploration of ultra-low precision posit training for richer datasets.\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}        \\label{sec:intro}\nBeam polarisation is a key ingredient to the physics programme of future linear colliders.\nIn particular the International Linear Collider~\\cite{tdr} foresees in its baseline configuration\nlongitudinal polarisation of the electron and positron beams of $ \\ensuremath{\\polzraw}=\\pm 80\\,\\%$ and $\\pm 30\\,\\%$,\nrespectively. Upgrade options comprise a higher degree of positron polarisation of up to $\\pm 60\\,\\%$.\n\nDue to the chiral nature of weak interactions in the Standard Model, most cross-sections depend linearly on\nthe longitudinal beam polarisations, rendering the knowledge of the luminosity-weighted average polarisations\nat the $e^+e^-$ interaction point as important as the knowledge of the luminosity itself. This does primarily\napply to electroweak precision measurements and indirect searches for new physics, e.g.\\ via anomalous\ncouplings of the top quark~\\cite{Amjad:2013tlv, powerReport}. Moreover, it also concerns direct searches for\nsmall signals above large irreducible backgrounds, e.g.\\  WIMP Dark Matter searches in the mono-photon\nsignature~\\cite{Bartels:2012ex}. While the luminosity is expected to be measured to a few permille based on\nsmall-angle Bhabha scattering~\\cite{Bozovic-Jelisavcic:2013aca}, permille-level precision is challenging for\npolarimetry. The design goal is thus to reach $\\delta\\ensuremath{\\polzraw}\/\\ensuremath{\\polzraw}=0.25\\,\\%$, while physics would profit\nfurther if $0.1\\,\\%$ could be reached.\n\nThe overall concept to determine the beam polarisation is based on the interplay of several complementary\napproaches: Laser-Compton polarimeters will monitor the instantaneous polarisation in some distance from the\n$e^+e^-$ interaction point (IP). Their fast measurements give important feed-back to the accelerator\noperators, but also track time variations on longer time scales as well as possible patterns within a\nbunch train. On each beam, there will be one polarimeter $\\sim 1650$\\,m upstream, and a second one\n$\\sim 150$\\,m downstream of the IP. While the upstream polarimeter measures the initial polarisation under\nvery clean conditions, the downstream polarimeter serves a double purpose: In absence of collisions, the\npolarimeters can be cross-calibrated, if the spin transport between both locations can be predicted with\nsufficient precision, i.e.\\ to $0.1\\,\\%$ or better.\nIn collision mode, the depolarising effects of the beam-beam interaction and the luminosity-weighted average\npolarisation at the IP could be monitored. This enterprise, however, depends crucially on a sufficiently\nprecise understanding of all effects of the beam-beam interaction and the spin transport, which is the main\nobjective of this publication.\n\nThe luminosity-weighted average polarisation at the IP can be directly accessed from $e^+e^-$ collision data\nthemselves. Several approaches have been studied for this in the\npast~\\cite{Moenig, thesisIvan, EPSprocedings}, comprising schemes operating on the\nmeasurements of total cross-sections for various polarisation configurations, as well as on single- and\ndouble-differential distributions of $W^+W^-$ production. All these approaches will finally yield a very\nimportant long-term scale calibration of the luminosity-weighted average polarisation when appropriately\ncompared with the polarimeter measurements. However, they all will take years of data taking before reaching\npermille-level precision. Furthermore, they assume that the helicity reversal is exact, i.e.\\ that the\nabsolute value of the polarisation is the same for all helicity configurations. Any deviation from this\nassumption can only be corrected for based on the polarimeter measurements and their propagation to the IP.\nIt has been shown that the need for such a correction could limit the precision of the collision data methods at the precision of the\npolarimeters~\\cite{thesisIvan}. The difference in absolute polarisation values between data sets, however, can\nbe minimized if the helicity of both beams can be reversed quickly and independently, e.g.\\ on a\npulse-by-pulse basis~\\cite{Riemann:2009wy}. For the electron beam, this is readily provided by switching the\nhelicity of the source laser. For the positron beam, currently a scheme based on switching between two spin\nrotator beam lines as proposed in~\\cite{Riemann:2012zza} is foreseen~\\cite{tdr}.\n\nThis paper is organised as follows: Sections~\\ref{sec:bds} and~\\ref{sec:sim} introduce the beam delivery\nsystem of the ILC  and the spin-tracking formalism, respectively. The design and the capabilities of the\nLaser-Compton polarimeters will be summarised in section~\\ref{sec:polarimetry}. In section~\\ref{sec:crosscal}\nwe will study the achievable precision for cross-calibration of the polarimeters in the absence of\ncollisions, while section~\\ref{sec:lumipol} discusses the extraction of the luminosity-weighted average\npolarisation in the presence of collisions. We conclude in section~\\ref{sec:conclusions} with the prospects\nfor reaching the goal of a few permille precision and give an outlook on studies required in the future.\n\n\\section{Accelerator Environment}       \\label{sec:bds}\nIn this section the parts of the ILC most relevant for the determination of the luminosity-weighted average\npolarisation at the $e^+e^-$ interaction point will be introduced. This includes the beam delivery systems\nand the extraction lines as well as the different beam parameter sets used in the simulation study.\nSince most of the aspects studied in this paper are identical between electrons and positrons, we will use\nthe term ``electron'' for both beams, unless specific differences need to be pointed out.\n\n\\subsection{The ILC Beam Delivery System and Extraction Line} \\label{subsec:ilcbds}\n\\begin{figure}[hbt]\n   \\centerline{\n      \\includegraphics*[width=0.75\\textwidth]{figures\/elec-bds}\n   }\n   \\caption[]{Layout of the electron beam delivery system (BDS). The positron BDS is a mirror image.\n              From figure~2.12 in~\\cite{tdr32}.}\n   \\label{fig:bds-SB2009}\n\\end{figure}\n\nThe beam delivery system (BDS) is an about $2$\\,km long set of beamlines which serves the final preparation of the\nfully accelerated beams for collisions. It hosts beam diagnostics, skew correction, betatron and energy\ncollimation as well as the final focus system, and has been carefully designed to minimise the emittance\ngrowth. This applies in particular to the vertical direction, since flat beams are essential to reconcile\nhigh luminosity with minimal beamstrahlung~\\cite{thesisschulte}. Therefore, all bends in the BDS are in the horizontal plane.\nFigure~\\ref{fig:bds-SB2009} shows a sketch of the BDS for the electron beam up to the $e^+e^-$ IP. The\nupstream polarimeter is located directly behind the branch-off to the tune-up dump. The positron BDS is\na mirror image, apart from the positron production system (left part of figure~\\ref{fig:bds-SB2009}).\n\n\\begin{figure}[hbt]\n   \\centerline{\n   \\includegraphics*[width=0.75\\textwidth]{figures\/downstreamPolChicane}\n   }\n   \\caption[]{Schematic view of the extraction line up to the downstream polarimeter.\n   From~\\cite{boogert}.}\n   \\label{fig:downstreamPolChicane}\n\\end{figure}\n\nFigure~\\ref{fig:downstreamPolChicane} shows the extraction line, which guides the spent beams to the dumps\nand provides post-collision diagnostics. In particular, the downstream polarimeter is located at a secondary\nfocus point of the optics, in order to provide the best possible beam conditions at the Compton IP of the\ndownstream polarimeter.\n\n\\subsection{Beam Parameters and Interaction Region}       \\label{subsec:beampars}\n\n\\begin{table}[hbt]\n   \\centering\n   \\begin{tabular}{lllcc}\n      Parameter                       & \\multicolumn{2}{l}{Symbol}              & RDR        & TDR         \\\\\n      \\hline\n      Bunches per train               &                       &                 & $2\\,625$   & $1\\,312$    \\\\\n      Train frequency                 &                       & [Hz]            & $5$        & $5$         \\\\\n      Horizontal bunch size           & $\\sigma_{xe}$         & [nm]            & $639$      & $474$       \\\\\n      Vertical   bunch size           & $\\sigma_{ye}$         & [nm]            & $5.7$      & $5.9$       \\\\\n      Horizontal angular spread       & $\\theta_x$            & [$\\mu$rad]      & $32$       & $43$        \\\\\n      Vertical   angular spread       & $\\theta_y$            & [$\\mu$rad]      & $14$       & $12$        \\\\\n      Horizontal norm. emittance      & $\\gamma\\varepsilon_x$ & [$\\mu$m]        & $10$       & $10$        \\\\\n      Vertical   norm. emittance      & $\\gamma\\varepsilon_y$ & [nm]            & $40$       & $35$        \\\\\n      Horizontal disruption parameter & $D_x$                 &                 & $0.17$     & $0.3$       \\\\\n      Vertical   disruption parameter & $D_y$                 &                 & $19.4$     & $24.6$      \\\\\n      Beam energy spread ($e^-, e^+$) & $\\sigma_E\/E$          & [$10^{-3}$]     & $1.4, 1.0$ & $1.24, 0.7$ \\\\\n      $e^+e^-$ luminosity             & $\\cal L$  & [$10^{34}$\\,cm$^{-2}$\\,s$^{-1}$] & $2$   & $1.8$       \\\\\n   \\end{tabular}\n   \\caption[]{Selected beam parameters at the IP for $E_{cm}=500\\,$GeV according to RDR~\\cite{rdr3} (nominal parameter\n              set) and TDR~\\cite{tdr32} (baseline parameters).}\n   \\label{tab:beamParametersDesign}\n\\end{table}\n\n\nTable~\\ref{tab:beamParametersDesign} lists the design beam parameters at the IP according to the Reference\nDesign Report (RDR) \\cite{rdr} and the more recent Technical Design Report (TDR) \\cite{tdr}. Most of the\nstudies presented in this paper have been performed with the RDR parameters, since they match the available lattice\\footnote{The \\textit{lattice} describes the layout of a beamline, e.g.\\ positions and strengths of\nthe magnets.}.\nFor the current TDR parameter set, the number of bunches per train is reduced with respect to the RDR.\nIn order to restore the luminosity, the beams are focussed more strongly at the IP. This results in more intense collisions, which might also affect the polarsation. Therefore,\nthe collision effects and the spin transport to the downstream polarimeter have been studied for both\nparameter sets. The electron and positron beam parameters are identical apart from the beam energy spread\n$\\sigma_E\/E$, which is slightly increased for the electrons during their passage through the undulator of\nthe positron source.\n\nThe ILC design foresees a horizontal crossing angle of $14\\,$mrad between the $e^-$ and the $e^+$ beamlines at the IP.\nIn order to maximise the luminosity, the bunches are correspondingly rotated by $7\\,$mrad using\ncrab cavities~\\cite{Adolphsen:2007zza}.\n\nThe collider experiments at the IP contain two types of magnets which also influence the beams. These are the\nmain solenoid for the tracking system and the anti-DID (detector-integrated dipole) to guide\nelectron-positron pairs produced by beamstrahlung photons into the outgoing beam pipe~\\cite{Seryi:2006ja}. Due to the crossing\nangle, the detector magnets at the IP are rotated by $7\\,$mrad with respect to each of the beamlines.\nThese magnets are not yet present in the official lattice files, but have been included in our simulations\nby hand.\n\n\\subsection{Magnet Misalignments and Orbit Correction}       \\label{subsec:misalignments}\nMisalignments of the magnets in an accelerator have various effects, among others beam jitter (deviations of\nthe beam orbit from the design orbit) or a detuned focussing. There are static misalignments due to the\nlimited precision to which the beamline elements can be adjusted at their designated positions, and\ntime-dependent misalignments from various sources, e.g.\\ from seismic noise, traffic, or cooling water pumps.\n\nTo minimise the beam jitter and prevent a beam loss in the worst case, the ILC will be equipped with several\nfeedback orbit correction systems. These systems consist of dipole magnets to perform the orbit correction\nand of beam position monitors, which measure the current beam position from which the required field\nstrengths for the correction magnets are recalculated. They operate on a timescale of a tenth of a second,\ni.e.\\ the measurements from one bunch train are used to correct the orbit for the next bunch train. At the IP,\na higher precision is required to bring the beams with vertical sizes of few nanometers to collision with the\nenvisaged luminosity. Therefore, an additional fast-feedback system is foreseen at the IP, which operates\non a timescale of nanoseconds, i.e.\\ bunch-to-bunch. The interplay of ground motion and the fast-feedback\nsystem and their effects on luminosity and polarisation have been studied\npreviously~\\cite{RestaLopez:2008zza, Bailey:2011ey}. It has been shown in particular that without correction,\nground-motion-induced misalignments and the resulting depolarisation become significant only at time scales\nof a day. Thus, the feedback required for maintaining luminosity is considered sufficient to also maintain\npolarisation, and we will not study time-dependent misalignments in this publication.\n\n\n\\section{Spin Transport and Collision Effects}       \\label{sec:sim}\nIn this section, the basic effects on the polarisation which occur in the beam delivery system, the\ninteraction region, and the extraction line as well as their implementation in the simulation are introduced.\n\n\\subsection{Spin Transport}       \\label{subsec:spinTransport}\nThe spins of particles in an accelerator are subject to spin precession in electromagnetic fields and to\nspin-flips from the emission of photons. For the bunch propagation through the ILC beam delivery system,\nit turns out that the effects of spin-flips can be neglected (cf.\\ section~\\ref{sec:crosscal}). The energy\nloss due to synchrotron radiation, however, affects the particle trajectories and the spin precession\ndownstream.\n\nThe change of a particle spin vector $\\vec{S}$ with time under the influence of electromagnetic fields is\ndescribed by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation~\\cite{Thomas:1926dy, Bargmann:1959gz}.\nSince neither the beam delivery system nor the extraction line contains components with sizable electric\nfields, the full T-BMT equation simplifies to\n\\begin{equation}\n\\label{eqn:TBMT}\n   \\dfrac{d}{dt} \\vec{S} = \\vec{\\Omega}_B\\br{\\vec{B}, \\vec{r},\\vec{p},t} \\times \\vec{S}\n                         = -\\dfrac{q}{m\\gamma}\\br{ \\br{1+a\\gamma}\\vec{B}\n                           - \\dfrac{a\\,\\vec{p}\\cdot\\vec{B}}{(\\gamma+1)\\, m^2 c^2}\\,\\vec{p} } \\times \\vec{S}.\n\\end{equation}\n\nHere, $\\vec{B}\\br{\\vec{r},t}$ denotes the magnetic field, $\\gamma$ the relativistic Lorentz factor,\n$c$ the vacuum speed of light and $a \\equiv (g-2)\/2$ the anomaly of the gyro-magnetic moment, with\n$a \\approx 0.001159652$ for electrons.\n\nThe expression for $\\vec{\\Omega}_B$ in equation~\\ref{eqn:TBMT} can be decomposed in two parts for the field\ncomponents, $\\vec{B}_\\parallel$ parallel to $\\vec{p}$ and $\\vec{B}_\\perp$ perpendicular to it:\n\\begin{equation}\n   \\vec{\\Omega}_B\\br{\\vec{B}, \\vec{r},\\vec{p},t} = -\\dfrac{q}{m\\gamma}\\brbigg{\\br{1+a\\gamma}\\vec{B}_\\perp\\;\\; +\\;\\;\\br{1+a}\\vec{B}_\\parallel}\n   \\label{eq:tbmtOmegaBdecomposed}\n\\end{equation}\n\nIn presence of only perpendicular magnetic fields, the momentum $\\vec{p}$ and the spin $\\vec{S}$ behave very\nsimilarly:\n\\begin{align}\n   \\dfrac{d}{dt} \\vec{p} =& -\\dfrac{q}{m\\gamma}\\br{\\phantom{\\br{1+a\\gamma}}\\vec{B}_\\perp} \\times \\vec{p}\\\\\n   \\dfrac{d}{dt} \\vec{S} =& -\\dfrac{q}{m\\gamma}\\br{         \\br{1+a\\gamma} \\vec{B}_\\perp} \\times \\vec{S}\n\\end{align}\n\nThus, the spin vector precesses in a perpendicular magnetic field about $\\vec{B}$ by the angle\n\\begin{equation}\n   \\xis = \\br{1+a\\gamma}\\,\\xio,\n   \\label{eq:tbmtTheta}\n\\end{equation}\nwhere \\xio\\ is the deflection angle of the particle~\\cite{hoffstaetter,thesisjeff}.\nFor an electron beam with an energy of $250\\,$GeV, the amplification factor is\n$\\br{1+a\\gamma}\\approx 568$; for an energy of $500\\,$GeV, it rises to $\\approx 1136$.\n\nFor a particle beam with a spatial extension, an angular divergence and an energy spread in an inhomogeneous\nmagnetic field, the spin precessions for different particles can vary significantly.\nOne possible pattern emerging from non-uniform spin precession is referred to as ``spin fan-out'' in the following\nand occurs in inhomogeneous magnetic fields or in the presence of a beam energy spread.\nFigure~\\ref{fig:spinFanOut} illustrates spin fan-out and its reversability using the example of a hypothetical two-particle bunch traversing\nquadrupole magnets: Both particles are deflected into different directions in the first quadrupole and the\nspin vectors precess correspondingly according to equation~\\ref{eq:tbmtTheta}. Thus, the longitudinal\npolarisation decreases, but the transverse polarisation remains zero since the transverse components of the\ntwo spin vectors cancel each other. Consequently, the magnitude of the polarisation decreases:\n$|\\polvec'|<|\\polvec|$. This can in principle be reversed by a second quadrupole which rotates the spin\nvectors back to the original orientation.\n\n\\begin{figure}[ht]\n   \\begin{center}\n      \\includegraphics[width=0.5\\columnwidth]{figures\/spinFanOut}\n   \\end{center}\n   \\caption{Spin fan-out at the example of two hypothetical particles. This sketch only illustrates the\n            possible effects of quadrupole magnets on the beam polarisation. It does not describe realistic\n            focussing, the two shown quadrupole magnets do not have the same strength, and betatron\n            oscillations in two dimensions are not indicated. \\cite{thesisMoritz}}\n   \\label{fig:spinFanOut}\n\\end{figure}\n\nIn a beamline consisting of F0D0 cells\\footnote{Standard set of focussing (F) and defocussing (D) quadrupoles\ninterleaved with drift spaces (0).}, however, these spin precessions occur alternatingly in the horizontal and\nthe vertical plane. Rotations in two dimensions do generally not commute, but this fact can be neglected if\nthe rotations are sufficiently small. In that case, the spin fan-out can be set into relation to the angular\ndivergence \\ensuremath{\\thrraw}\\ (i.e.\\ to the fan-out of the momentum vectors) of the bunch by the following function,\nassuming a longitudinally polarised beam, for which the maximum polarisation $\\pol_\\text{max}$ is obtained\nfor $\\ensuremath{\\thrraw}=0$ (as in figure~\\ref{fig:spinFanOut}):\n\\begin{equation}\n   f(\\ensuremath{\\thrraw})=\\pol_\\text{max}\\cdot\\cos\\brBig{\\brbig{1+a\\mean{\\gamma}}\\cdot\\ensuremath{\\thrraw}},\n   \\label{eq:fThr}\n\\end{equation}\nwhere $\\mean{\\gamma}$ denotes the average relativistic Lorentz factor.\n\nIn presence of a beam energy spread, a similar behavior occurs (also in homogeneous magnetic fields),\nsince the ``amplification factor'' $\\br{1+a\\gamma}$ is energy-dependent.\nThis fan-out is in principle reversible as well.\n\nSo it might seem that polarisation can be generated by spin fan-out, but that is actually not the case.\nSpin fan-out allows only to restore an existing ordering in the spin orientation which is preserved in a\ncorrelation between the spin orientation and the particle energy or a particle coordinate. It is not\npossible to restore polarisation which got lost in stochastic processes like radiative depolarisation,\nwhich will be introduced in the following section.\n\nThe description of the spin fan-out by equation~\\ref{eq:fThr}, however, relies on sufficiently small rotations.\nVerifying the validity of this assumption for the ILC BDS is one of the objectives of a full spin tracking study.\n\n\\subsection{Beam-Beam Collisions and Luminosity-Weighted Average Polarisation}\n\\label{subsec:beambeamCollisions}\nIn the beam-beam collisions at the IP, the colliding bunches distort each other by their electromagnetic\nfields, while only a few particles undergo hard interactions. The polarisation of these particles is the\nluminosity-weighted polarisation for a single collision of an electron bunch with a positron bunch, which\nwill be denoted here by the symbol \\ensuremath{\\vec{\\cal P}^\\text{lumi,1}}. It is luminosity-weighted with respect to the local distribution\nof the luminosity during a collision.\n\n\nThe decisive quantity for interpreting collision data is the luminosity-weighted polarsation averaged over\ntime \\ensuremath{\\vec{\\cal P}^\\text{lumi}}\\ for each of the beams, defined as\n\\begin{equation}\n   \\ensuremath{\\vec{\\cal P}^\\text{lumi}} = \\dfrac{\\int \\lumi(t) \\ensuremath{\\vec{\\cal P}^\\text{lumi,1}}(t)\\,dt}{\\int \\lumi(t)\\,dt},\n   \\label{eqn:pollumi}\n\\end{equation}\nand in particular its longitudinal component $\\ensuremath{{\\polzraw^\\text{lumi}}}$. Depending on the chiral structure of the studied\nobservable, it might also be advantageous to consider the time average of one of the effective\npolarisations~\\cite{Moenig} listed in table~\\ref{tab:effpol} instead, which can all be defined in analogy to\nequation~\\ref{eqn:pollumi}.\n\\begin{table}[hbt]\n   \\centering\n   \\begin{tabular}{c|l|l}\n   effective polarisation & observable & type of interaction\\\\\n   \\hline\n   $\\frac{\\ensuremath{\\polraw_{e^-}} + \\ensuremath{\\polraw_{e^+}}}{1 + \\ensuremath{\\polraw_{e^-}} \\ensuremath{\\polraw_{e^+}}}$ & $A_{LR}$ & $s$-channel vector exchange \\\\\n   $\\ensuremath{\\polraw_{e^-}} \\ensuremath{\\polraw_{e^+}}$ & cross-section & $s$-channel vector exchange \\\\\n   $\\ensuremath{\\polraw_{e^-}} + \\ensuremath{\\polraw_{e^+}} - \\ensuremath{\\polraw_{e^-}} \\ensuremath{\\polraw_{e^+}}$ & cross-section & $t$-channel $W$ or $\\nu_e$ production \\\\\n   \\hline\n   \\end{tabular}\n   \\caption[]{Effective polarisations minimising the impact of polarisation uncertainties on various observables.}\n   \\label{tab:effpol}\n\\end{table}\nThe gain in error reduction when using the appropriate effective polarisations depends on the\ndegree of correlation between the measurements of electron and positron polarisation. In this\nrespect, it is reasonable to assume that the instrumental uncertainties of the polarimeters\nas well as the influence of misalignments in the BDS are uncorrelated. Collision effects on\nthe other hand have to be assumed to be highly correlated, since the intensity of the collision is reciprocal.\n\nIn head-on collisions, the already focussed bunches attract each other even further due to their mutual electrical fields (pinch effect~\\cite{thesisschulte}).\nThe T-BMT precession due to this mutual focussing of the bunches leads to a spin fan-out like in a quadrupole\nmagnet. For flat, longitudinally polarised beams ($\\sigma_{xe}\\gg\\sigma_{ye}$, $\\ensuremath{\\polzraw} = \\pol$) and a small\nhorizontal disruption parameter ($D_x\\ll 1$), the polarisations before ($\\pol^\\text{bef}$) and after ($\\pol^\\text{aft}$) the collision as well as the luminosity-weighted polarisation \\ensuremath{\\pol^\\text{lumi,1}}\\ are related as follows\n(equation~16 in \\cite{yokokaChen}):\n\\begin{equation}\n   \\pol^\\text{bef}-\\ensuremath{\\pol^\\text{lumi,1}} = 0.273 \\br{\\pol^\\text{bef}-\\pol^\\text{aft}}\n   \\label{eq:yokoyaChen16}\n\\end{equation}\n\nIf the angular divergence of the bunches before the collision is negligible in terms of spin fan-out, the\nspin fan-out during the collision can be related to the angular divergence $\\thrraw^\\text{aft}$ after the\ncollision (equation~31 in \\cite{yokokaChen}, see also equation~\\ref{eq:fThr}):\n\\begin{equation}\n   \\pol^\\text{bef}-\\pol^\\text{aft} \\approx \\dfrac{1}{2} \\pol^\\text{bef} \\cdot\\br{1+a\\gamma}^2 \\cdot(\\thrraw^\\text{aft})^2\n   \\label{eq:yokoyaChen31}\n\\end{equation}\n\nMerging these two equations, one obtains:\n\\begin{equation}\n   \\pol^\\text{bef}-\\ensuremath{\\pol^\\text{lumi,1}} \\approx \\dfrac{1}{2} \\pol^\\text{bef} \\cdot\\br{1+a\\gamma}^2 \\cdot\\br{\\dfrac{\\thrraw^\\text{aft}}{2}}^2\n   \\label{eq:mikesPaper}\n\\end{equation}\nAs explained in \\cite{mikesPaper} and \\cite{moenig}, one can interpret this as about half of the T-BMT precession occurring\nbefore the hard interaction. A comparison to equation~\\ref{eq:yokoyaChen31} implies that one can reproduce\nthe luminosity-weighted polarisation at a point behind the IP where the angular divergence has to be reduced\nby a factor $1\/2$ with respect to the divergence at the IP after the collision.\n\nDuring the mutual distortion of the bunches, the deflected particles also emit beamstrahlung photons, which\ncan cause a flip of the electron spin by the Sokolov-Ternov effect~\\cite{Sokolov:1963zn}. Unlike for the\nsynchrotron radiation in the BDS, the radiative depolarisation in the collision could reach a non-negligible level. Therefore, dedicated simulations of the bunch-bunch interaction are needed in\norder to verify whether equation~\\ref{eq:mikesPaper} holds for realistic ILC beam parameters.\nA detailed description of the beamstrahlung-related effects on the polarisation can be found in section~5.9 of \\cite{cainManual}.\n\n\n\n\\subsection{Simulation Framework {\\tt STALC}}      \\label{subsec:simulationFramework}\nIn order to study the spin transport in the beam delivery system of a linear collider, the simulation framework\n{\\tt STALC} (Spin Transport at Linear Colliders) has been developed \\cite{thesisMoritz}. As sketched in\nfigure~\\ref{fig:simulationFramework}, it interfaces dedicated programs to simulate the beam transport\nincluding the polarisation through the BDS, the collisions at the IP and the measurements of the\npolarimeters, respectively:\n\n\\begin{figure}[ht]\n   \\includegraphics[width=\\columnwidth]{figures\/simulationFramework}\n   \\caption{Program flow in {\\tt STALC}. UP\/DP denotes the up-\/downstream polarimeter.}\n   \\label{fig:simulationFramework}\n\\end{figure}\n\n{\\tt STALC} generates electron\/positron bunches and tracks them through a given lattice via\n{\\tt Bmad}~\\cite{bmad,bmadmanual}, a subroutine library for particle simulations in high-energy accelerators,\nwhich can take also the polarisation into account. At the polarimeters, the particle information can be\npassed to {\\tt LCPolMC}, which simulates the polarisation measurement in a Compton polarimeter from the\nCompton scattering process to the detector response. At the IP, the particle information from one electron\nbunch and one positron bunch is passed to {\\tt Guinea-Pig++}~\\cite{guineapig++} to simulate the beam-beam\ncollisions at the IP. {\\tt Guinea-Pig++} is an extension of {\\tt Guinea-Pig}~\\cite{guineapig}, which takes\ninto account the polarisation and simulates T-BMT precession and Sokolov-Ternov effects. The spent bunches\nafter the collision are passed back to the accelerator simulation and tracked to the downstream polarimeter.\nThe final analysis is based on {\\tt ROOT}~\\cite{root}.\n\n{\\tt STALC} is not inherently limited to the ILC, but can be run on any lattice and beam parameter set.\n\n\\section{Compton Polarimetry at the ILC}       \\label{sec:polarimetry}\nIn this section, we will summarise the working principle and the capabilities of the Laser-Compton polarimeters\nat the ILC. A more detailed description of both polarimeters can be found in~\\cite{boogert}.\n\nThe single differential cross section $d \\sigma \/ dE$ for Compton scattering~\\cite{fano} contains a term\nproportional to the product of the longitudinal beam polarisation and the circular laser polarisation, which\ncan be isolated by measuring the count-rate asymmetry with respect to the laser helicity.\n\nIn order to quickly gather statistical precision, both polarimeters operate in a multi-event\nmode with $\\mathcal{O}(10^3)$ Compton interactions per bunch.\n\n\\begin{table}[hbt]\n   \\centering\n   \\begin{tabular}{lccc}\n                              & $e^+\/e^-$ beam      & \\multicolumn{2}{c}{Laser beam} \\\\\n                              &                     & Upstream         & Downstream  \\\\\n      \\hline\n      Energy                  & $45.6$ - $500$\\,GeV &  $2.33$\\,eV      & $2.33$\\,eV  \\\\\n      Bunch charge\/energy     & $2\\cdot10^{10}\\,e$  &  $35\\,\\mu$J      & $100$\\,mJ   \\\\\n      Bunches per train       & $1312$ - $2625$     &  $1312$ - $2625$ & $1$         \\\\\n      Bunch length $\\sigma_t$ & $1.3$\\,ps           &  $10$\\,ps        & $2$\\,ns     \\\\\n      Average power           &                     &  $0.2$-$0.5$\\,W  & $0.5$\\,W    \\\\\n   \\end{tabular}\n   \\caption[]{Selected parameters of the electron and laser beams at the up- and downstream polarimeter.}\n   \\label{tab:polbeampars}\n\\end{table}\n\nTable~\\ref{tab:polbeampars} summarises specifications of the electron and laser beams relevant for the\nluminosity calculation. A striking difference between the up- and downstream polarimeter lasers is the energy\nper bunch, which differs by more than three orders of magnitude.\nAt the location of the downstream polarimeter, significant amounts of background are expected, at the level\nof ${\\cal{O}}(10^3)$ photons and ${\\cal{O}}(10^2)$ charged particles per bunch\ncrossing~\\cite{Ken:EPWS08proceedings}. In order to maintain a suitable signal-to-background ratio in\npresence of this background, a significantly higher laser power per shot is needed, at the price of a much\nlower repetition rate and longer pulse durations. This allows to shoot at one electron bunch per train, or\na few if several laser are employed. In the nearly background-free conditions of the upstream polarimeter,\nit is possible to cover every bunch in a train, e.g.\\ by employing similar lasers as for the electron\nsource~\\cite{teslaPolarimeter}.\n\n\\begin{table}[hbt]\n   \\centering\n   \\begin{tabular}{lccccccc}\n                                          &           & \\multicolumn{2}{c}{UP} & \\multicolumn{2}{c}{DP} &  DP with collisions\\\\\n                                          &           & $e^+$   & $e^-$        & $e^+$ & $e^-$          & $e^+\/e^-$          \\\\\n      \\hline\n      Horizontal bunch size $\\sigma_{xe}$ &[$\\mu$m]   & $24$    & $32$         & $7$   & $15$           & $\\sim 3000$        \\\\\n      Vertical   bunch size $\\sigma_{ye}$ &[$\\mu$m]   &  $3$    &  $3$         & $33$  & $39$           & $\\sim 1200$        \\\\\n      Beam energy spread $\\sigma_E\/E$     &[$10^{-3}$]& $0.7$   & $1.2$        & $0.7$ & $1.3$          & $\\sim 44$          \\\\\n   \\end{tabular}\n   \\caption[]{Size and energy spread of electron and positron beams at the up- and downstream polarimeter locations obtained from {\\tt STALC}.}\n   \\label{tab:polbeamsizes}\n\\end{table}\n\nThe relevant electron beam sizes at the polarimeter location as obtained from {\\tt STALC} are shown in\ntable~\\ref{tab:polbeamsizes}.\nIt should be noted that after collisions, the beams are highly disrupted and have large non-Gaussian tails,\nas will be discussed in more detail in section~\\ref{subsec:measurablePolarisation}.\nThus, the Gaussian beam sizes listed here should be considered as indicative only.\n\n\\begin{table}[hbt]\n   \\centering\n   \\begin{tabular}{lccc}\n                                                          &                                & Upstream      & Downstream      \\\\\n      \\hline\n      Beam crossing angle  $\\alpha$                       & [mrad]                         & $10$          & $15.5$          \\\\\n      Crossing plane                                      &                                & horizontal    & vertical        \\\\\n      Laser beam size $\\sigma_{x\\gamma}=\\sigma_{y\\gamma}$ & [$\\mu$m]                       & $50$          & $100$           \\\\\n      Luminosity \/ bunch                                  & [$10^{4}$\\,b$^{-1}$]           & $1.0$         & $18$            \\\\\n      Luminosity                                          & [$10^{29}$\\,cm$^{-2}$s$^{-1}$] & $630$         & $9.1$           \\\\\n      $\\delta \\ensuremath{\\polzraw} \/ \\ensuremath{\\polzraw}$ (stat)                       &                                & $<1\\,\\% \/$\\,s & $<1\\,\\% \/$\\,min \\\\\n      $\\delta \\ensuremath{\\polzraw} \/ \\ensuremath{\\polzraw}$ (sys)                        &                                & $0.25\\,\\%$    & $0.25\\,\\%$      \\\\\n   \\end{tabular}\n   \\caption[]{Selected parameters of the up- and downstream polarimeters. For the downstream polarimeter, the luminosities \n   are based on the electron beam parameters in absence of \n   collisions at the $e^+e^-$ IP.}\n   \\label{tab:polpars}\n\\end{table}\n\nTable~\\ref{tab:polpars} finally shows the actual luminosities per bunch and per second, calculated for laser\ncrossing under a small angle $\\alpha$ in the vertical plane according to\n\\begin{equation}\n{\\cal{L}} = \\frac{N_e N_{\\gamma}}{2\\pi \\sqrt{(\\sigma_{xe}^2 + \\sigma_{x\\gamma}^2)}\\sqrt{(\\sigma_{ye}^2 + \\sigma_{y\\gamma}^2)+(\\sigma_{ze}^2 + \\sigma_{z\\gamma}^2)(\\alpha\/2)^2}}.\n\\end{equation}\n\nAs desired, the luminosity per bunch turns out to be a factor~$20$ larger at the downstream polarimeter than\nat the upstream polarimeter, at the price of a much lower instantaneous luminosity. This in turn leads to a\nlonger time required to reach the same statistical precision. Nevertheless, both measurements\nare expected to be systematically limited after a very short time.\n\nSince the electron beam is much more energetic than the (optical) laser photons, all scattered particles are\nforward directed within a narrow cone of $10-20\\,\\mu$rad. For the relevant photon and electron energies, the\ntotal Compton cross section amounts to $154$ ($118$)\\,mb for anti-parallel (parallel)  laser and electron\npolarisation. For the luminositiy of the upstream\n(downstream) polarimeter, this results in ${\\cal O}(10^3)$ (${\\cal O}(10^4)$)  Compton scatterings per bunch.\nThe Compton scattered electrons are momentum-analysed by a set of dipole magnets, and their flux is measured\nas a function of position. The count-rate asymmetry expected for a fully polarised beam is called analysing\npower. The magnets are arranged as part of a chicane such that the undisturbed beam resumes its original\ntrajectory after passing the whole polarimeter section, while the Compton-scattered electrons should be\nkicked out sufficiently far from the main beam axis to allow detection.\n\nIn case of the upstream polarimeter, the chicane consists of four symmetric sets of dipole magnets, with\nthe Compton interaction point in the middle and the detector behind the forth set of dipoles. Such a\nconfiguration offers the additional advantage of decoupling the detector acceptance from the initial beam\nenergy, since the Compton spectrum is projected onto the same area in the detector plane for all beam\nenergies. Instead the dispersion in the middle of the chicane and thus the position of the Compton IP\nchanges according to beam energy.\n\nIn case of the downstream polarimeter, the chicane is based on six dipoles, again with the detector behind\nthe forth dipole. In this case, the third and forth dipole are operated at a larger magnetic field in order\nto kick the Compton signal sufficiently far out of the synchrotron radiation fan from upstream magnets.\nThe fifth and sixth dipole then bring the main beam back to its original trajectory. Detailed specifications\nfor both chicanes can be found in~\\cite{boogert}.\n\nThe lattice between the IP and the downstream polarimeter has been designed such that the angular divergence\nof the beam at the downstream polarimeter is by a factor two smaller than at the end of the collision\\footnote{\nI.e.\\ that the relevant element of the transfer matrix between $e^+e^-$ IP and the downstream polarimeter\nCompton IP is $|R_{22}|=0.5$.}. Thus, the downstream polarimeter measurement is expected to relate directly\nto the luminosity-weighted average polarisation during collisions as explained in\nsection~\\ref{subsec:beambeamCollisions}. Whether this picture holds e.g.\\ in presence of beamstrahlung will\nbe investigated in section~\\ref{sec:lumipol}.\n\nBuilding on the successful experience at SLD~\\cite{Abe:2000dq}, the baseline foresees an array of $20$\ngas-Cherenkov detectors read-out by photomultiplier tubes~\\cite{Bartels:2010eb}. The advantages of this\ntechnology comprise radiation hardness, robustness against low-energetic backgrounds due to a Cherenkov\nthreshold in the MeV-regime and a low number of channels to be read-out.\n\nThe limiting systematic uncertainties of the SLD polarimeter were the knowledge of the analysing power and\nthe photo detector non-linearities. Both aspects have been studied for the ILC polarimeters in a recent R\\&D effort: The dominating\ncontribution to the analysing power, i.e.\\ the alignment of the detector with respect to the beam, has been studied\nin a testbeam campaign with a prototype~\\cite{Bartels:2010eb}, and a calibration system for monitoring and\ncorrecting non-linearities to a sufficient level has been developed~\\cite{Vormwald:thesis}. Based on this\nexperience, the uncertainty budget listed in table~\\ref{tab:polsys} seems achievable. Details on the online\nmonitoring of the laser polarisation can be found in~\\cite{Ken:EPWS08proceedings}. The total systematic\nuncertainty on the polarimeter measurements amounts to $\\delta \\ensuremath{\\polzraw}\/\\ensuremath{\\polzraw} = 0.25\\,\\%$.\n\n\\begin{table}[hbt]\n   \\centering\n   \\begin{tabular}{lc}\n      Source of uncertainty            & $\\delta \\ensuremath{\\polzraw} \/ \\ensuremath{\\polzraw}$ \\\\\n      \\hline\n      Detector analysing power         & $0.15 - 0.2\\,\\%$       \\\\\n      Detector linearity               & $0.1\\,\\%$              \\\\\n      Laser polarisation               & $0.1\\,\\%$              \\\\\n      Electronic noise and beam jitter & $0.05\\,\\%$             \\\\\n      \\hline\n      Total                            & $0.25\\,\\%$             \\\\\n   \\end{tabular}\n   \\caption[]{Uncertainty budget for the Compton polarimeters.}\n   \\label{tab:polsys}\n\\end{table}\n\n\nAll the contributions in table~\\ref{tab:polsys} are expected to be uncorrelated between all four\npolarimeters. Thus, additional precision could be gained from a cross calibration of the polarimeters in\nabsence of collisions, if the spin transport from one location to the other is sufficiently well understood.\nIn presence of collisions, both polarimeters fulfil very complementary tasks: While the downstream\npolarimeter in principle gives access to the depolarisation in collision, the upstream polarimeter provides\na clean measurement of the initial polarisation and resolves possible time dependent patterns, e.g.\\ inside\na bunch train.\n\n\\section{Cross calibration of the Polarimeters}       \\label{sec:crosscal}\nIn absence of collisions at the $e^+e^-$ IP, the polarisation measurement at the upstream polarimeter can be\npropagated by spin tracking along the BDS and predict the expected polarisation\nat the downstream polarimeter. Provided that the spin propagation effects are well under control,\nthe two polarimeters can thus be cross calibrated. However, it is crucial to understand how deviations\ndue to instrumental effects of the polarimeters can be disentangled from effects caused by\nthe spin transport. Therefore, it is important to understand the individual impact of the various \neffects which could influence the spin transport between the polarimeters. Both the\ncross calibration of the polarimeters and the understanding of the spin transport are crucial\nto finally assess the collision effects and to extract the luminosity-weighted polarisation.\n\nWe studied the spin transport with {\\tt STALC} based on the SB2009\\_Nov10 lattice \\cite{SB2009lattice}, which\ndiffers from the RDR lattice with respect to the new location of the upstream polarimeter\\footnote{Due to its\nseparation from the emittance measurement following earlier recommendations~\\cite{Kafer:2008ku}.}. For the\npositrons, the tracking starts at the beginning of the BDS (dashed line in figure~\\ref{fig:bds-SB2009}).\nOn the electron side, the simulation starts at the beginning of the positron production system,\nthus including the ``dogleg''\nto bypass the positron source target area. In both cases, the beams are initialised according to the twiss parameters of the lattice and the TDR\nbeam energy spreads. However, the final focus system of the SB2009\\_Nov10 lattice was not\nyet adapted to the TDR design and thus produces the RDR beta functions at the $e^+e^-$ IP.\nIn absence of collisions, no significant impact on the spin transport is expected due to this issue.\n\nTo avoid numerical problems, some magnets have been sliced into smaller units as described in chapter~7 of\n\\cite{thesisMoritz}. In each studied configuration, 1000 individual bunches have been tracked, each\nbunch with 40\\,000 macroparticles\\footnote{Thus, for a design bunch population of $N_{e}= 2\\cdot 10^{10}$,\none macroparticle represents $5\\cdot10^5$ electrons.}. To each macroparticle, a polarisation vector with a\nlength of $\\pol_\\text{max}=0.8$ is assigned.\nAt the beginning of the tracking, the polarisation vectors are assumed to be oriented such that they are\nparallel to the orbit if the macroparticle trajectory is parallel to the orbit as well, and rotated\nanalogously to T-BMT precession otherwise. In this case, the spin fan-out is expected to be described by\nequation~\\ref{eq:fThr}. This spin configuration has already implicitly been assumed in earlier studies\n\\cite{yokokaChen, spinConfigExample}. The effects of a slightly different spin configuration will be\nexamined in section~\\ref{sec:lumipol}.\n\n\\begin{figure}[ht]\n   \\includegraphics[width=\\columnwidth]{figures\/PolSIdealExPRDR}\n   \\caption{Longitudinal polarisation \\ensuremath{\\polzraw}\\ (black, solid), magnitude of the polarisation vector \\pol\\\n            (red, dashed) and the function $f(\\ensuremath{\\thrraw})$ of the angular divergence defined in\n            equation~\\ref{eq:fThr} (green, dash-dotted) of the electron beam at the upstream polarimeter\n            (UP, left part) and between $e^+e^-$ IP and downstream polarimeter (DP, right part).}\n   \\label{fig:PolSIdeal}\n\\end{figure}\n\nFigure~\\ref{fig:PolSIdeal} shows the polarisation \\pol\\ and its longitudinal component \\ensuremath{\\polzraw}\\ of the electron\nbeam at the upstream polarimeter and between the $e^+e^-$ IP and the downstream polarimeter for perfect\nalignment and in absence of detector magnets and crab cavities. It shows that the final focus magnets (in\nfront of the IP) and the extraction line quadrupoles affect \\pol\\ and \\ensuremath{\\polzraw}\\ likewise, while the polarisation\nvector precesses in the dipole chicanes, but returns to a longitudinal configuration behind a pair of dipoles\nwith opposite fields. The influence of the quadrupoles is well described by the spin fan-out prediction\n$f(\\ensuremath{\\thrraw})$ (equation~\\ref{eq:fThr}) which is drawn as green dash-dotted line. The residual differences stem\nmost likely from the beam energy spread, which is not taken into account by $f(\\ensuremath{\\thrraw})$, or from the betatron\noscillations.\n\n\\begin{table}[hbt]\n   \\renewcommand{\\arraystretch}{1.05}\n   \\centering\n   \\begin{tabular}{llll@{\\hspace{7mm}}l@{\\hspace{7mm}}l@{\\hspace{7mm}}l}\n      Simulation      &              & \\multicolumn{1}{c}{UP} & \\multicolumn{2}{c}{$e^+e^-$ IP}     & \\multicolumn{2}{c}{DP}\\\\\n      \\hline\n      \\ensuremath{\\thrraw}            & [$\\mu$rad]   & $0.986 \\pm 0.004$      & \\multicolumn{2}{l}{$35.4 \\pm 0.1$}  & \\multicolumn{2}{l}{$16.27 \\pm 0.06$} \\\\\n      $(0.8-\\ensuremath{\\polzraw})$   & [$10^{-6}$]  & $0.2 \\pm \\varepsilon$  & \\multicolumn{2}{l}{$143.0 \\pm 0.8$} & \\multicolumn{2}{l}{$48.4 \\pm 0.3$}   \\\\\n      $(0.8-\\pol)$    & [$10^{-6}$]  & $0.2 \\pm \\varepsilon$  & \\multicolumn{2}{l}{$143.0 \\pm 0.8$} & \\multicolumn{2}{l}{$48.4 \\pm 0.3$}   \\\\\n      $(0.8-f(\\ensuremath{\\thrraw}))$ & [$10^{-6}$]  & $0.1 \\pm \\varepsilon$  & \\multicolumn{2}{l}{$162 \\pm 1$}     & \\multicolumn{2}{l}{$34.2 \\pm 0.2$}   \\\\[3mm]\n      \\multicolumn{3}{l}{  }                                  & \\multicolumn{2}{c}{$e^+e^-$ IP}     & \\multicolumn{2}{c}{DP}               \\\\\n      \\multicolumn{3}{l}{Design values}                       & RDR       & TDR                     & RDR        & TDR                     \\\\\n      \\hline\n      \\ensuremath{\\thrraw}            & \\multicolumn{2}{l}{[$\\mu$rad]}        & $35$      & $45$                    & $17$       &  $23$                   \\\\\n      $(0.8-f(\\ensuremath{\\thrraw}))$ & \\multicolumn{2}{l}{[$10^{-3}$]}       & $0.157$   & $0.25$                  & $0.039$    &  $0.064$                \\\\\n   \\end{tabular}\n   \\caption[]{Upper table: Angular divergence \\ensuremath{\\thrraw}, longitudinal polarisation \\ensuremath{\\polzraw}, magnitude of the\n              polarisation vector \\pol\\ and the function $f(\\ensuremath{\\thrraw})$ as defined in equation~\\ref{eq:fThr}\n              of the electron beam at the polarimeters (UP\/DP = up-\/downstream polarimeter) and the\n              $e^+e^-$ IP. Lower table: \\ensuremath{\\thrraw}\\ and $f(\\ensuremath{\\thrraw})$ calculated from the design values\n              (table~\\ref{tab:beamParametersDesign}). The accuracy of the simulation is estimated to be\n              $10^{-7}$ at best. Therefore, uncertainties below $0.5\\cdot 10^{-7}$ are denoted as\n              $\\varepsilon$. Uncertainties between $0.5\\cdot 10^{-7}$ and $1\\cdot 10^{-7}$ are rounded up.}\n   \\label{tab:beamParameters}\n\\end{table}\n\nTable~\\ref{tab:beamParameters} lists selected bunch parameters from the simulation (and calculated from the\ndesign values, as far as possible) at the polarimeters and at the $e^+e^-$ IP, where the latter serves\nmainly for illustration here, since it does not affect the cross calibration of the polarimeters. In the\ncontext of the envisaged precision of $0.1\\,\\%$, the simulation results are in good agreement with the\nvalues predicted based on the angular divergence \\ensuremath{\\thrraw}. On this basis, the consequences of the stronger\nfocussing for the TDR parameters can be estimated as well. Assuming the spin fan-out at the upstream\npolarimeter to be negligible, the relative difference in \\ensuremath{\\polzraw}\\ at the two polarimeters due to spin\nfan-out for TDR parameters amounts to $\\Delta\\ensuremath{\\polzraw}\/\\ensuremath{\\polzraw}=8.0\\cdot 10^{-5}$. This is $64\\,\\%$ larger\nthan for RDR parameters, but still very small compared to the aim of $10^{-3}$.\n\nWe also studied the effect of a finite knowledge of the beam parameters, in particular the emittances.\nA variation of $10\\,\\%$ leads to a negligible change in polarisation of $3\\cdot 10^{-5}$.\n\nThe effects arising from the emission of synchrotron radiation in the beamline magnets have been\nfound to be negligible as well: The change in polarisation due to spin-flips between the two polarimeters\nhas been calculated to be $<10^{-6}$. The energy loss due to emission of synchrotron radiation and the\nresulting changes in the particle trajectories change the polarisation by $5\\cdot 10^{-6}$. For more\ndetails, see section 9.1 of \\cite{thesisMoritz}.\n\n\n\n\\subsection{Beam Alignment at the Polarimeters and at the $e^+e^-$ IP}  \\label{subsec:beamAlignment}\nIn order to avoid measuring values for \\ensuremath{\\polzraw}\\ at the polarimeters and at the $e^+e^-$ IP which differ\na priori due to T-BMT precession, there should be no relative angle between the orbits at these points.\nTherefore, no effective bending angle between these points is foreseen in the BDS design.\n\nMisalignments, however, might lead to relative incident angles $\\Delta\\thb$, which translates to a precession of the\npolarisation vector by an angle of $\\br{1+a\\gamma}\\cdot\\Delta\\thb$ (equation~\\ref{eq:tbmtTheta}) if the energy spread can\nbe neglected \\cite{Woods:2004qu}. The design requirement on the relative beam alignment between the polarimeter locations is\n  $\\Delta\\thb \\leq 50\\,\\mu$rad, driven by polarimetry demands.\nTo achieve the maximum longitudinal polarisation, spin rotators in front of the main linac are used\nto adjust the polarisation vector parallel to the beam at the upstream polarimeter. This is expected\nto be possible with an uncertainty of $\\Delta\\thp=25\\,$mrad%\n\\footnote{A possible scheme to measure the alignment of the polarisation vector is presented in \\cite{moenig}.}.\n\nFor a beam energy of $250$\\,GeV, these two contributions give the following total uncertainty on the\npolar angle of the polarisation vector:\n\\begin{equation}\n   \\Delta\\thp^\\text{tot} = \\sqrt{ \\Delta\\thp^2 + \\brbig{\\br{1+a\\gamma}\\cdot \\Delta\\thb}^2} = 38\\,\\text{mrad}\n\\end{equation}\nThe corresponding uncertainty of the longitudinal polarisation amounts to\n\\begin{equation}\n   \\Delta \\ensuremath{\\polzraw} \/ \\ensuremath{\\polzraw} = 1-\\cos\\br{\\Delta\\thp^\\text{tot}} = 0.72\\cdot 10^{-3}.\n   \\label{eq:tabAngleCorrection1}\n\\end{equation}\n\nThe contribution from the incident angles increases with the beam energy. For a beam energy of $500\\,$GeV\ninstead of $250\\,$GeV, $\\Delta\\thp^\\text{tot}$ rises to $62\\,$mrad and $\\Delta\\ensuremath{\\polzraw}\/\\ensuremath{\\polzraw}$ to\n$1.9\\cdot 10^{-3}$. This is by far dominated by the contribution from the beam alignment: even for perfect\nalignment of the polarisation vector, $\\Delta\\ensuremath{\\polzraw}\/\\ensuremath{\\polzraw}$ would still amount to $1.6\\cdot 10^{-3}$.\nThus, an upgrade to a collision energy of $1\\,$TeV would require an improved beam alignment in order\nto achieve the precision goal of $0.1\\,\\%$ for the spin transport in the BDS.\n\nA computational correction of the measured \\ensuremath{\\polzraw}\\ for known incident angles would in principle be possible,\nbut gives an additional contribution to the uncertainty of \\ensuremath{\\polzraw}. As shown in section~7.4 of\n\\cite{thesisMoritz}, such a correction seems extremely difficult in view of the precision goal of $0.1\\,\\%$,\nsince the contribution to $\\Delta \\ensuremath{\\polzraw} \/ \\ensuremath{\\polzraw}$ grows with the angle itself as well\nas with its uncertainty. Therefore, the beam orbits at the polarimeters and at the $e^+e^-$ IP have to be\naligned by correction magnets. This correction method gives an additional contribution to the uncertainty\nof \\ensuremath{\\polzraw}, which is discussed in the following.\n\n\n\n\n\n\n\n\\subsection{Residual Effects from the Beam Orbit Correction}       \\label{subsec:orbitCorrectionEffects}\nMisalignments of magnets as described in section~\\ref{subsec:misalignments} lead to misalignments of the\nbeams. To keep the beams close to the design orbit and bring the bunches to collision at the $e^+e^-$ IP,\nthe BDS is equipped with a number of correction dipoles. However, these dipoles can not prevent the beam\nfrom leaving the design orbit in the first place, but only bend it back to the design orbit. While it was\nassumed in the previous section that a re-alignment of the beam orbit would also fully restore the\nalignment of the polarisation vector, the accuracy of this assumption will be tested in this section.\n\n\n\\begin{table}[hb]\n   \\centering\n   \\begin{tabularx}{\\textwidth}{lX}\n      Sample name & Description                                                               \\\\\n      \\hline\n      \\samnom   & No misalignments (same data as for figure~\\ref{fig:PolSIdeal}).             \\\\\n      \\sammfive & RMS size of all offsets  $5\\,\\mu$m, RMS size of all rotations  $5\\,\\mu$rad. \\\\\n      \\sammten  & RMS size of all offsets $10\\,\\mu$m, RMS size of all rotations $10\\,\\mu$rad. \\\\\n \\end{tabularx}\n   \\caption[]{Simulated samples for the investigation of the effects of magnet misalignments.}\n   \\label{tab:dataSamplesMisalignments}\n\\end{table}\n\nFor this purpose, random misalignments of the beamline magnets, i.e.\\ offsets and rotations in all three\ndimensions have been introduced into the simulation. Three data samples with different RMS sizes of\nmisalignments have been produced as listed in table~\\ref{tab:dataSamplesMisalignments}. For each of the\n1000 bunches per sample, new misalignments are generated.\nCorrelations in space or time have not been taken into account for the misalignments.\nTo correct the beam orbit, the corrector magnets foreseen in the lattice are used. The magnet strengths\nare computed from the beam positions at the beam position monitors (BPMs)~\\cite{Boogert:2013sea} such that\nthe beam position offsets measured by the BPMs are minimised. For the current study, negligible BPM\nresolutions have been assumed.\nAt the $e^+e^-$ IP, the fast-feedback correction is emulated which adjusts the beam position and incident\nangle according to the foreseen tolerances. The correction procedure is described in detail\nin \\cite{thesisMoritz}.\n\n\n\n\\begin{figure}[ht]\n   \\includegraphics[width=\\columnwidth]{figures\/ZPolSMis5ExPRDR}\n   \\caption{Longitudinal polarisation \\ensuremath{\\polzraw}\\ of the electron beam at the upstream polarimeter (UP, left part) and between $e^+e^-$ IP and downstream polarimeter (DP, right part) for the data samples listed in table~\\ref{tab:dataSamplesMisalignments}. The uncertainty bands correspond to the RMS spread of all runs. For the perfectly aligned case, the band is too small to be visible.}\n   \\label{fig:PolSIdealMisal}\n\\end{figure}\n\n\\begin{table}[hbt]\n   \\renewcommand{\\arraystretch}{1.05}\n   \\centering\n   \\begin{tabular}{l|lll@{\\hspace{5mm}}l@{\\hspace{5mm}}l}\n                  &                 &              & \\multicolumn{1}{c}{\\samnom} & \\multicolumn{1}{c}{\\sammfive} & \\multicolumn{1}{c}{\\sammten} \\\\\n      \\hline\n      Upstream    & \\thb            & [$\\mu$rad]   & $0.006 \\pm 0.004$           & $2 \\pm 1$                     & $3 \\pm 2$                    \\\\\n      polarimeter & \\ensuremath{\\thrraw}            & [$\\mu$rad]   & $0.986 \\pm 0.004$           & $0.989 \\pm 0.005$             & $1.00 \\pm 0.01$              \\\\[2mm]\n                  & $(0.8-\\ensuremath{\\polzraw})$   & [$10^{-6}$]  & $0.2 \\pm \\varepsilon$       & $34 \\pm 34$                   & $132 \\pm 128$                \\\\\n                  & $(0.8-\\pol)$    & [$10^{-6}$]  & $0.2 \\pm 0.1$               & $0.2 \\pm \\varepsilon$         & $0.3 \\pm 0.1$                \\\\\n                  & $(0.8-f(\\ensuremath{\\thrraw}))$ & [$10^{-6}$]  & $0.1 \\pm \\varepsilon$       & $0.1 \\pm \\varepsilon$         & $0.1 \\pm \\varepsilon$        \\\\\n      \\hline\n      Downstream  & \\thb            & [$\\mu$rad]   & $0.21 \\pm 0.08$             & $0.7 \\pm 0.4$                 & $1.2 \\pm 0.6$                \\\\\n      polarimeter & \\ensuremath{\\thrraw}            & [$\\mu$rad]   & $16.27 \\pm 0.06$            & $16.4 \\pm 0.4$                & $16.7 \\pm 0.9$               \\\\[2mm]\n                  & $(0.8-\\ensuremath{\\polzraw})$   & [$10^{-6}$]  & $48.4 \\pm 0.3$              & $86 \\pm 37$                   & $192 \\pm 138$                \\\\\n                  & $(0.8-\\pol)$    & [$10^{-6}$]  & $48.4 \\pm 0.3$              & $49 \\pm 2$                    & $50 \\pm 5$                   \\\\\n                  & $(0.8-f(\\ensuremath{\\thrraw}))$ & [$10^{-6}$]  & $34.2 \\pm 0.2$              & $35 \\pm 2$                    & $36 \\pm 4$                   \\\\\n   \\end{tabular}\n   \\caption[]{Incident angle \\thb, angular divergence \\ensuremath{\\thrraw}, longitudinal polarisation \\ensuremath{\\polzraw}, magnitude of\n              the polarisation vector \\pol\\ and the function $f(\\ensuremath{\\thrraw})$ as defined in equation~\\ref{eq:fThr}\n              of the electron beam at the polarimeters for the data samples listed in\n              table~\\ref{tab:dataSamplesMisalignments}. For $\\varepsilon$, see the explanation in\n              the caption of table~\\ref{tab:beamParameters}}\n   \\label{tab:beamParametersMisalignments}\n\\end{table}\n\nFigure~\\ref{fig:PolSIdealMisal} shows the longitudinal polarisation \\ensuremath{\\polzraw}\\ at the upstream polarimeter and\nbetween the $e^+e^-$ IP for the same data samples. In addition, table~\\ref{tab:beamParametersMisalignments}\nlists selected bunch parameters of the electron beam at the polarimeters for different sizes of misalignments\nfor the data samples listed in table~\\ref{tab:dataSamplesMisalignments}.\nThe uncertainties in the table and the shaded areas in the figure denote the RMS spread of \\ensuremath{\\polzraw}\\ of\nthe simulated bunches. These spreads reflect the possible variations depending on\nthe exact misalignments of the individual magnets, which lead to different trajectories for each bunch.\n\nWhile the norm of the polarisation vector is hardly affected by the misalignments, there is a larger\neffect on the longitudinal component \\ensuremath{\\polzraw}. The values of \\pol\\ remain consistent with $f(\\ensuremath{\\thrraw})$\n(cf.\\ section~\\ref{subsec:spinTransport}); thus, the behaviour of \\pol\\ is still fully explained\nby spin fan-out.\nThe increases in the mean value and the uncertainty of \\thb\\ do, however, not suffice to explain the\ndecrease in \\ensuremath{\\polzraw}\\ by precession of the polarisation vector according to equation~\\ref{eq:tbmtTheta}:\n$\\Delta\\ensuremath{\\polzraw}=2\\cdot 10^{-4}$ would correspond to $\\thb=39\\,\\mu$rad, while $\\thb=5\\,\\mu$rad would yield\nonly $\\Delta\\ensuremath{\\polzraw}=4\\cdot 10^{-6}$. This discrepancy appears after one of the correction magnets in the\n``dogleg'' where the electron beam by-passes the photon target area of the positron source (cf.\\ fig.~\\ref{fig:bds-SB2009}), possibly due to non-commutation of rotations about different axes: while all deflections of the bunch by misaligned magnets are compensated for with the corresponding deflections at the correction magnets, this compensation does not work perfectly for the polarisation vector which performs rotations by the $(1+a\\gamma)$-fold deflection angles.\n\n\\begin{figure}[ht]\n   \\centerline{\\includegraphics[width=0.5\\columnwidth]{figures\/ZpolDPvsUPEM10}}\n   \\caption{Difference of the longitudinal polarisation \\ensuremath{\\polzraw}\\ of the electron beam at the upstream polarimeter (UP) and the downstream polarimeter (DP) for the misalignment sample ``\\sammten''.}\n   \\label{fig:ZpolDPvsUPEM10}\n\\end{figure}\n\nIn case of purely static misalignments, the decisive quantity for the cross calibration of the polarimeters\nis the residual difference of the longitudinal polarisation at both locations. Figure~\\ref{fig:ZpolDPvsUPEM10} shows this difference in terms of $\\Delta\\ensuremath{\\polzraw}({\\mathrm{UP,DP}}):= (\\ensuremath{\\polzraw}^{{\\mathrm{UP}}}-\\ensuremath{\\polzraw}^{{\\mathrm{DP}}})\/\\sqrt{2}$\nfor the sample  ``\\sammten''. It follows an approximately Gaussian distribution with an RMS spread of $0.04\\cdot 10^{-3}$. This is about a factor $10$ smaller than the variations of \\ensuremath{\\polzraw}\\ at any individual location, i.e. the uncertainty band in figure~\\ref{fig:PolSIdealMisal}.\nNevertheless, we conservatively estimate the uncertainty from the magnet misalignments and the beam orbit correction\nbased on figure~\\ref{fig:PolSIdealMisal}, in order to allow for\n time-dependent misalignments which have not been explicitely studied here.\nThus, we define this uncertainty as the relative deviation of the lower bound of the spread for \\ensuremath{\\polzraw}\\ for the samples with misalignments\nfrom the mean value of \\ensuremath{\\polzraw}\\ for the sample without misalignments (both values at\nthe downstream polarimeter, neglecting the decrease in \\ensuremath{\\polzraw}\\ at the upstream polarimeter).\nThis yields a relative uncertainty of $0.09\\cdot 10^{-3}$ for the sample ``\\sammfive'' and\n$0.35\\cdot 10^{-3}$ for the sample ``\\sammten''.\n\nThus, we conclude that misalignments of beamline elements are not the dominating source of\nsystematic uncertainty, but still can give a sizable contribution.\nEspecially the impact of time-dependent effects should be studied in more detail based on\nrealistic ground-motion models of the selected ILC site, also taking into account finite BPM resolutions.\n\n\n\n\\subsection{Detector Magnets and Crab Cavities}       \\label{subsec:commissioning}\nIn this section we investigate more closely the interaction region devices and their impact on\nthe spin transport. When operating the machine without collisions, the detector magnets and the\ncrab cavities could be switched on or off. Thus, their impact on the polarisation measured at the\ndownstream polarimeter could rather easily be disentangled from instrumental differences between the\nup- and downstream polarimeter measurements.\n\n\\subsubsection{Detector Magnets}       \\label{subsubsec:detectorMagnets}\nThe beams enter the detector under an angle of 7\\,mrad, which is half the beam-beam crossing angle.\nHence, both the main solenoid and the anti-DID feature magnetic field components parallel and\nperpendicular to the beam.\n\nThe parallel field components might destroy the synchronisation between the angles of the beam orbit\nand the polarisation vector, such that \\ensuremath{\\polzraw}\\ takes different values at the $e^+e^-$ IP and at the\npolarimeters even if the beam orbit is aligned parallel at these three locations. This effect has,\nhowever, turned out to be negligible ($\\Delta\\ensuremath{\\polzraw}\\sim 10^{-5}$) \\cite{thesisMoritz} despite the\nstrong longitudinal field of the solenoid. This can be understood since the transverse components\nof the polarisation vector, which would undergo the precession about the longitudinal axis, are rather\nsmall, and since the precession about the longitudinal axis does not speed up with rising particle\nenergies (cf.\\ equation~\\ref{eq:tbmtOmegaBdecomposed}). The effect of the longitudinal field components\nwould have to be re-evaluated for physics operation of the ILC with transverse beam polarisation.\n\nThe perpendicular field components act like dipole magnets on the beam. The resulting deflection would\nalter the beam position by several micrometers at the $e^+e^-$ IP, such that no collisions would occur,\nand several millimeters at the downstream polarimeter. To correct the beam position, additional dipole\nmagnets in front of and behind the detector are foreseen, which are not contained in the lattice files\n\\cite{SB2009lattice} yet. For establishing luminosity, only the relative angles of the beams to each\nother are relevant \\cite{antididAndrei}. However, polarimetry requires also the angles of the beam\norbits at the polarimeters to be aligned parallel to the beam orbits at the $e^+e^-$ IP to a level of\n$50\\,\\mu$rad (cf.\\ section~\\ref{subsec:beamAlignment}) in presence of collisions. Without any beam\norbit correction, the fields of the detector magnets would induce a deflection angle of\n$\\thb=185\\,\\mu$rad at the $e^+e^-$ IP, which leads to a decrease in longitudinal polarisation by\n$\\Delta\\ensuremath{\\polzraw}=4.5\\cdot 10^{-3}$ due to T-BMT precession \\cite{thesisMoritz}. As explained in the\nprevious section, such a deflection angle at the $e^+e^-$ IP can only be corrected for if it can be\nmeasured to significantly better than $10\\,\\%$. If for instance the above $\\thb=185\\,\\mu$rad are known\nto $15\\,\\mu$rad ($8\\,\\%$), the resulting uncertainty on the polarisation from this correction alone\namounts to $0.1\\,\\%$. Thus, it is very important to foresee sufficient orbit correction possibilities\nsuch that not only the relative, but also the absolute angles of the beams at the $e^+e^-$ IP can be\nadjusted in presence of the detector magnets.\n\n\n\n\\subsubsection{Crab Cavities}       \\label{subsubsec:crabCavities}\nTo initiate the rotation of the particle bunches, the crab cavities generate a time-dependent\nelectromagnetic field. In the centre of the cavity and at the time of the bunch passages, this field\ncan be approximated by a time-dependent magnetic dipole field, which deflects the bunch particles\nhorizontally by an angle proportional to the longitudinal distance from the bunch centre. This deflection\ngoes along with the corresponding spin precession (equation~\\ref{eq:tbmtTheta}), which results in a spin\nfan-out along the $z$-axis and thus in a decrease of the polarisation. The effect on the polarisation\ncan, however, be expected to be small, since the fan-out vanishes at the bunch centre, where the majority\nof the particles is located. Simulations have shown that the effect on the polarisation is smaller than\n$10^{-5}$ (section 8.1 in \\cite{thesisMoritz}).\n\n\n\\subsection{Cross Calibration of the Polarimeters}       \\label{subsec:summaryCrossCalib}\nTable~\\ref{tab:uncertainties} summarises the uncertainties on the spin transport between the polarimeters\nfor the cross calibration in absence of collisions and for a beam energy of $250\\,$GeV. The major\ncontribution comes from the alignment precision of the beam and the polarisation vector\n(section~\\ref{subsec:beamAlignment}). A second large contribution comes from the beam orbit correction\n(section~\\ref{subsec:orbitCorrectionEffects}), while all other contributions listed are negligible.\nAll contributions sum up to an uncertainty of $0.080\\,\\%$, which matches the goal of at most $0.1\\,\\%$.\nFor higher beam energies, many of the contributions can be expected to increase. The contribution from\nthe alignment precision alone rises to $1.9\\cdot 10^{-3}$ for a beam energy of $500\\,$GeV, which\nnecessitates a better beam alignment.\n\n\n\n\\begin{table}[hbt]\n  \\renewcommand{\\arraystretch}{1.05}\n   \\centering\n   \\begin{tabularx}{\\textwidth}{Xr}\n      Contribution                                                            & $\\delta\\ensuremath{\\polzraw}\/\\ensuremath{\\polzraw}\\ [10^{-3}]$ \\\\\n      \\hline\n      Beam and polarisation alignment at polarimeters                         & $0.72$                         \\\\\n      (assuming $\\Delta\\thb=50\\,\\mu$rad, $\\Delta\\thp=25\\,$mrad) &                                \\\\\n      Random misalignments ($10\\,\\mu$m$\/\\mu$rad) with beam orbit correction   & $0.35$                         \\\\\n      Variation in beam parameters ($10\\,\\%$ in the emittances)               & $0.03$                         \\\\\n      Longitudinal precession in detector magnets                             & $0.01$                         \\\\\n      Bunch rotation to compensate the beam crossing angle                    & $<0.01$                        \\\\\n      Emission of synchrotron radiation                                       & $0.005$                        \\\\\n      \\hline\n      Total                                                                   & $0.80$                         \\\\\n   \\end{tabularx}\n   \\caption{Contributions to the uncertainty of the spin transport from the upstream to the downstream polarimeter for a beam energy of $250$\\,GeV in the absence of collisions.\n   \\label{tab:uncertainties}}\n\\end{table}\n\n\\section{Luminosity-Weighted Average Polarisation}       \\label{sec:lumipol}\nIn this section, we discuss the relevant quantity for the interpretation of collision data,\ni.e.\\ the luminosity-weighted average polarisation at the IP. We studied the direct depolarisation\nin collisions as well as for the first time the effects which arise\nat the downstream polarimeter location and their interplay with polarimetry. Finally, we\nlay out a strategy how to access the luminosity-weighted average polarisation from polarimeter\nmeasurements and how to compare this to collision data.\n\nIn addition to the RDR beam parameter set used in section~\\ref{sec:crosscal}, also collisions with\nTDR beam parameters have been simulated (cf.\\ section~\\ref{subsec:ilcbds}). As explained in\nsection~\\ref{sec:crosscal}, the current lattice features a final focus according to RDR parameters\nat the $e^+e^-$ IP.\nHowever, no significant changes in the spin tracking upstream of the $e^+e^-$ IP are expected.\nOn the other hand, the extraction line lattice is suitable for both parameter sets. Thus, the beams are\ngenerated at the $e^+e^-$ IP, passed to the simulation of the collisions, and then propagated to\nthe downstream polarimeter.\n\nThese two data samples named ``\\samrdr'' and ``\\samtdr'' are both generated with the spin configuration\nas described in the beginning of section~\\ref{sec:crosscal}.\nThis is well motivated since the spin transport is dominated by spin fan-out as shown in the previous\nsection. To illustrate nevertheless the effects of a different initial spin configuration, a third\nsample ``\\samtdrstar'' has been generated:\nall macroparticle\npolarisation vectors are aligned along the beam axis at the $e^+e^-$ IP, instead of being fanned out\naccording to the focussing of the beam. Such a different spin configuration could e.g.\\ be the result\nof the betatron oscillations (non-commutating rotations, cf.\\ section~\\ref{subsec:orbitCorrectionEffects}).\n\n\nTo be able to distinguish the contributions from spin precession and from energy losses, each of the\nthree data samples has been resimulated with the particle energies fixed to $E=250\\,$GeV\n(\\samrdrfixedEnergy, \\samtdrfixedEnergy, TDR*$\\hspace{-1.7mm}_0$). In other words, the bunches are\ngenerated without initial energy spread, and no energy loss due to synchrotron radiation or beamstrahlung\n(and thus no depolarisation due to Sokolov-Ternov effects) is simulated, so that spin precession\nis the only source of changes in polarisation. The definitions of all samples are summarised in\ntable~\\ref{tab:dataSamples}.\n\n\\begin{table}[hb]\n   \\centering\n   \\begin{tabularx}{\\textwidth}{lX}\n      Sample name & Description    \\\\\n      \\hline\n      \\samrdr     & Beam parameters according to the RDR (cf.\\ table~\\ref{tab:beamParametersDesign}),\n                    but with the energy spread according to the TDR (same parameters as in\n                    section~\\ref{sec:crosscal}).\\\\\n      \\samtdr     & Beam parameters according to the TDR (cf.\\ table~\\ref{tab:beamParametersDesign}).\\\\\n      \\samtdrstar & Like TDR, but with a different initial spin configuration: all macroparticle\n                    polarisation vectors are aligned along the $z$-axis at the IP, instead of being\n                    fanned out according to the focussing of the beam.\\\\\n      \\samrdrfixedEnergy, \\samtdrfixedEnergy, TDR*$\\hspace{-1.7mm}_0$ & Samples like above, but all\n                    particles with fixed energy $E=250\\,$GeV (no energy spread, no synchrotron\n                    radiation or beamstrahlung).\\\\\n \\end{tabularx}\n   \\caption[]{Simulated samples for the investigation of the beam-beam collision effects.}\n   \\label{tab:dataSamples}\n\\end{table}\n\nThe luminosities of the simulated collisions amount to $(2.02\\pm 0.02)\\cdot 10^{34}$\\,cm$^{-2}$\\,s$^{-1}$\nand $(1.52\\pm 0.03)\\cdot 10^{34}$\\,cm$^{-2}$\\,s$^{-1}$ for the samples \\samrdr\\ and \\samtdr, respectively,\nwhich is in good agreement with the design luminosities (cf.\\ table~\\ref{tab:beamParametersDesign}).\n\n\n\n\\subsection{Beam Properties at Downstream Polarimeter}     \\label{subsec:measurablePolarisation}\nIn absence of collisions, the bunches at the Compton IPs of the polarimeters are smaller than the laser\nspots or of similar size. In collision mode, the beams get disrupted by the beam-beam interaction.\nAlthough the downstream polarimeter is placed at a secondary focus of the extraction line optics,\nthe refocussing is hampered by the larger emittance and the larger energy spread of the disrupted beam\nafter the collision.\n\n\\begin{figure}[htb]\n   \\begin{center}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/XEDPE}\n         \\subcaption{}\n      \\end{subfigure}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/YEDPE}\n         \\subcaption{}\n      \\end{subfigure}\n      \\ \\\\\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/XEDPCollE}\n         \\subcaption{}\n      \\end{subfigure}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/YEDPCollE}\n         \\subcaption{} \\label{fig:dispersionyDPSubfigB}\n      \\end{subfigure}\n      \\ \\\\\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/XEDPCollEZoom}\n         \\subcaption{\n      \\end{subfigure}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/YEDPCollEZoom}\n         \\subcaption{} \\label{fig:dispersionyDPSubfigC}\n      \\end{subfigure}\n   \\end{center}\n   \\caption{Horizontal and vertical particle positions vs the energy at the downstream polarimeter\n            for 1000 electron bunches without collisions (a,b), after (c,d) the collision. (e,f) show zooms into (c,d).\n            The black solid curve in (b,d,f) shows the dispersion relation between $y$-position and energy.\n  \n   }\n   \\label{fig:dispersionxDP}\n\\end{figure}\n\nThis is illustrated by figure~\\ref{fig:dispersionxDP}, which shows the horizontal and vertical\nparticle positions vs. the particle energies for the sample \\samtdr\\ at the downstream polarimeter\nwithout collisions (a,b) and after a collision (c-f).\nSince the downstream polarimeter is located in a vertical magnet chicane\n(figure~\\ref{fig:downstreamPolChicane}), dispersion also contributes to the vertical beam size.\nThe expected offset due to dispersion is indicated as a black line for comparison.\nIn absence of collisions, the particle bunch sizes are well matched by the design laser spot size\nof $\\sigma_{x\\gamma}=\\sigma_{y\\gamma}=100\\,\\mu$m (cf.\\ section~\\ref{sec:polarimetry}). In contrast,\nthe disrupted beam after a collision extends over centimeters, although a large fraction of particles\nhas lost no or only little energy and is still well focussed.\nThese particles are confined to a spot of $\\sim 100\\,\\mu$m size, while the particles which have\nlost more energy are spread out much further and are not covered by the laser spot of the polarimeter.\n\n\\begin{figure}[hbt]\n   \\begin{center}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/SzXDPCollE}\n         \\subcaption{} \\label{fig:corrPolEYDPSubfigA}\n      \\end{subfigure}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/SzYDPCollE}\n         \\subcaption{} \\label{fig:corrPolEYDPSubfigB}\n      \\end{subfigure}\n     \\ \\\\\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/SzXDPCollEZoom}\n         \\subcaption{} \\label{fig:corrPolEYDPSubfigC}\n      \\end{subfigure}\n      \\begin{subfigure}{.475\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/SzEDPCollE}\n         \\subcaption{} \\label{fig:corrPolEYDPSubfigD}\n      \\end{subfigure}\n   \\end{center}\n   \\caption[]{Longitudinal polarisation of the macroparticles versus the horizontal particle position\n              $x$ (a) (and zoom (c)), the verticle particle position $y$ (b) and the particle energy $E$ (d)\n              at the downstream polarimeter.\n  \n   }\n   \\label{fig:corrPolEYDP}\n\\end{figure}\n\nAs illustrated by figure~\\ref{fig:corrPolEYDP}, the longitudinal polarisation in the electron beam\n(sample \\samtdr, after collision) correlates strongly with the position (a-c) and the energy (d) of the\nmacroparticles, since particles which have been deflected more strongly by electromagnetic fields during\nthe collision experience more spin precession and emit more beamstrahlung.\n\nFor a measurement using a laser spot of $\\sim 100\\,\\mu$m size, this implies that the measurement will\nnot only be rather sensitive to the correct positioning of the laser, but also that the outcome of\nthe measurement depends on the exact size of the laser spot at the Compton IP. As a first estimate of\nthe effect on the downstream polarisation measurement, we investigated the \\textit{measurable polarisation},\nwhich we define as the average longitudinal polarisation of the subset of macroparticles which would\nbe hit by the laser, i.e.\\ within a slice through the bunch defined by the laser spot size and\nthe vertical crossing angle.\n\n\n\n\\subsection{Collision Effects on Polarisation and Polarimetry}\nAs discussed in section~\\ref{subsec:beambeamCollisions}, the luminosity-weighted average polarisation\nduring collisions can be restored at the location of the downstream polarimeter if the extraction line\nquadrupoles halve the divergence angle compared to its value at the $e^+e^-$ IP. This behaviour, however,\nis only obtained if spin fan-out is the dominating effect on the polarisation,\nwhich should be the case for the samples with pure T-BMT precession, \\samrdrfixedEnergy,\n\\samtdrfixedEnergy\\ and TDR*$\\hspace{-1.7mm}_0$.\n\n\\begin{figure}[hbt]\n   \\begin{center}\n      \\includegraphics[width=0.75\\textwidth]{figures\/PolSEIPDPColldE0}\n   \\end{center}\n   \\caption[]{Magnitude of the polarisation vector of the electron beam along the BDS in the absence\n              of any energy loss (data sample \\samrdrfixedEnergy). Shown are the propagation in the\n              case of no collisions, the different propagation after the collision, the luminosity-weighted\n              polarisation in the collision and $f(\\ensuremath{\\thrraw})$ after the collision.}\n  \n   \\label{fig:PolSCollNodE}\n\\end{figure}\n\nFigure~\\ref{fig:PolSCollNodE} shows the polarisation \\pol\\ of the electron beam between the IP and the\ndownstream polarimeter for the case of \\samrdrfixedEnergy\\ after the collision and without collisions.\nDue to the disruption in the collision, the divergence angle of the beam is enlarged from $33\\,\\mu$rad\nto $205\\,\\mu$rad at the IP, which leads to a decrease in polarisation of $0.5\\,\\%$ by spin fan-out.\nThe luminosity-weighted polarisation in the collision, indicated by the red dot at the IP, is lower\nthan the incoming polarisation by $0.27\\,\\%$, which is in accordance with equation~\\ref{eq:yokoyaChen16}.\nBehind the IP, the full tracking result is in perfect agreement with the expectation from pure\nspin precession, $f(\\ensuremath{\\thrraw})$. At the location of the downstream polarimeter, the divergence angle\nis halved to $102\\,\\mu$rad as desired. Thus, the value of the luminosity-weighted polarisation\nis reproduced at the downstream polarimeter to a level of $0.01\\,\\%$ despite spin fan-out of\nseveral percent along the way. This larger spin fan-out occurs since the disrupted beam is more divergent\nand hence refocussed more strongly than a non-colliding beam.\nIn summary, the design concept for the spin transport in the extraction line works perfectly if only\nT-BMT precession is taken into account, but no energy losses or radiative depolarisation.\n\nIn reality, however, energy loss and radiative depolarisation are non-negligible and need to be included\nin any realistic study. Therefore, all six\ncases have been simulated in analogy to figure~\\ref{fig:PolSCollNodE}, and the results\nare summarised in figure~\\ref{fig:PolOverview7}, by displaying the obtained polarisation\nvalues only at the points of interest. For the longitudinal polarisation, the\n{\\itshape measurable polarisation} is indicated in addition for four different assumptions on the laser\nspot size $\\sigma_{x\\gamma}$ ($=\\sigma_{y\\gamma}$), always assuming a perfect centering of laser\nand electron beam.\n\n\\begin{figure}[hbt]\n   \\begin{center}\n      \\begin{subfigure}{0.75\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/ZpolEOverview7}\n         \\subcaption{} \\label{fig:PolOverview7SubfigA}\n      \\end{subfigure}\n       \\ \\\\\n      \\begin{subfigure}{0.75\\linewidth}\n         \\includegraphics[width=\\textwidth]{figures\/PolEOverview7}\n         \\subcaption{} \\label{fig:PolOverview2SubfigB}\n      \\end{subfigure}\n   \\end{center}\n   \\caption{Electron beam polarisation along the BDS at the IP and at the polarimeters. Shown are\n            the longitudinal polarisation \\ensuremath{\\polzraw}\\ (a) and the magnitude of the polarisation vector\n            \\pol\\ (b) of the electron beam at the IP before and after the collision as well as\n            the luminosity-weighted values and the values at the downstream polarimeter (DP).\n            In (a), also the measurable longitudinal polarisation for different laser spot sizes\n            $\\sigma_{x\\gamma}$ ($=\\sigma_{y\\gamma}$) is shown.}\n   \\label{fig:PolOverview7}\n\\end{figure}\n\nThe first column, \\samrdrfixedEnergy , corresponds directly to figure~\\ref{fig:PolSCollNodE},\nshowing again the perfect recovery of the luminosity-weighted average polarisation at the\ndownstream polarimeter. Since the electron beam spot is fully hit by the laser, the\n{\\itshape measurable polarisation} is identical to the longitudinal polarisation of the full bunch.\n\nThe next two columns show the results for TDR parameters, still for T-BMT precession only, but two\ndifferent start configurations of the spins. Interestingly enough, the luminosity-weighted average\npolarisation and the polarisation at the downstream polarimeter\nstart to deviate by $0.06$ - $0.15\\,\\%$ even without energy loss, both in magnitude and in longitudinal\ncomponent. This could be due to the larger horizontal disruption parameter of the TDR parameter set\n($D_x=0.3$ compared to $D_x=0.17$ in the RDR case), since equation~\\ref{eq:yokoyaChen16} assumes\nthe horizontal disruption parameter to be small ($D_x \\ll 1$). Moreover, the size of this deviation\ndepends on the initial spin configuration.\nThis indicates that\na full start-to-end spin tracking simulation from the electron and positron sources is\nhighly desirable for the future.\n\nThe realistic results including energy loss and Sokolov-Ternov depolarisation are shown in\nthe remainder of the figure. Compared to the corresponding sample without energy loss,\nthe depolarisation in collisions increases in all three cases as expected. More importantly,\nalso the differences between the polarisation at the downstream polarimeter and the luminosity-weighted\naverage one increase to tolerable $0.07\\,\\%$ in the \\samrdr\\ case, while it amounts to $0.17$ - $0.24\\,\\%$\nin the \\samtdr\\ case, which exceeds the aim of $0.1\\,\\%$.\n\nHowever, most strikingly the energy loss limits the refocussing of the beams, resulting in significantly\nincreased beam sizes at the location of the downstream polarimeter\ndiscussed in the previous section. Now, the {\\itshape measurable polarisation} deviates\nsignificantly from the total average at the downstream polarimeter by up to $0.3\\,\\%$, and changes with\nthe assumed size of the laser spot by up to $0.07\\,\\%$. The difference between\nthe measureable polarisation and the luminosity-weighted average at the IP is somewhat\nsmaller due to a partial cancellation. In particular in the nominal \\samtdr\\ case, the net difference\namounts to $0.15\\,\\%$ for a nominal laser spot size of $100\\,\\mu$m. This number depends significantly\non the actual beam parameters at the IP: for the \\samrdr\\ case, the\nanaloguous difference is only $0.08\\,\\%$.\n\nFor the positron beam, a different behaviour could be expected due to the different degree of polarisation,\nthe different initial beam energy spread and the absence of the positron production system,\nwhich only exists in the electron beamline. However, the most dominant effect, T-BMT precession,\nscales linearly with \\pol. Thus, the relative changes in the (longitudinal) polarisation of\nthe positron beam differ only slightly from those of the electron beam \\cite{thesisMoritz}.\n\n\nNot yet included in the above numbers are additional contributions from non-perfect centering of\nthe laser beam onto the electron beam nor any effect of the energy and position distributions of\nthe incoming electrons on the analysing power of the downstream polarimeter,\nwhich needs to be addressed in future studies. The luminosity-weighted average polarisations above\nare expressed in the laboratory system, already accounting for the crossing angle of the beamlines,\nbut not boosted to the centre-of-mass system.\n\n\n\n\n\n\\subsection{Accessing the Luminosity-Weighted Average Polarisation}\nIn view of the above results, assuming the downstream polarimeter to directly measure the\nluminosity-weighted average polarisation at the electron-positron interaction point \\ensuremath{{\\polzraw^\\text{lumi}}}\\ seems\nsufficient for percent-level, but not for permille-level precision. Instead, a more sophisticated\nstrategy based on a detailed understanding of the collisions as a function of time is suggested\nin the following.\n\n\\begin{enumerate}\n\\item{\\bf Cross-calibration of the polarimeters:} The very first step, to be repeated regularly,\ne.g.\\ during maintainance of the main detectors, is the cross-calibration of the polarimeters without\ncollisions. The beam time requirement is given by the downstream polarimeter, which\nreaches a statistical precision of below $0.1\\,\\%$ after about one hour\\footnote{It should be\ninvestigated in the future whether in absence of collisions the downstream measurement could collect\nsufficient statistics on similar timescales as the upstream polarimeter.}. Systematic effects on the spin transport\nare expected to be $<0.1\\,\\%$, provided the\norbit alignment goals at the two polarimeter locations can be reached.\n\\item{\\bf Upstream polarimeter:} During collisions, the upstream polarimeter is essential to determine\nthe initial polarisation value, which is expected to be $0.25\\,\\%$ (RDR) to $0.4\\,\\%$ (TDR) above \\ensuremath{{\\polzraw^\\text{lumi}}},\nand to track time variations. Both aspects are needed to predict the expectation at the\ndownstream polarimeter, which measures just one (or a few) bunches out of each train.\n\\item{\\bf Knowledge of collision properties:} In order to predict a) the depolarisation in collisions\nand b) the beam properties at the downstream polarimeter, all possible means to\nmonitor the beam parameters before, during, and after collisions should be employed.\nThis includes the energy spectrometers~\\cite{boogert}, the monitoring of the instantaneous luminosity,\nthe beamstrahlung and the pair background in the forward calorimeters of\nthe main detectors~\\cite{Abramowicz:2010bg} and the Gamma Calorimeter in the downstream polarimeter chicane.\nFrom their combined information, the beam parameters, in particular $\\sigma_{xe}$ and $\\sigma_{ye}$\ncan be determined to about $10\\,\\%$~\\cite{Grah:2008zz}. For comparison, the main difference between the\nRDR and TDR beam parameter sets studied here is a reduction of $\\sigma_{xe}$ by $25\\,\\%$. Thus\nwe estimate that the depolarisation in collisions could be predicted to about $0.1\\,\\%$ based on\nthe measured beam parameters.\n\\item{\\bf Collision simulation:} Based on theoretical understanding of the intense-field QED environment\nof the collisions, the knowledge of initial beam parameters and the post-collision diagnostics,\nsimulations can be employed to predict \\ensuremath{{\\polzraw^\\text{lumi}}}\\ itself, as well as the energy and position distributions\nand the polarisation of the spent beam at the downstream polarimeter location. Redundancy in instrumentation\nis a key to validate the simulation tools.\n\\item{\\bf Downstream polarimeter:} The prediction of the beam properties\nat the downstream polarimeter location then needs to be employed to determine the changes in analysing power\ncompared to the well defined situation without collisions. Furthermore, the effect of uncertainties\nin the laser alignment and focussing need to be assessed. The possibility to defocus the laser on purpose\nand\/or scan the outer regions of the electron beam should be considered in order to ``map out''\nthe spent beam and verify the effects predicted by simulations.\n\\item {\\bf Comparison of polarimeters and simulations:} Based on the previous steps,\nthe measured polarisation value at the downstream polarimeter can be extrapolated to\nthe luminosity-weighted average at the IP. This needs to happen in a time-dependent manner,\nsince the intensities of the bunch-bunch collisions might change during a bunch train and on\nlonger time scales. This value can then be compared to the prediction for \\ensuremath{{\\polzraw^\\text{lumi}}}\\ based on the\nupstream polarimeter measurement and the beam parameter estimates. Only if sufficient agreement\nis found in this step, permille-level accuracy on \\ensuremath{{\\polzraw^\\text{lumi}}}\\ can be claimed.\nIt should be noted that once agreement is established, the extrapolated upstream polarimeter measurement\nwill be the main ``working horse'' due to its ability to simultanously measure\nthe polarisation for each bunch position in a train, while for the downstream polarimeter it takes\norder of days to sample each bunch position with permille-level statistical precision.\n\\item {\\bf Polarisation values for physics analyses:} The relevant effective polarisation\nfor the physics process under study (cf.\\ section~\\ref{subsec:beambeamCollisions}) then needs\nto be calculated from values of \\ensuremath{{\\polzraw^\\text{lumi}}}\\ obtained for both beams from the beam-parameter-dependent\nextrapolation of the polarimeter measurements. Finally a luminosity-weighted average of the effective\npolarisations can be formed for each particular data set, including\ne.g.\\ the analysis-specific run selection based on sub-detector availabilities. Here it should be noted\nthat instrumental uncertainties of the polarimeters as well as effects of misalignments should be\nuncorrelated between both beams, while in particular the collision\neffects are due to the mutual influence of the beams on each other and thus are correlated to a large extent.\n\\item {\\bf Comparison with collision data:} After accumulation of a significant amount of collision data,\nthe long-term average of \\ensuremath{{\\polzraw^\\text{lumi}}}\\ can be extracted also from the collision data themselves.\nTypically several $100$\\,fb$^{-1}$ distributed over all four helicity configurations are needed\nfor sub-percent statistical precision~\\cite{Moenig, thesisIvan, EPSprocedings}, where again\ntime-dependencies and correlations can only be resolved based on\npolarimeter information. Nevertheless, polarisation-sensitive Standard Model processes with\nsufficiently large cross sections will provide an essential verification of the absolute\npolarisation scale.\n\\end{enumerate}\n\n\n\n\\section{Conclusions}      \\label{sec:conclusions}\nThe polarimetery concept for the ILC is based on the combination of two complementary Laser-Compton\npolarimeters per beam and the long-term average polarisation determined\nfrom collision data. These three ingredients can only be exploited coherently in conjunction\nwith detailed simulations of the spin transport and the beam-beam collision effects. For this purpose,\nthe simulation framework {\\tt STALC} has been developed. It is not intrinsically\nlimited to the ILC, but can be run on any lattice and beam parameter set.\n\nIn absence of collisions, various effects which could influence the spin transport between\nthe polarimeters and, thus,\ne.g.\\ their cross-calibration have been evaluated. The dominating uncertainty stems from\nthe relative beam alignment at the two polarimeter locations. Here, the design goal of\n$\\Delta \\thb \\leq 50\\,\\mu$rad is just sufficient for permille-level precision for a beam energy\nof $250$\\,GeV, while for the upgrade to $\\sqrt{s}=1$\\,TeV an improvement to about\n$\\Delta \\thb \\sim 25\\,\\mu$rad would be required in order to maintain the same level of\nprecision on the polarisation.\n\nIn presence of collisions, additional effects have to be considered with respect to the polarisation.\nThis includes both the depolarisation in collision and thus the luminosity-weighted average polarisation\nduring the collision as well as the properties of the spent beam which influence\nthe downstream polarimeter measurement. It has been shown that reducing the angular divergence at\nthe downstream polarimeter with respect to the IP by a factor of two can only partially restore the\nluminosity-weighted average polarisation at the downstream polarimeter location, since\nfor ILC beam parameters the energy loss and Sokolov-Ternov spin flips due to the emission\nof beamstrahlung cannot be fully neglected anymore. The influence of the spent beam properties\non the downstream polarimeter measurement itself have been estimated roughly by\ntaking into account the finite laser spot size and the crossing angle. The variations found\nare again at a level relevant for permille-level precision. Thus, it will be crucial to monitor\nthe collision parameters in real-time at least to the $10\\,\\%$-level by the forward calorimeters\nof the main detectors and additional diagnostics like the GamCal and the extraction-line energy spectrometer.\nWith this knowledge, the luminosity-weighted average\npolarisation at the IP can be extracted from both the upstream and the downstream polarimeter measurements\nwith complementary systematic uncertainties.\n\nIn the future, the influence of the spent beam properties on the downstream polarimeter measurement\nas well as the possibilities to realise the long-range orbit alignment to the tens of micrometer level\nshould be investigated in more detail. Eventually, random misalignments of the beamline elements\nshould be replaced by a proper ground motion model\nfor the actual ILC site in the north of Japan. Many studies performed here depend on the beam energy.\nIn general it is expected that maintaining a permille-level polarisation measurement\nbecomes more difficult with increasing beam energy,\nwhile at lower energies, e.g.\\ at the Higgs or $t\\bar{t}$ threshold, the impact of the collision effects\nis reduced. This, however, needs to be quantified. Finally, in order to eliminate the uncertainty\nfrom assuming a certain spin configuration at the starting point of the simulation,\na full start-to-end spin simulation of the ILC would be highly desirable.\n\n\n\n\\section{Appendix}      \\label{sec:appendix}\nAll results presented in this paper are based on \\cite{thesisMoritz}. Some simulations have been rerun with the following modifications:\n\\begin{itemize}\n   \\item The initial spin configurations are slightly different (see section~7.1.2 in \\cite{thesisMoritz} and section~\\ref{sec:crosscal} in this paper). The discrepancy is however much smaller than the difference between the samples \\samtdr\\ and \\samtdrstar\\ in section~\\ref{sec:lumipol}, since the angular divergence at the beginning of the lattice is only 1\\,mrad.\n   \\item While the collisions are simulated with the TDR beam parameters in this paper (samples \\samtdr\\ and \\samtdrstar), the corresponding simulations in \\cite{thesisMoritz} are performed with RDR beam parameters and an increased bunch charge, such that the amount of beamstrahlung produced in the collisions is approximately the same as for collisions with TDR beam parameters (sample $\\Upsilon_{TDR}$).\n   \\item For the collision effects, some data samples have been simulated without beamstrahlung and synchrotron radiation. In \\cite{thesisMoritz}, the initial beam energy spreads are still present (sample no BS,SR), while in this paper also the initial beam energy spreads are set to zero (samples \\samrdrfixedEnergy, \\samtdrfixedEnergy\\ and TDR*$\\hspace{-1.7mm}_0$).\n   \\item Misalignments: In contrast to this paper, the corresponding simulations in \\cite{thesisMoritz} include bunch rotation at the IP (crab cavities) and energy losses due to synchrotron radiation.\n   \\item Measurable Polarisation: In \\cite{thesisMoritz}, the crossing angle of the laser beam has not been taken into account. Therefore, it is defined by a cut on the radius around the bunch center rather than by a cut on the horizontal distance.\n\\end{itemize}\n\n\n\\section*{Acknowledgement}\nWe thankfully acknowledge the support by the BMBF Verbundforschung ``Spin Optimierung'' and by the DFG via\nthe Emmy-Noether-Grant Li\/1560-1. We thank Karsten B\\\"u\\ss er, Mathias Vogt,\nPeter Sch\\\"uler, Anthony Hartin, Desmond Barber, Ken Moffeit, Mike Woods and Yuri Nosochkov for many helpful discussions over the last years.\n\n\\input{appendix}\n\n\\input{references}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPerovskite cerates and zirconates are a class of materials that, with\nappropriate doping, exhibit ionic conductivity both of O vacancies and\nprotons, and therefore are suitable as solid electrolytes for fuel cells,\ngas sensors and other electrochemical devices. Dissolution of H is achieved\nin two steps:\\cite{Kre97}\\ the material is doped with lower valence cations,\n\\textit{e.g.} BaCe$_{1-x}$Y$_{x}$O$_{3-x\/2}$ (BCY) where the partial\nsubstitution of Ce$^{4+}$ with Y$^{3+}$ introduces charge compensating O\\\nvacancies; the material is then exposed to a humid atmosphere at high\ntemperature, so that the H$_{2}$O molecules may dissociate, each of them\nfilling an O vacancy and contributing with two H atoms. Infrared spectroscopy%\n\\cite{GPB02} and diffraction\\cite{Kni00} experiments indicate that H is\nbound to an O, forming an OH$^{-}$ ion, but also makes some hydrogen-bonding\nwith a next-nearest O atom. The proton diffusion is believed to consist of a\nrapid rotation about the O\\ atom to which is associated, and slower jumps to\none of the eight next-nearest neighbor O atoms with which an instantaneous\nhydrogen bond is established (see Fig. \\ref{fig7}a below). It has also been\nproposed, however, that in some distorted zirconates and titanates the\nrotational barrier may be higher than the transfer barrier.\\cite{GGJ05} In\nvarious cubic perovskites ABO$_{3}$ there are four equilibrium orientations\nof the OH$^{-}$ complex, with an OH separation of $0.9-1.0$\\emph{~}\\AA\\ and\nH pointing in the $\\left\\langle 100\\right\\rangle $ directions perpendicular\nto the B-O-B bond;\\cite{MSK96,INI07} in the presence of dopants or non cubic\ndistortions such positions would be shifted.\\cite{Kni00} It is very likely\nthat the fast local motion of H\\ about the same O is dominated by tunneling,\nbut so far no quantitative measurements of the associated correlation times\nhave appeared, except for quasi-elastic neutron scattering experiments on\nhydrated Ba(Ca$_{0.39}$Nb$_{0.61}$)O$_{2.91}$, where a fast local motion has\nbeen detected above room temperature with an apparent activation energy of $%\n\\sim 0.1$~eV.\\cite{PMS97} Also in SrCe$_{0.95}$Yb$_{0.05}$O$_{2.97}$ a broad\nquasielastic component has been attributed to fast proton rotation,\\cite%\n{MSK96b} but no reliable measurement of the associated rate was possible.\n\nThere is some controversy on the effect of dopants on the proton mobility,\nas reviewed in Ref. \\onlinecite{BSW07}: on the one hand there are several\nindications for trapping,\\cite{HKM95,HSH98,GLD07, BSW07} with formation of\nstable dopant-H complexes, but it has also been proposed that the excess\ndoped charge distributes over all O sites, causing an increase of the\nhopping barrier for the proton over the whole lattice.\\cite{MUM04}\n\nAnelastic and dielectric spectroscopies may contribute to answer to such\nissues, since both an electric and elastic dipole are associated with a OH$%\n^{-}$ ion or Y-OH$^{-}$ complex, and each type of jump or reorientation with\ncharacteristic time $\\tau \\left( T\\right) $ causes a maximum of the losses\nat the temperature $T$ and frequency $\\omega \/2\\pi $ such that $\\omega \\tau\n\\left( T\\right) \\simeq 1$. Anelastic relaxation is particularly useful,\nsince it is almost insensitive to electronic conduction and is not affected\nby charged interface layers. We present anelastic and dielectric\nspectroscopy measurements on protonated, deuterated and outgassed BaCe$%\n_{1-x} $Y$_{x}$O$_{3-x\/2}$, where different relaxation processes are\nascribed to H migration, reorientation about an Y dopant and tunneling about\na same O atom; the focus will be on the last type of motion.\n\n\\section{Experimental}\n\nThe starting powders of BCY were prepared by auto-combustion synthesis,\nwhich is an easy and convenient solution-based method for the preparation of\nnanometric mixed-oxide powders. The method\\cite{DMD08} is an improvement of\nthat for metal citrates described in the literature.\\cite{XHC04}\nStoichiometric amounts of highly purified metal nitrates were dissolved in\ndistilled water and mixed with citric acid which acted both as metal ions\ncomplexant and as fuel. The citric acid to metal nitrates ratio was\nmaintained to 2, ammonium nitrate was added to regulate the fuel to oxidant\nratio (citric acid\/total nitrate ions ratio) to 0.4 and ammonia solution\n(30\\%wt) was added to regulate the pH value at 6. The water solution was\nleft to evaporate at 80~$^{\\mathrm{o}}$C under constant stirring in a beaker\nimmersed in a heated oil bath, until a whitish and sticky gel was obtained.\nThe temperature was then raised to 200~$^{\\mathrm{o}}$C until the gel became\ncompletely black and dry. The beaker was then put directly on the hot-plate\nat $250-300$~$^{\\mathrm{o}}$C until the auto-combustion reaction occurred\nleaving the powdered product. Crystallization was completed by firing the\ncombusted powders in stagnant air at 1000~$^{\\mathrm{o}}$C for 5~h. No\nweight loss occurred during the synthesis and no oxide segregation has been\ndetected by X-ray diffraction, so that we assume the nominal compositions of\nBaCe$_{1-x}$Y$_{x}$O$_{3-x\/2}$ with $x=0.1$ (BCY10)\\ and $x=0.15$ (BCY15).\nThe nanopowders were first uniaxially pressed at 50~MPa and then\nisostatically pressed at 200~MPa obtaining $60\\times 7\\times 6$~mm bars,\nwhich were sintered at 1500~$^{\\mathrm{o}}$C for 10~h. The bars where cut as\nthin reeds about 4~cm long and 1~mm thick, whose major faces where covered\nwith Ag paint.\n\nThe maximum molar concentration $c_{\\mathrm{H\/D}}$ of H and D was measured\nfrom the change of weight when the sample state was changed between fully\noutgassed (up to 730~$^{\\mathrm{o}}$C in vacuum $<10^{-5}$~mbar) and\nhydrated for 1-2~h at 520~$^{\\mathrm{o}}$C in a static atmosphere of $50-100$%\n~mbar H$_{2}$O or D$_{2}$O followed by slow cooling; it was found $c_{%\n\\mathrm{H}}=0.14$ for $x=0.15$ and $c_{\\mathrm{H}}\\lesssim 0.086$ for $%\nx=0.10 $, slightly less than the theoretical maximum $c_{\\mathrm{H}}=x$.\n\nThe elastic compliance $s\\left( \\omega ,T\\right) =s^{\\prime }-is^{\\prime\n\\prime }$ was measured by electrostatically exciting the flexural modes of\nthe bars suspended in vacuum on thin thermocouple wires in correspondence\nwith the nodal lines; the 1st, 3rd and 5th modes could be measured, whose\nfrequencies are in the ratios $1:5.4:13.3$, the fundamental frequencies of\nsamples being $\\omega \/2\\pi \\simeq 2.8$\\ kHz. The elastic energy loss\ncoefficient, or the reciprocal of the mechanical quality factor,\\cite{NB72} $%\nQ^{-1}\\left( \\omega ,T\\right) =$\\ $s^{\\prime \\prime }\/s^{\\prime }$ was\nmeasured from the decay of the free oscillations or from the width of the\nresonance peak. The elastic compliance $s$\\ is the mechanical analogue of\nthe dielectric susceptibility $\\chi $, with $Q^{-1}$\\ corresponding to $\\tan\n\\delta $.\n\nThe dielectric permittivity $\\varepsilon =\\varepsilon ^{\\prime\n}-i\\varepsilon ^{\\prime \\prime }$\\ was measured with a HP 4284A impedance\nbridge with a four wire probe between 3 and 100~kHz in the same cryostat\nused for the anelastic measurements. After depositing the Ag electrodes, an\nintense dielectric relaxation process, identified with the motion of charge\ncarriers within a Schottky barrier at the electrode interface, completely\nmasks the true bulk relaxation. Such an effect was suppressed by applying\n40~V up to $\\sim 500$~K, switching the direction of the dc current in order\nto avoid electromigration of O vacancies or protons.\\cite{Cra08}\n\nA defect hopping or reorienting with characteristic time $\\tau $ and causing\na change $\\Delta \\lambda $ of its elastic quadrupole and $\\Delta p$ of its\nelectric dipole contributes with a Debye peak to the imaginary parts of the\nelastic compliance $s$ and dielectric permittivity $\\varepsilon $ as:\\cite%\n{NH65}\n\\begin{equation}\ns^{\\prime \\prime }=\\frac{c~\\left( \\Delta \\lambda \\right) ^{2}}{3v_{0}k_{%\n\\text{B}}T\\cosh ^{2}\\left( E\/2T\\right) }\\frac{\\omega \\tau }{1+\\left( \\omega\n\\tau \\right) ^{2}}\\text{ and}  \\label{Ds_anel}\n\\end{equation}\n\n\\begin{equation}\n\\varepsilon ^{\\prime \\prime }=\\frac{c~\\left( \\Delta p\\right) ^{2}}{%\n3\\varepsilon _{0}v_{0}k_{\\text{B}}T\\cosh ^{2}\\left( E\/2T\\right) }\\frac{%\n\\omega \\tau }{1+\\left( \\omega \\tau \\right) ^{2}}~,  \\label{De_diel}\n\\end{equation}%\nwhere $c$ is the molar concentration of relaxing defects, $v_{0}$ the\nmolecular volume, and $E$ is the energy difference between the states\nparticipating to relaxation. When $E~\\gtrsim T$, the higher energy state\nbecomes less populated and therefore there is a reduced change of the defect\npopulations under application of the probe field with respect to the case $%\nE=0$; the consequent reduction of the relaxation strength is described by\nthe factor sech$^{2}\\left( E\/2T\\right) $.\\cite{33}\n\n\\section{Results}\n\n\\subsection{Anelastic spectra}\n\nFigure \\ref{fig1} presents the anelastic spectra of BaCe$_{0.9}$Y$_{0.1}$O$%\n_{3-\\delta }$ measured at the fundamental frequency of 2.8~kHz: 1)\\ in the\nas prepared state, therefore nearly saturated with H$_{2}$O; 2) after\noutgassing H$_{2}$O at 900~$^{\\mathrm{o}}$C in a flux of pure O$_{2}$ for\n2.5~h; 3)\\ after keeping in a static atmosphere of $\\sim 70$~mbar D$_{2}$O\nat 520~$^{\\mathrm{o}}$C followed by cooling at 1~$^{\\mathrm{o}}$C\/min; 4)\\\nafter having measured in vacuum up to 500~$^{\\mathrm{o}}$C with partial loss\nof D$_{2}$O.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig1.EPS}\n\\caption{Anelastic spectra of BCY10 measured in different conditions of\nhydration with H$_{2}$O and D$_{2}$O.}\n\\label{fig1}\n\\end{figure}\n\nThere are at least three relaxation processes clearly due to the presence of\nH or D: the peak labeled P$_{\\mathrm{Y}}$ at $\\sim 300$~K, P$_{\\mathrm{h}}$\nat $\\sim 200$~K and P$_{\\mathrm{t}}$ at $\\sim 30$~K; above 400~K start the\ncontributions due to O\\ vacancies. All these peaks shift to higher\ntemperature when measured at higher frequency, and a preliminary analysis\nyields activation energies of 0.58~eV with $\\tau _{0}\\sim $ $3\\times\n10^{-14} $~s for P$_{\\mathrm{Y}}$ and $\\sim 0.4$~eV for P$_{\\mathrm{h}}$;\nthe latter is likely composed of different processes. These peaks are quite\nbroader than pure Debye relaxations and the possible difference in\nrelaxation parameters between H and D is not apparent without an accurate\nanalysis; they must be due to hopping of H between different O\\ atoms. Peak P%\n$_{\\mathrm{t}}$ is the focus of the present work and shifts from 29 to 38~K\nwhen H\\ is replaced with D, clearly indicating that it is due to the fast\nmotion of H about a same O atom with a dynamics dominated by tunneling. An\nadditional peak at $\\sim 100$~K might also be due to H and possibly shifts\nto higher temperature after substitution of H with D, but its nature is not\nas clear as for the other processes and we will not discuss it further.\n\n\\subsection{Comparison between anelastic and dielectric spectra}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig2.EPS}\n\\caption{Dielectric (left-hand scale) and anelastic (right-hand scale)\nspectra of BCY: $\\protect\\varepsilon ^{\\prime \\prime }$ of hydrated BCY15 at\n3, 10 and 30~kHz (continuous lines) and $\\protect\\varepsilon ^{\\prime \\prime\n}$ of hydrated BCY10 at 3~kHz (curve 1). The dotted lines are $Q^{-1}$ of\nhydrated (2) and outgassed (3) BCY10 measured at 2.8~kHz.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig3.EPS}\n\\caption{Low-temperature region of the dielectric and anelastic spectra of\nBCY of Fig. \\protect\\ref{fig2}. The $\\protect\\varepsilon ^{\\prime \\prime }$\n(left) and $Q^{-1}$ (right) scales are chosen in order to emphasize the near\ncoincidence of dielectric and anelastic P$_{\\mathrm{t}}$ (curves 1 and 2).}\n\\label{fig3}\n\\end{figure}\n\nAll the peaks appearing in the anelastic spectrum are present also in the\ndielectric one, which also displays an additional maximum P$_{\\mathrm{d}}$\nat $\\sim 60$~K. Figure \\ref{fig2} shows $\\varepsilon ^{\\prime \\prime }$ of\nhydrated BCY15 and BCY10 ($c_{\\mathrm{H}}=0.078$, curve 1); the latter is\ncompared with the elastic counterpart $Q^{-1}$ in the hydrated (curve 2)\\\nand outgassed (curve 3) states, measured at the same frequency of 3~kHz. It\nappears that the intensities of peaks P$_{\\mathrm{t}}$, P$_{\\mathrm{h}}$ and\nP$_{\\mathrm{Y}}$ span more than four orders of magnitude in $\\varepsilon\n^{\\prime \\prime }$ but only two in $Q^{-1}$, meaning that the fast tunneling\nmotion of H produces a much smaller change in the electric dipole than in\nthe elastic dipole, compared to the hopping motion. On the other hand, the\npeak temperatures and activation energies appear to be practically the same\nin the dielectric and anelastic losses. These dielectric measurements on\nBCY15 have been made after full elimination of the charge relaxation at the\nelectrodes, so that the intensities of P$_{\\mathrm{Y}}$ and P$_{\\mathrm{h}}$\nare reliable. Figure \\ref{fig3} shows $\\varepsilon ^{\\prime \\prime }$ of\nboth hydrated (thick lines) and deuterated (thin lines)\\ BCY15. While peak P$%\n_{\\mathrm{t}}$ shifts to higher temperature after the isotope substitution\nof H with D as for the anelastic case, peak P$_{\\mathrm{d}}$ remains\ncompletely unaffected; this fact, together with its absence in the anelastic\nlosses, indicate that P$_{\\mathrm{d}}$ is not connected with the H motion,\nbut is likely of electronic origin. Curves $1-3$ in Fig. \\ref{fig3} are the\nsame dielectric and anelastic curves of BCY10 appearing in Fig. \\ref{fig2},\nbut this time the two scales are chosen in order to yield the same intensity\nof P$_{\\mathrm{t}}$. Unfortunately, experimental difficulties connected with\nthe elimination of the metal-semiconductor barrier at the electrodes did not\nallow us to obtain significant dielectric spectra in the outgassed state, so\nthat it is possible to determine that the intensity of P$_{\\mathrm{d}}$\nalmost triplicates in the hydrated state passing from 10\\%\\ to 15\\%Y doping,\nbut nothing can be said on its dependence on the content of H or O vacancies.\n\n\\section{Discussion}\n\n\\subsection{Proton hopping and trapping}\n\nIn what follows we will identify peak P$_{\\mathrm{Y}}$\\ with hopping of H\namong the O atoms of YO$_{6}$ octahedra, namely with the reorientation of\nthe Y-H complex, and P$_{\\mathrm{h}}$\\ with hopping over CeO$_{6}$\noctahedra, with the possible contribution of the formation\/dissociation of\nY-H complexes. These assignments are suggested by the fact that it is\nnatural to assume that (OH)$^{+}$ ions, having a formal charge +1 with\nrespect to O$^{2-}$, may form relatively stable complexes with trivalent\ndopants Y$^{3+}$, having formal charge -1 with respect to Ce$^{4+}$. Then, P$%\n_{\\mathrm{Y}}$ with higher intensity and activation energy should be due to\nthe more numerous H atoms associated with Y, while P$_{\\mathrm{h}}$ should\nbe due to the faster jumps of the less numerous H atoms not associated with\nY. The assignment of anelastic relaxation peaks to dopant-H complexes has\nalready been proposed for BaCe$_{1-x}$Nd$_{x}$O$_{3-\\delta }$\\cite{Du94} and\n(Ba,Sr)Ce$_{1-x}$Yb$_{x}$O$_{3}$,\\cite{ZBS95} although the argument that the\n(OH)$^{-}$ ion has the same symmetry as the crystal and therefore cannot\nproduce anelastic relaxation unless forming defect complexes\\cite{Du94} is\nincorrect. Additional experimental indications that H is trapped by\ntrivalent dopants in BaCe$_{1-x}$Y$_{x}$O$_{3}$ are the analysis in terms of\ntwo components of the quasielastic neutron scattering peak,\\cite{HKM95} and\nthe observation with EXAFS of an enhancement of the disorder in the\nenvironment of Y after hydration.\\cite{GLD07} Also first-principle\ncalculations and Monte-Carlo simulations of the proton diffusion indicate\nthat dopants act as traps.\\cite{BSW05,BSW07}\n\nOn the other hand, there are also experiments suggesting the absence of\nsignificant trapping, like an NMR investigation\\cite{MUM04} on BaCe$_{1-x}$Y$%\n_{x}$O$_{3}$\\ with $x=0.01$\\ and 0.1, where the correlation time deduced\nfrom the $^{1}$H NMR relaxation is the same at both doping levels and\nreproduces the conductivity with a simple hopping model without trapping. At\nthis stage, it cannot be completely excluded from our data that there is\nindeed very little trapping effect from Y dopants, and the two main\nanelastic relaxation processes P$_{\\text{Y}}$ and P$_{\\text{h}}$ are\nassociated with H hopping among O1 and O2 atoms of the orthorhombic\nstructure, having different symmetries. It is possible that H binds to O1\nand to O2 with different probabilities, as neutron diffraction experiments\nand molecular dynamics simulations suggest,\\cite{MKA99,Kni00} and jumps\nwithin the respective sublattices with different rates, so giving rise to\npeaks with distinct intensities and temperatures. In the discussion we will\nalso mention the possibility that in the distorted orthorhombic structure\nthe reorientation of the OH ions is much slower than found in the higher\ntemperature phases, and P$_{\\text{h}}$\\ is due to such slow reorientation.\n\n\\subsection{Fit of the anelastic P$_{\\mathrm{t}}$}\n\nThe relaxation modes of the fast reorientation of OH among four positions\ncan be found by solving the rate equations for classical hopping between the\nfour H sites or the quantum mechanical problem with tunneling between\nnearest neighbor orientations. In the classical symmetric case there are two\nmodes: one active in the dielectric relaxation and another in the anelastic\nrelaxation with a rate twice larger; in the case that the site energies are\ndifferent, however, the modes become three, all contributing to both\nanelastic and dielectric relaxation, and with possibly widely changing\nrates, depending on the type of asymmetry and the degree of coherence of the\neigenstates of H plus polaron-like distortion of the surrounding atoms. At\nvariance with the geometrically analogous case of the reorientation of Zr-H\ncomplexes in Nb,\\cite{36,45,85,CCC98} in highly doped and orthorhombic BCY\nit is not possible to distinguish different peaks arising from these modes,\nand therefore we will limit ourselves to fit P$_{\\mathrm{t}}$ with a single\nrelaxation time $\\tau $ plus broadening, as\n\\begin{equation}\nQ^{-1}\\left( \\omega ,T\\right) =\\frac{\\Delta _{0}}{T~\\text{cosh}^{2}\\left(\nE\/2k_{\\text{B}}T\\right) }\\frac{\\sqrt{\\alpha \\beta }}{\\left( \\omega \\tau\n\\right) ^{\\alpha }+\\left( \\omega \\tau \\right) ^{-\\beta }},  \\label{eq Q0}\n\\end{equation}%\nwhere the parameters $\\alpha ,\\beta \\leq 1$ produce broadening of the low-\nand high-temperature sides of the peak, respectively; when $\\alpha =\\beta =1$\nthe above expression reduces to a Debye peak. The parameter $E$ is the\nenergy difference between the configurations involved in relaxation, and\nmust be introduced in order to reproduce the enhancement of the peak height\nat higher frequency.\\cite{33} The relaxation rate was modeled as\n\\begin{eqnarray}\n\\tau ^{-1} &=&\\tau _{\\mathrm{1ph}}^{-1}\\coth \\left( E\/2k_{\\text{B}}T\\right)\n+\\left( T\/T_{\\mathrm{2ph}}\\right) ^{n}+  \\label{eq t0} \\\\\n&&+\\tau _{0}^{-1}\\exp \\left( -W\/T\\right) .\n\\end{eqnarray}%\nSuch an expression is similar to that used by Kuskowsky, Lim and Nowick,\\cite%\n{KLN99} for the low temperature dielectric relaxation in Ba$_{1-x}$Nd$_{x}$%\nCeO$_{3-x\/2}$; it does not rely on a model of the interaction between a\nprecise defect geometry and the actual phonon bath, but is able to describe\nthe main hopping regimes, including tunneling in an insulating crystal. One\nexpects that, starting from low temperature, the transitions between the\ndefect eigenstates occur through processes involving one-phonon, then\ntwo-phonons and finally several phonons or semiclassical hopping.\\cite{CCC98}\nIn the one-phonon regime the transition rate is $\\tau _{\\text{1ph}%\n}^{-1}\\coth \\left( E\/2k_{\\text{B}}T\\right) $, which becomes temperature\nindependent when $k_{\\text{B}}T\\ $is smaller than the separation $E$ between\nthe eigenstates; the latter has a form of the type $E\\simeq \\sqrt{t^{2}+a^{2}%\n}$ where the tunneling matrix element $t$ is expected to be smaller than the\ntypical energy asymmetry $a$ between the site energies due to the\northorhombic distortion and disordered nature of the BCY solid solution; in\nthe fit $E=a$. The two-phonon relaxation rate approximately depends on\ntemperature through a power law,\\cite{FS70,Kli83,SBH94} with $5\\leq n\\leq 9$\ndepending on the type of interaction with acoustic phonons and on the energy\ndifference between H\\ sites. For interaction with optical phonons of\nfrequency $\\omega _{0}$ an Arrhenius-like $\\exp \\left( -\\hbar \\omega _{0}\/k_{%\n\\text{B}}T\\right) $ dependence is predicted.\\cite{Kli83} At sufficiently\nhigh temperatures an Arrhenius-like temperature dependence is always found,\nalthough the pre-exponential term $\\tau _{0}$ is not necessarily related to\nthe frequency of a local H vibration mode, but depends also on the overlap\nof the wave functions of H\\ in adjacent sites, and the barrier $W$ may\ncontain significant corrections due to phonon fluctuations with respect to\nthe static potential.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig4.EPS}\n\\caption{Fit of the anelastic P$_{\\mathrm{t}}$ of hydrated and deuterated\nBCY10, measured at 2.9 and 15~kHz.}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig5.EPS}\n\\caption{Relaxation rates $\\protect\\tau^{-1}$ used in the fits of Fig.\n\\protect\\ref{fig4}. The rate for deuterium is decomposed into one- ,\ntwo-phonon and Arrhenius-like contributions.}\n\\label{fig5}\n\\end{figure}\n\nWe found that all the three contributions to $\\tau ^{-1}$ are necessary to\nobtain a good fit: the one-phonon term reproduces the low temperature\nbroadening of P$_{\\mathrm{t}}$ without resorting to extremely small values\nof $\\alpha $. The introduction of the power low significantly improves the\nfit quality around the maximum. The Arrhenius contribution is not essential\nto obtain good fits, since its suppression can be partially compensated by\nan increase of $\\beta $, namely by diminishing the broadening at high\ntemperature; however, if one wants to keep $\\alpha \\simeq \\beta $ the\nArrhenius contribution must be included. The condition $\\alpha \\simeq \\beta $\nappears desirable, if such parameters should describe the broadening of a\nDebye peak due to lattice disorder, and not to other types of interactions,\nlike collective interactions among H atoms.\n\nThe continuous lines in Fig. \\ref{fig4} are fits of anelastic P$_{\\mathrm{t}%\n} $ of BCY10 both hydrated and deuterated with the above expressions plus a\nlinear background. In view of the partial duplication of the effects of some\nparameters, like the high temperature broadening $\\beta $ and the Arrhenius $%\n\\tau _{0}$ and $W$, these fits are not unique, and we tried to obtain a\nphysically sound combination of the parameters. The mean asymmetry energy is\ndetermined quite precisely as $E\/k_{\\text{B}}=64$~K from the temperature\ndependence of the peak intensities. Regarding broadening, it is possible to\nobtain good fits with $\\alpha =\\beta =0.5$ for both H and D, although $\\beta\n=0.38$ for H gives a slightly better interpolation, as in Fig. \\ref{fig4};\nthese values of $\\alpha $ and $\\beta $ imply a broadening that is\nconspicuous but expected, in view of the high lattice disorder. The other\nparameters are: $\\tau _{0}=$ $7.7\\times 10^{-14}$~s ($1.1\\times 10^{-13}$), $%\nW=744$~K (820), $T_{\\mathrm{2ph}}=$ $6.7$~K (8.4), $n=7$, $\\tau _{\\mathrm{1ph%\n}}^{-1}=$ 7300~s$^{-1}$ (990) for H (D). The resulting $\\tau _{\\mathrm{H,D}%\n}^{-1}\\left( T\\right) $ are plotted in Fig. \\ref{fig5} and it turns out that\n$\\tau _{\\mathrm{H}}^{-1}\/\\tau _{\\mathrm{D}}^{-1}\\simeq 8$ at all\ntemperatures; such a ratio certainly indicates a non-classical effect of the\nisotope mass on the H dynamics. The barrier $W\\simeq 800$~K of the\nArrhenius-like contribution is close to the activation energy for rotational\ndiffusion obtained from quantum molecular dynamics simulations,\\cite%\n{MKA99,note} and to that extracted from quasi-elastic neutron scattering in\nBa(Ca$_{0.39}$Nb$_{0.61}$)O$_{2.91}$.\\cite{PMS97} It cannot be excluded,\nhowever, that it rather originates from two-phonon interaction with optical\nphonons,\\cite{Kli83} considering that the infrared absorption bands in\nvarious cerates range from $\\hbar \\omega \/k_{\\text{B}}=$ 600~K to 1000~K\n(400 to 700~cm$^{-1}$).\\cite{MOK06} The power law with $n=7$ instead is\ntypical of two-phonon transitions of asymmetric states interacting with\nacoustic phonons; lower values of $n$ yield definitely worse fits, while $%\nn\\simeq 7.8$ is found if the Arrhenius contribution is omitted.\n\n\\subsection{Fit of dielectric P$_{\\mathrm{t}}$ and P$_{\\mathrm{d}}$}\n\nThe most reliable fit is on BCY10, where peak P$_{\\mathrm{d}}$ has a reduced\nintensity. Below 100~K, $\\varepsilon ^{\\prime }\\left( \\omega ,T\\right) $ has\nalready approached the limiting high frequency value $\\varepsilon _{\\infty\n}=20.1$, so that the $\\varepsilon ^{\\prime \\prime }\\left( T\\right) =$ $%\n\\varepsilon _{\\infty }\\times \\tan \\delta \\left( T\\right) $ and in Fig. \\ref%\n{fig6} we show $\\tan \\delta $ at 10, 30 and 100~kHz; at lower frequency the\nnoise was too large to add significant information. Note that, due the\nimperfect compensation of the cables impedance at low temperature, and the\nvery small values of the dielectric losses, the $\\tan \\delta $ curves can be\narbitrarily shifted in the ordinate scale. In fact, it was verified by\nswitching on and off the cable compensation that below 150~K such a\ncorrection became independent of temperature and therefore introduced only a\nshift of the $\\tan \\delta $ curves.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig6.EPS}\n\\caption{Fit of the dielectric spectrum of BCY10.}\n\\label{fig6}\n\\end{figure}\n\nThe shapes and temperatures of dielectric and anelastic P$_{\\mathrm{t}}$\nneed not to be identical, since the various relaxation modes contributing to\nthem may have different strengths for anelastic and dielectric relaxation.\nYet, we chose not to let all the parameters of P$_{\\mathrm{t}}$ to vary\nfreely, due to the still overwhelming presence of P$_{\\mathrm{d}}$ at higher\ntemperature and possibly to the presence of another relaxation at lower\ntemperature. Therefore, all the parameters of P$_{\\mathrm{t}}$ where set to\nthe values of the anelastic fit of Fig. \\ref{fig4}, with a factor\nmultiplying the anelastic $\\tau ^{-1}$ as the only degree of freedom: $\\tau\n_{\\mathrm{diel}}^{-1}=r~\\tau ^{-1}$; it turns out that $r=10.6$ yields a\ngood fit. Peak P$_{\\mathrm{d}}$, appearing only in the dielectric spectrum,\nhas an intensity $\\Delta \\varepsilon _{\\mathrm{d}}$ that clearly rises with\ntemperature (see Fig. \\ref{fig3}) and there are two main possible causes for\nsuch a behavior: \\textit{i)} P$_{\\mathrm{d}}$ is due to the relaxation of a\nthermally excited state with energy $E$ over the ground state, \\textit{e.g.}\ncharges from a ionized defect, and therefore the relaxation strength\ncontains the population of the ionized state as a factor, $\\Delta\n\\varepsilon _{\\mathrm{d}}\\propto $ $1\/\\cosh (E\/2T)$; \\textit{ii)} relaxation\noccurs between two states differing in energy by $a$, so that $\\Delta\n\\varepsilon _{\\mathrm{d}}\\propto $ $n_{1}n_{2}=$ $1\/\\cosh ^{2}\\left(\na\/2T\\right) $. It is also possible that both mechanisms are present, but the\ndata do not allow to distinguish between these possibilities, and we will\nconsider the case $a\\neq 0$, $E=0$; note that the two mechanisms give\nsimilar temperature dependence of $\\Delta \\varepsilon _{\\mathrm{d}}$ in a\nbroad temperature range if $E\\sim 1.5a$. The shape of P$_{\\mathrm{d}}$ could\nbe better reproduced with the Cole-Cole expression for broadening, so that\nthe expression for fitting P$_{\\mathrm{d}}$ was\n\\begin{equation*}\n\\varepsilon _{\\mathrm{d}}^{\\prime \\prime }=\\frac{\\Delta _{\\mathrm{d}}}{T~%\n\\text{sech}^{2}\\left( a_{\\mathrm{d}}\/2T\\right) }\\frac{\\sin \\left( \\pi \\alpha\n_{\\mathrm{d}}\/2\\right) }{\\cosh \\left[ \\alpha _{\\mathrm{d}}\\ln \\left( \\omega\n\\tau _{\\mathrm{d}}\\right) \\right] +\\cos \\left( \\pi \\alpha _{\\mathrm{d}%\n}\/2\\right) }\n\\end{equation*}%\nand $\\tau _{\\mathrm{d}}=\\tau _{\\mathrm{d0}}\\exp \\left( W_{\\mathrm{d}%\n}\/T\\right) $. The thick curves in Fig. \\ref{fig6} are the resulting fit with\nlinear backgrounds, and also P$_{\\mathrm{t}}$ and P$_{\\mathrm{d}}$ are\nshown. The parameters of P$_{\\mathrm{d}}$ are: $a_{\\mathrm{d}}=$ 150~K\n(150), $\\tau _{\\mathrm{d0}}=$ $1\\times 10^{-15}$~s ($6\\times 10^{-15}$), $W_{%\n\\mathrm{d}}=$ 1270~K (1270) and $\\alpha _{\\mathrm{d}}=$ 0.26 (0.38), where\nthe values in parenthesis are obtained from the measurements on BCY15 (data\nof Fig. \\ref{fig3}). The small values of $\\tau _{\\mathrm{d0}}$ are in\nagreement with the electronic origin of the relaxation, but, considering the\nextreme peak broadening (small $\\alpha _{\\mathrm{d}}$), it cannot be\nexcluded that a correlated dynamics is present, which may be better\ndescribed by a Vogel-Fulcher type $\\tau \\left( T\\right) $ rather than\nArrhenius.\n\n\\subsection{The intensity of P$_{\\mathrm{t}}$}\n\n\\subsubsection{The electric dipole of the OH group}\n\nThe dependence of the intensity of the anelastic P$_{\\mathrm{t}}$ on\nthe H content and the marked shift to higher temperature with the\nheavier D\\ isotope mass leave little doubt that anelastic\nP$_{\\mathrm{t}}$ is due to the fast motion of H around a same O with\na dynamics dominated by tunneling.\nIt is also clear that P$_{\\mathrm{t}}$ has its dielectric counterpart (Fig. %\n\\ref{fig3}, curves 1 and 2). A puzzling feature of the dielectric P$_{%\n\\mathrm{t}}$ is its very small intensity, and therefore we discuss now about\nthe estimated strengths of dielectric and anelastic relaxation. The\ndielectric relaxation strength associated with various types of H jumps\nshould be relatively easy to estimate in a medium which is not particularly\nhighly polarizable like BCY. Let us first consider P$_{\\mathrm{Y}}$,\ninterpreted as the reorientation of an effective dipole Y$^{\\prime }-$OH$%\n^{\\bullet }$ where the Kr\\\"{o}ger-Vink notation expresses the fact that Y$%\n^{3+}$ has an excess $-e$ charge with respect to Ce$^{4+}$ of the perfect\nlattice and OH$^{-}$ has a $+e$ charge with respect to O$^{2-}$. On the time\nscale of the H reorientation among different faces of the cube containing Y,\nthe fast motion around each O is completely averaged out with barycenter\nnear O, so giving rise to an effective dipole $p^{\\mathrm{Y}}=ea\/2$, where $%\na $ is the lattice constant of the cubic perovskite (Fig. \\ref{fig7}a). A\njump to a different cube face will cause a change of the dielectric dipole\nby $\\Delta p=e\\sqrt{2}a$; setting\\cite{Kni00} $a=4.39$~\\AA , Eq. (\\ref%\n{De_diel}) gives a dielectric relaxation strength $\\Delta \\varepsilon ^{%\n\\mathrm{Y}}=c_{\\mathrm{Y-H}}\\times 1060$ at 300~K; therefore the intensity $%\n\\Delta \\varepsilon \\simeq 30$ of P$_{\\mathrm{Y}}$ at $x=0.15$ (Fig. \\ref%\n{fig2}) is obtained setting $c_{\\mathrm{Y-H}}\\simeq 0.03$, namely that at\nthis temperature $\\sim 20\\%\\ $of H is trapped by Y. It is likely, however,\nthat $c_{\\mathrm{Y-H}}$ is closer to $c_{\\mathrm{H}}$ but the effective\ndipole $p^{\\mathrm{Y}}$ is reduced by the surrounding polarization and\ndistortion. The dipole involved in P$_{\\mathrm{t}}$ instead, is associated\nwith the OH ion in its four equilibrium positions (Fig. \\ref{fig7}b):\n\\begin{eqnarray}\np_{a}^{\\mathrm{OH}} &=&-p_{c}^{\\mathrm{OH}}=qd\\mathbf{\\hat{x},}  \\label{pOH}\n\\\\\np_{b}^{\\mathrm{OH}} &=&-p_{d}^{\\mathrm{OH}}=-qd\\mathbf{\\hat{y}~,}  \\notag\n\\end{eqnarray}%\nwhere $d\\sim 1$~\\AA\\ is the O-H separation. The actual charge on O and H\\ is\nlikely reduced by polarization effects, and it has been calculated by\natomistic simulations\\cite{GIN99} as $-1.426e$ and $0.426e$, so giving $%\nq=0.426e$. A\\ reorientation of the OH ion by 90$^{\\mathrm{o}}$ causes $%\n\\Delta p^{\\mathrm{OH}}=q\\sqrt{2}d$ and, setting $E_{a}=64$~K from the fit,\nthe resulting relaxation strength of peak P$_{\\mathrm{t}}$ at $T\\simeq 35$~K\nshould be $\\Delta \\varepsilon ^{\\mathrm{Y}}=c_{\\mathrm{Y-H}}\\times 41$. It\nresults that P$_{\\mathrm{t}}$ at 35~K should be $\\sim 26$ times smaller than\nP$_{\\mathrm{Y}}$ at 300~K, but instead it is nearly 1000 times smaller,\ntherefore nearly $40$ times smaller than expected.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=8.5 cm]{fig7.EPS}\n\\caption{(a) Network of the H sites in the cubic structure of BCY; Ba =\nwhite, O = gray, H sites = black, Y\/Ce at the centers of the octahedra. (b)\nElectric dipoles $\\mathbf{p}_{i}^{\\mathrm{OH}}$ and elastic quadrupoles $%\n\\protect\\lambda _{i}^{\\mathrm{OH}}$ associated with the four orientations of\nthe OH$^{-}$ ion . (c) Distorted environment of a set of four sites around a\nO2 atom in the orthorhombic structure (atomic positions except H from Ref.\n\\protect\\cite{Kni00} in Pnma setting).}\n\\label{fig7}\n\\end{figure}\n\n\\subsubsection{The elastic quadrupole of the OH group}\n\nThe anelastic case is different, since the elastic quadrupole\n$\\lambda $ corresponds to the long range component of the strain\nassociated with the OH ion and cannot be estimated in an obvious\nmanner as the electric dipole. It\ncan only be said that the OH ion in the cubic configuration of Fig. \\ref%\n{fig7} has orthorhombic symmetry and therefore in a $xy$ cube face it is\n\\begin{equation}\n\\lambda _{a}^{\\mathrm{OH}}=\\lambda _{c}^{\\mathrm{OH}}=\\left(\n\\begin{array}{ccc}\n\\lambda _{1}^{\\mathrm{OH}} & 0 & 0 \\\\\n0 & \\lambda _{2}^{\\mathrm{OH}} & 0 \\\\\n0 & 0 & \\lambda _{3}^{\\mathrm{OH}}%\n\\end{array}%\n\\right)  \\label{lOH}\n\\end{equation}%\nand $\\lambda _{b,d}^{\\text{OH}}$ have the $x$ and $y$ components exchanged.\nThe reorientation of OH by 90$^{\\mathrm{o}}$ causes a change $\\Delta \\lambda\n^{\\mathrm{OH}}=\\left\\Vert \\lambda _{1}^{\\mathrm{OH}}-\\lambda _{2}^{\\mathrm{OH%\n}}\\right\\Vert $. On the longer time scale of relaxation P$_{\\mathrm{Y}}$,\nthe four positions of the face $\\perp z$ produce a tetragonal quadrupole\n\\begin{equation}\n\\lambda ^{z}=\\left(\n\\begin{array}{ccc}\n\\frac{1}{2}\\left( \\lambda _{1}^{\\mathrm{OH}}+\\lambda _{2}^{\\mathrm{OH}%\n}\\right) & 0 & 0 \\\\\n0 & \\frac{1}{2}\\left( \\lambda _{1}^{\\mathrm{OH}}+\\lambda _{2}^{\\mathrm{OH}%\n}\\right) & 0 \\\\\n0 & 0 & \\lambda _{3}^{\\mathrm{OH}}%\n\\end{array}%\n\\right) ~,\n\\end{equation}%\nplus, in case of Y-OH complex, a possible additional distortion\n\\begin{equation}\n\\lambda ^{\\mathrm{Y}z}=\\left(\n\\begin{array}{ccc}\n\\lambda _{2}^{\\mathrm{Y}} & 0 & 0 \\\\\n0 & \\lambda _{2}^{\\mathrm{Y}} & 0 \\\\\n0 & 0 & \\lambda _{1}^{\\mathrm{Y}}%\n\\end{array}%\n\\right) .\n\\end{equation}%\nThen, the anelastic peak P$_{\\mathrm{Y}}$ associated with the reorientation\nof the tetragonal defect complex Y-OH, has $\\Delta \\lambda ^{\\mathrm{Y}%\n}=\\left( \\lambda _{1}^{\\mathrm{Y}}-\\lambda _{2}^{\\mathrm{Y}}\\right) +\\lambda\n_{3}^{\\mathrm{OH}}-\\frac{1}{2}\\left( \\lambda _{1}^{\\mathrm{OH}}+\\lambda\n_{2}^{\\mathrm{OH}}\\right) $. Lacking any method for estimating these\ncomponents of the elastic quadrupoles associated with OH and Y-OH complexes,\nwe cannot exclude that $\\Delta \\lambda ^{\\mathrm{OH}}$ is $\\sim 10$ times\nsmaller than $\\Delta \\lambda ^{\\mathrm{Y}}$, so determining a reduction of\nthe intensity of P$_{\\mathrm{t}}$ by two orders of magnitude with respect to\nP$_{\\mathrm{Y}}$. Then, the small intensity of the anelastic P$_{\\mathrm{t}}$\ndoes not necessarily constitute a problem as the dielectric intensity does.\n\n\\subsubsection{Anelastic and dielectric relaxation strengths of tunneling states in a cubic environment}\n\nThe explanation why the intensity of the dielectric P$_{\\mathrm{t}}$ is so\nmuch smaller than expected from a simple estimate of the magnitude of dipole\nchange would be easy if BCY was cubic or very close to cubic. In fact, in\ncase of coherent tunneling among four nearly equivalent positions, the\neffective dipole strength for transitions between H eigenstates would be\nmuch smaller than for hopping among the same positions, at variance with the\nanelastic case. The Hamiltonian of the symmetric four-level tunnel system\n(FLS) of H may be written in the basis $\\left\\vert i\\right\\rangle $ where H\nis localized in each site $i=a-d$ as:\n\\begin{equation}\nH_{\\mathrm{cub}}=\\frac{1}{2}\\left[\n\\begin{array}{cccc}\n0 & t & 0 & t \\\\\nt & 0 & t & 0 \\\\\n0 & t & 0 & t \\\\\nt & 0 & t & 0%\n\\end{array}%\n\\right] ,  \\label{Hcub}\n\\end{equation}%\nwhere $t$ is the effective tunneling matrix element between adjacent sites\ndressed with the interaction with phonons and the site energies are all set\nto 0. The eigenstates of $H_{\\mathrm{cub}}$ are%\n\\begin{equation}\n\\left\\vert 1,4\\right\\rangle \\propto \\left[\n\\begin{array}{c}\n\\pm 1 \\\\\n1 \\\\\n\\pm 1 \\\\\n1%\n\\end{array}%\n\\right] ,\\;\\left\\vert 2\\right\\rangle \\propto \\left[\n\\begin{array}{c}\n0 \\\\\n-1 \\\\\n0 \\\\\n1%\n\\end{array}%\n\\right] ,\\;\\left\\vert 3\\right\\rangle \\propto \\left[\n\\begin{array}{c}\n-1 \\\\\n0 \\\\\n1 \\\\\n0%\n\\end{array}%\n\\right] ~;  \\label{Psicub}\n\\end{equation}%\ntwo states are delocalized over all sites, with energies $E_{1,4}=\\mp t\/2$,\nand two are delocalized over either pair of opposite sites, with energies $%\nE_{2,3}=0$. The associated electric dipoles $\\mathbf{p}^{\\mathrm{FLS}}$ and\nelastic quadrupoles $\\lambda ^{\\mathrm{FLS}}$ may be obtained taking the\nmatrix elements $\\lambda _{i}^{\\mathrm{FLS}}=\\left\\langle i\\right\\vert\n\\lambda ^{\\mathrm{OH}}\\left\\vert i\\right\\rangle $ and $\\mathbf{p}_{i}^{%\n\\mathrm{FLS}}=\\left\\langle i\\right\\vert \\mathbf{p}^{\\mathrm{OH}}\\left\\vert\ni\\right\\rangle $, with $\\lambda ^{\\mathrm{OH}}$ and $\\mathbf{p}^{\\mathrm{OH}%\n} $ given above. It is evident that $\\mathbf{p}^{\\mathrm{OH}}$ averaged over\nany of the above states is null, since opposite sites are equally occupied\nand cancel out the respective dipoles. Then, no dielectric relaxation is\nexpected from H tunneling in a cubic environment, or also next to a Y\ndopant, which leaves the fourfold symmetry of the FLS. The elastic\nquadrupoles, instead, being centrosymmetric as in Eq. (\\ref{lOH}), are equal\nwithin pairs of opposite sites, and transitions between the two intermediate\neigenstates cause a change of elastic quadrupole by $\\Delta \\lambda ^{%\n\\mathrm{OH}}=\\left\\Vert \\lambda _{1}^{\\mathrm{OH}}-\\lambda _{2}^{\\mathrm{OH}%\n}\\right\\Vert $; transitions between states 2,3 and 1,4 cause a change by $%\n\\Delta \\lambda ^{\\mathrm{OH}}\/2$ and those between 1 and 4 no anelastic\nrelaxation. Therefore, the formation of FLS with weak deviations from\nfourfold symmetry would explain why the dielectric relaxation from H\ntunneling is much more effectively suppressed than the anelastic one, with\nrespect to classical reorientation.\n\nA picture like this, with H and D performing coherent tunneling within\nnearly symmetric FLS,\\ has been thoroughly studied by anelastic relaxation\nin Nb with substitutional traps.\\cite{36,45,85,CCC98}\\emph{\\ }Also in that\ncase H tunnels within rings of four equivalent tetrahedral sites on the\nfaces of the \\textit{bcc} cells with a dopant in the center, and performs\noverbarrier jumps to the neighboring rings, which form a network exactly as\nin the cubic perovskite (see Fig. \\ref{fig7}a). In Nb$_{1-x}$Zr$_{x}$ single\ncrystals with $x=0.0013$ it was also possible to distinguish well separated\nrelaxation processes arising from transitions involving eigenstates with\ndifferent symmetries, only moderately perturbed by interactions among\ndopants;\\cite{85} in addition, the dependence of the anelastic relaxation\ndue to the slower reorientation among different cube faces on the symmetry\nof the excitation stress\\cite{45} provides evidence that the symmetry of the\nFLS persists at least up to 150~K, implying that the H eigenstates maintain\ncoherence up to that temperature. It should be noted that the measured\neffective tunneling matrix element between tetrahedral sites in Nb is $t\\sim\n0.2$~meV for H and 0.02~meV for D, corresponding to $t\/k_{\\text{B}}=2$ and\n0.2~K or $t\/h=4\\times 10^{10}$ and $4\\times 10^{9}$~s$^{-1}$ respectively;\nit appears therefore that coherence is maintained both at temperatures\norders of magnitude larger than $t\/k_{\\text{B}}$ and with tunneling\nfrequencies of the order of phonon frequencies.\n\nA theoretical basis for the persistence of quantum coherence at such high\ntemperatures comes from analytical and numerical analysis of the\ncentrosymmetric FLS, where coherent oscillations of the H\\ populations are\nfound even for strong interaction with the thermal bath,\\cite{WG00} and from\nthe analysis of the dynamics of polarons using the dynamical mean-field\napproximation,\\cite{FC03} which is a non-perturbative approach. It is shown\nthat the coherence of the state including tunneling particle and surrounding\npolaronic distortion is maintained to temperatures up to a substantial\nfraction of the energy of the phonon coupled with the particle, $k_{\\text{B}%\n}T\\sim 0.2\\hbar \\omega _{0}$, also in the case of strong coupling, where the\nenergy for the polaron formation is $E_{p}>\\hbar \\omega _{0}$. We assume $%\nE_{p}\\simeq 1$~eV from the theoretical estimate\\cite{SBW07} of the\nself-trapping energy of H in BaZrO$_{3}$, and that the strongest coupling is\nwith the O-Ce-O bending mode (in BaCeO$_{3}$ $\\hbar \\omega _{0}=$ 41~meV\\cite%\n{CPL99}) modulating the distance with the neighboring O atoms and therefore\nthe hydrogen bonds with them. Then we are in the limit of strong coupling,\nand the dynamical mean-field analysis\\cite{FC03} ensures us that coherent\nstates may be maintained up to $0.2\\hbar \\omega _{0}\/k_{\\text{B}}\\simeq $\n100~K; this holds for both H\\ and D, since\\ the smallness of the dressed\ntunneling matrix element should not be a problem, until the tunneling\nfrequency is larger than the measuring frequency.\n\n\\subsubsection{Tunneling states in the orthorhombic lattice}\n\nThe main problem with the above explanation of the smallness of the\ndielectric relaxation strength of P$_{\\mathrm{t}}$ is that BCY at low\ntemperature is not cubic but orthorhombic, and the FLS should be far from\nsymmetric. Figure \\ref{fig7}c shows an undistorted ring, as in the cubic\ncase, between two octahedra tilted as in the orthorhombic $Pnma$ low\ntemperature structure of BaCe$_{0.9}$Y$_{0.1}$O$_{3-\\delta }$.\\cite{Kni00}\nThe O\\ atoms split in two types: O1 at the vertices of the octahedra along\nthe $b$ axis and O2 near the $ac$ plane; neutron diffraction indicates that,\nat 4.2~K, H in BCY occupies a site near the one labeled as $d$ in Fig. \\ref%\n{fig7}c, which is also the one with the largest distances from the next\nnearest neighbor O\\ atoms. It is therefore reasonable to assume that in the\ndistorted orthorhombic structure H occupies sites slightly displaced from\nthose in the cubic cell, and that the site energy is mainly determined by\nthe mean distance $\\overline{l}$ from the two next nearest neighbor O atoms,\nwith which some hydrogen bonding can take place.\\cite{BSW07} For the sites\nlabeled $a-d$ in Fig. \\ref{fig7}b it is $\\overline{l}=$ 2.66, 2.71, 2.53,\n2.50 respectively, so that the site energies may be written as $E_{a}\\simeq\nE_{b}=a\/2$, $E_{c}\\simeq E_{d}=-a\/2$, and the Hamiltonian of the tunnel\nsystem becomes:%\n\\begin{equation}\nH_{\\mathrm{ortho}}=\\frac{1}{2}\\left[\n\\begin{array}{cccc}\na & t & 0 & t \\\\\nt & a & t & 0 \\\\\n0 & t & -a & t \\\\\nt & 0 & t & -a%\n\\end{array}%\n\\right] ,\n\\end{equation}%\nwith eigenstates%\n\\begin{equation}\n\\left\\vert 1,2\\right\\rangle \\propto \\left[\n\\begin{array}{c}\na-\\delta \\\\\n\\mp \\left( a-\\delta \\right) \\\\\n\\mp t \\\\\nt%\n\\end{array}%\n\\right] ,\\;\\left\\vert 3,4\\right\\rangle \\propto \\left[\n\\begin{array}{c}\na+\\delta \\\\\n\\mp \\left( a+\\delta \\right) \\\\\n\\mp t \\\\\nt%\n\\end{array}%\n\\right] ,\n\\end{equation}%\nand energies $E_{1,2}=\\left( -\\delta \\mp t\\right) \/2,$ $E_{3,4}=\\left(\n\\delta \\mp t\\right) \/2,$ where $\\delta =\\sqrt{a^{2}+t^{2}}$. This means\nthat, when $a\\gg t$, there are two low energy eigenstates with H\\ mainly\ndelocalized over sites $c$ and $d$ and two higher energy eigenstates\ndelocalized over sites $a$ and $b$; these states have an electric dipole $%\np\\simeq ed\/\\sqrt{2}$ oriented roughly midway between the two occupied sites.\nTherefore, while in the limit $a\\ll t$ , valid for the cubic ideal case, the\nH atom is delocalized over all sites and the electric dipole is averaged out\nto almost zero, H in BCY at low temperature is expected to be in the\nopposite limit $a\\gg t$ where the averaging effect of the electric dipole\noccurs only within the pairs $ab$ and $cd$ of low- and high-energy states,\nwithout a suppression of the dipole magnitude.\n\nThe present data do not allow an estimate of $t$, neither do we know of\nestimates of the energy asymmetry $a$ due to the orthorhombic distortion,\nbut based on the comparison with the better known case of Zr-H complexes in\nNb we should be in the limit $a\\gg t$. In fact, $t$ should be smaller than\nin Nb, since, assuming an O-H bond $\\simeq 0.93$~\\AA\\ long, the distance\nbetween neighboring sites is $\\sim 1.3$~\\AA\\ in BCY while it is only $1$~%\n\\AA\\ in Nb; in addition, the maximum of the anelastic relaxation is shifted\nto higher temperature with respect to Nb$_{1-x}$Zr$_{x}$H$_{y}$, indicating\nslower transition rates; therefore it should be $t\/k_{\\text{B}}\\ll 1$~K, the\nvalue found in Nb. On the other hand, we expect $a\/k_{\\text{B}}\\gtrsim\n10^{2} $~K, considering that the random strains due to $<1$ at\\% of\nimpurities in Nb cause $a\/k_{\\text{B}}$ of tens of kelvin,\\cite{CCC98} and\nthe strain associated with the octahedral tilts in orthorhombic BCY is\ncertainly larger. Another indication that $a$ due to the orthorhombic\ndistortion is large comes from the theoretical estimate\\cite{SBW07} $%\nE_{p}\\simeq 1$~eV of the self-trapping energy of H in BaZrO$_{3}$.\n\n\\subsubsection{Symmetric tunneling states within twin walls}\n\nAfter these arguments, it is puzzling that the intensity of the dielectric P$%\n_{\\mathrm{d}}$ in orthorhombic BCY is so small. The situation is similar to\nthat found in hydrated BaCe$_{1-x}$Nd$_{x}$O$_{3-\\delta }$, where a low\ntemperature dielectric relaxation exists whose rate has a temperature\ndependence indicating tunneling; however, it was concluded that if all the\nprotons were responsible for such a relaxation, the intensity should be 50\ntimes larger, and therefore only special defect configurations contributed\nto that relaxation.\\cite{KLN99} In the present case, unless the OH dipole is\nsmaller than estimated,\\cite{GIN99} less than 3\\%\\ of the H atoms should\ncontribute to P$_{\\mathrm{t}}$ and it should be explained what kind of\nparticular configuration exhibits tunneling and what is the dynamics of the\nmajority H atoms. In fact, both neutron spectroscopy at high temperature\\cite%\n{MSK96b,PMS97} and simulations\\cite{BSW07} indicate that the reorientation\nof the OH ion is much faster than the hopping between different O atoms in\nperovskite cerates. A possible scenario is that the fast reorientation\noccurs only in high temperature phases that are cubic or less distorted than\nthe orthorhombic phase, whereas in the latter H is nearly localized at\nlowest energy site, close to site $d$ in Fig. \\ref{fig7}b. Then the\nreorientation of the OH ion would be slower than that producing P$_{\\mathrm{t%\n}}$ and might be identified with P$_{\\mathrm{h}}$ or some broad peak masked\nby the tail of P$_{\\mathrm{Y}}$ and by P$_{\\mathrm{h}}$, perhaps the one\naround 100~K in Fig. \\ref{fig1}. In this scenario the tunneling motion would\nappear only in particularly symmetric environments, \\textit{e.g.} at the\nboundaries between different structural domains. We are not aware of any\nstudy of the density, width and morphology of the domain boundaries in BCY,\nbut indirect support to this mechanism comes from a simulation on\northorhombic CaTiO$_{3}$, where the twin walls are found to be about 6\npseudocubic cells wide and to trap the O vacancies;\\cite{CDS03} it is also\nproposed that the diffusion of O vacancies should be faster within the twin\nplanes, which are more symmetric than the orthorhombic bulk. Also in the\ncase of H in BCY, the reorientation rate of H might be faster at the twin\nwalls, but the relevance of this effect to the long range mobility would be\nlimited, since the rate limiting step is not the OH reorientation but the\nhopping to a different O\\ atom. We think however that the effect of the\ngreater symmetry at the twin walls should be studied also in relation with\nthe hopping mechanism and in the high temperature phases of BCY. Also\nsimulations on BaZrO$_{3}$\\ and CaTiO$_{3}$\\ suggest that the octahedral\ntilting is essential in determining H site energies and diffusion paths.\\cite%\n{GGJ05}\n\nIf H tunneling in the orthorhombic phase indeed occurs only in the twin\nboundaries, than the intensity of peak P$_{\\mathrm{t}}$ would depend on\ntheir density, which in turn may depend on microstructure and thermal\nhistory. This might be the reason why the dielectric P$_{\\mathrm{t}}$ in\nBCY15 is nearly three times more intense than in BCY10, a feature otherwise\ndifficult to explain; additional measurements are necessary to clarify this\nissue.\n\nAnother factor that may contribute to lower the dielectric relaxation\nstrength of P$_{\\mathrm{t}}$ is the lattice polarization around the OH$^{-}$\nion; in fact, the actual dipole is not the bare OH$^{-}$ dipole with nominal\ncharges $\\pm 1$, but it is due to the OH$^{-}$ complex plus the shifted\nsurrounding atoms and with charge transfers with respect to the purely ionic\ncase. We estimated the intensity of dielectric P$_{\\mathrm{t}}$ assuming the\nOH dipole calculated in Ref. \\cite{GIN99}, but the actual dipole may be even\nsmaller.\n\n\\section{ Conclusions}\n\nThree main relaxation processes have been identified both in the anelastic\nand dielectric spectra as due to hopping of H around an Y dopant (peak P$_{%\n\\mathrm{Y}}$), hopping far from dopants (peak P$_{\\mathrm{h}}$) and\ntunneling within the four sites around a same O atom (peak P$_{\\mathrm{t}}$%\n). An additional dielectric relaxation maximum around 60~K (P$_{\\mathrm{d}}$%\n) does not involve H motion, but rather appears as a relaxation of\nelectronic origin like small polaron hopping. Peak P$_{\\mathrm{t}}$ can be\nfitted assuming a relaxation rate that is Arrhenius-like above $\\sim 50$~K,\npossibly due to two-phonon transitions with optical phonons rather than to\noverbarrier hopping, below 50~K exhibits a $T^{7}$ dependence typical of\ntwo-phonon transitions with acoustic phonons, and finally becomes nearly\nconstant below $\\sim 20$~K, as expected from one-phonon transitions. The\nisotopic substitution with D decreases the rate by a factor of $8$. The\ndielectric spectra are more difficult to analyze, due to the presence of\npeak P$_{\\mathrm{d}}$ and possibly other peaks at lower temperature, but\nreasonable fits are obtained using the same anelastic expression of P$_{%\n\\mathrm{t}}$ with a rate increased by $\\sim 10$ times; such a difference in\nthe rate may be due to the fact that, although not clearly distinguishable,\nthere are at least three anelastic and dielectric relaxation modes\ncontributing to P$_{\\mathrm{t}}$, having different strengths and rates.\n\nThe intensity of the dielectric P$_{\\mathrm{t}}$ is nearly 40 times smaller\nthan expected from a simple estimate of the reorientation of the electric\ndipole associated with the OH$^{-}$ ion. It is shown that, while this\nsuppression of the dielectric relaxation would be easily explained in terms\nof coherent tunneling of H around O in a cubic environment, the same seems\nnot to be true in the presence of the low-temperature orthorhombic\ndistortion. As alternative or concomitant explanations, it is proposed that\nH\\ tunneling may occur only in the more symmetric cells within twin walls,\nwhile slower semiclassical reorientation of OH$^{-}$ would occur within the\northorhombic domains; in addition, the total dipole of OH$^{-}$ ion and\nsurrounding lattice polarization may be smaller than expected.\n\n\\subsection*{Acknowledgments}\n\nWe wish to thank S. Ciuchi for useful discussions on the coherence of the\npolaron states, O. Frasciello for valuable suggestions on the dielectric\nmeasurements, and F. Corvasce, M. Latino, A. Morbidini for their technical\nassistance. This research is supported by the FISR Project of Italian MIUR:\n\"Celle a combustibile ad elettroliti polimerici e ceramici: dimostrazione di\nsistemi e sviluppo di nuovi materiali\".\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Concluding Remarks}\\label{sec:conclude\n \nIn this paper, we designed a procedure for generating chemical graphs\nas a new building block \nin Stage~4 of the framework for inferring a desired chemical graph.\nThe main task of Stage~4 is to find a feasible solution $\\C^\\dagger$ of an MILP that\nrepresents a feature function and a topological specification.\nIn the framework, isomers $\\C^*$ of $\\C^\\dagger$ are generated \nin Stage~5 by a dynamic programming algorithm.\nHowever, the number of isomers of $\\C^\\dagger$ is sometimes small.\nOur new procedure searches the neighbor of $\\C^\\dagger$ in a search space\n defined with a set of linear functions.\nWe divide the neighbor of $\\C^\\dagger$ into subspaces and\nsolve the MILP in Stage~4 for each subspace \nimposing a set of linear constraints that represents the subspace.\nFrom the results of our computational experiments, we observe that\nan additional number of  solutions $\\C^*$\n can be found in the neighbor of $\\C^\\dagger$ \nby our new procedure. \n\n\n\\subsection{Descriptor for the Number of  Adjacency-configurations}  \n\\label{sec:AC\n\nWe call a tuple $(\\ta,\\tb,m)\n\\in (\\Lambda\\setminus\\{{\\tt H}\\})\\times (\\Lambda\\setminus\\{{\\tt H}\\}) \\times[1,3]$\nan {\\em adjacency-configuration}.\nThe adjacency-configuration of an edge-configuration\n$(\\mu=\\ta d, \\mu'=\\tb d', m)$ is defined to be\n $(\\ta,\\tb,m)$.\nWe include constraints to compute the frequency of each adjacency-configuration\nin an inferred chemical graph $\\C$. \n \n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate}[leftmargin=*\n\\item[-] A set  $\\Gamma^\\inte$ of edge-configurations $\\gamma=(\\mu,\\mu',m)$ \nwith  $\\mu\\leq \\mu'$;\n\n\\item[-] \nLet  $\\overline{\\gamma}$ of an edge-configuration $\\gamma=(\\mu,\\mu',m)$\ndenote the  edge-configuration $(\\mu',\\mu,m)$; \n\n\\item[-] Let $\\Gamma_{<}^\\inte=\\{(\\mu,\\mu',m)\\in  \\Gamma^\\inte\\mid \\mu < \\mu' \\}$, \n$\\Gamma_{=}^\\inte=\\{(\\mu,\\mu',m)\\in  \\Gamma^\\inte\\mid \\mu= \\mu' \\}$\nand    $\\Gamma_{>}^\\inte=\\{\\overline{\\gamma}\\mid \n    \\gamma\\in  \\Gamma_{<}^\\inte  \\}$;\n    \n\\item[-] \nLet  $\\Gacs^\\inte$, $\\Gace^\\inte$ and $\\Gacl^\\inte$ \n denote  the sets of the adjacency-configurations of\nedge-configurations in the sets \n$\\Gamma_{<}^\\inte$, $\\Gamma_{=}^\\inte$ and $\\Gamma_{>}^\\inte$, \n respectively;\n \n\\item[-] \nLet  $\\overline{\\nu}$ of an adjacency-configuration $\\nu=(\\ta, \\tb,m)$\ndenote the  adjacency-configuration $(\\tb,\\ta,m)$; \n\n\\item[-] \n Prepare a coding of   the  set \n$\\Gac^\\inte \\cup \\Gacl^\\inte$  and let \n$[\\nu]^\\inte$   denote  \nthe coded integer of  an element $\\nu$ in $\\Gac^\\inte \\cup \\Gacl^\\inte$; \n\n\\item[-] \nChoose subsets   $\\tGacC,\\tGacT,\\tGacCT,\\tGacTC, \\tGacF, \\tGacCF , \\tGacTF\n   \\subseteq \\Gac^\\inte\\cup\\Gacl^\\inte$;  \n To compute the frequency     of adjacency-configurations exactly,  set \n  $\\tGacC:= \\tGacT :=\\tGacCT:=  \\tGacTC :=\\tGacF:=  \\tGacCF := \\tGacTF :=\n  \\Gac^\\inte\\cup\\Gacl^\\inte$; \n   \n\\item[-]  $\\ac_\\LB^\\inte(\\nu),  \\ac_\\UB^\\inte(\\nu) \\in [0,2\\nint_\\UB ], \n\\nu=(\\ta,\\tb,m)\\in \\Gac^\\inte$: \nlower and upper bounds on the number \n  of interior-edges  $e=uv$  with $\\alpha(u)=\\ta$, \n $\\alpha(v)=\\tb$ and $\\beta(e)=m$; \n\\end{enumerate}\n\n \n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n\\item[-]\n$\\ac^\\inte([\\nu]^\\inte) \\in [\\ac_\\LB^\\inte(\\nu), \\ac_\\UB^\\inte(\\nu)], \n\\nu\\in \\Gac^\\inte$: \nthe number of interior-edges  with  adjacency-configuration $\\nu$; \n\\item[-]\n$\\acC([\\nu]^\\inte)\\in [0,\\mC],  \\nu\\in \\tGacC$, \n$\\acT([\\nu]^\\inte)\\in [0,\\tT],   \\nu\\in \\tGacT$, \n$\\acF([\\nu]^\\inte)\\in [0,\\tF], \\nu\\in \\tGacF$:  \nthe number of  edges $\\eC\\in \\EC$ (resp.,  edges $\\eT\\in \\ET$\nand   edges $\\eF\\in \\EF$)  with  adjacency-configuration $\\nu$; \n\n\\item[-]\n$\\acCT([\\nu]^\\inte)\\in [0, \\min\\{\\kC,\\tT\\} ], \\nu\\in \\tGacCT$,\n$\\acTC([\\nu]^\\inte)\\in [0,\\min\\{\\kC,\\tT\\} ], \\nu\\in \\tGacCT$, \n$\\acCF([\\nu]^\\inte)\\in [0,\\widetilde{\\tC}],  \\nu\\in \\tGacCF$,\n$\\acTF([\\nu]^\\inte)\\in [0,\\tT], \\nu\\in \\tGacTF$: \nthe number of  edges   $\\eCT\\in \\ECT$  \n(resp.,   edges $\\eTC\\in \\ETC$\nand  edges $\\eCF\\in \\ECF$ and $\\eTF\\in \\ETF$)  with  adjacency-configuration $\\nu$; \n\\item[-]\n$\\dlacC(i,[\\nu]^\\inte)\\in [0,1], \n  i\\in [\\widetilde{\\kC}+1,\\mC]=\\Iw\\cup \\Iz\\cup\\Iew, \\nu\\in \\tGacC$, \n$\\dlacT(i,[\\nu]^\\inte)\\in [0,1],  i\\in [2,\\tT],   \\nu\\in \\tGacT$, \n$\\dlacF(i,[\\nu]^\\inte)\\in [0,1] , i\\in [2,\\tF],\\nu\\in \\tGacF$:\n$\\dlacX(i,[\\nu]^\\inte)=1$  $\\Leftrightarrow$\nedge  $\\eX_i$ has  adjacency-configuration $\\nu$; \n\\item[-]\n$\\dlacCT(k,[\\nu]^\\inte),\\dlacTC(k,[\\nu]^\\inte)\\in [0,1],\nk\\in [1, \\kC]=\\It\\cup \\Iw,  \\nu\\in \\tGacCT$: \n$\\dlacCT(k,[\\nu]^\\inte)=1$   (resp., $\\dlacTC(k,[\\nu]^\\inte)=1$)  $\\Leftrightarrow$\nedge  $\\eCT_{\\tail(k),j}$ (resp.,  $\\eTC_{\\hd(k),j}$) \nfor some $j\\in [1,\\tT]$ has  adjacency-configuration $\\nu$;\n\n\\item[-]\n$\\dlacCF(c,[\\nu]^\\inte)\\in [0,1],  c\\in [1,\\widetilde{\\tC}],\\nu\\in \\tGacCF$:\n$\\dlacCF(c,[\\nu]^\\inte)=1$    $\\Leftrightarrow$\nedge   $\\eCF_{c,i}$  for some $i\\in [1,\\tF]$ has  adjacency-configuration $\\nu$;\n\n\\item[-]\n  $\\dlacTF(i,[\\nu]^\\inte)\\in [0,1],  i\\in [1,\\tT],  \\nu\\in \\tGacTF$: \n   $\\dlacTF(i,[\\nu]^\\inte)=1$  $\\Leftrightarrow$\nedge   $\\eTF_{i,j}$\n for some $j\\in [1,\\tF]$ has  adjacency-configuration $\\nu$;   \n\\item[-]\n$\\aCT(k),\\aTC(k)\\in [0, |\\Lambda^\\inte|],   k\\in [1, \\kC]$: \n$\\alpha(v)$  of the edge $(\\vC_{\\mathrm{tail}(k)},v)\\in \\ECT$   \n (resp., $(v,\\vC_{\\mathrm{head}(k)})\\in \\ETC$) if any; \n \n\\item[-]\n$\\aCF(c)\\in [0, |\\Lambda^\\inte|], c\\in [1,\\widetilde{\\tC}]$: \n $\\alpha(v)$  of the edge $(\\vC_{c},v)\\in \\ECF$   if any; \n \n\\item[-]\n$\\aTF(i)\\in [0, |\\Lambda^\\inte|], i\\in [1,\\tT]$: \n $\\alpha(v)$  of the edge $(\\vT_{i},v)\\in \\ETF$   if any; \n\\item[-]\n$\\DlacCp(i),  \\DlacCm(i), \\in [0,|\\Lambda^\\inte|], \n  i\\in [\\widetilde{\\kC}+1,\\mC]$, \n$\\DlacTp(i),\\DlacTm(i)\\in [0,|\\Lambda^\\inte|],  i\\in [2,\\tT]$,\n$\\DlacFp(i),\\DlacFm(i)\\in [0,|\\Lambda^\\inte|] , i\\in [2,\\tF]$: \n$\\DlacXp(i)=\\DlacXm(i)=0$ (resp., \n $\\DlacXp(i)=\\alpha(u)$ and $\\DlacXm(i)=\\alpha(v)$) $\\Leftrightarrow$\nedge  $\\eX_i=(u,v)\\in \\EX$   is used in $\\C$ (resp., $\\eX_i\\not\\in E(G)$); \n\n\\item[-]\n$\\DlacCTp(k),\\DlacCTm(k)\\in [0,|\\Lambda^\\inte|],\nk\\in [1, \\kC]=\\It\\cup \\Iw$: \n$\\DlacCTp(k)=\\DlacCTm(k) =0$ \n(resp.,  $\\DlacCTp(k)=\\alpha(u)$ and $\\DlacCTm(k)=\\alpha(v)$) \n $\\Leftrightarrow$  \nedge  $\\eCT_{\\tail(k),j}=(u,v)\\in \\ECT$   \n  for some $j\\in [1,\\tT]$ is used in $\\C$ (resp., otherwise);  \n  \n\\item[-]\n$\\DlacTCp(k),\\DlacTCm(k)\\in [0,|\\Lambda^\\inte|],\nk\\in [1, \\kC]=\\It\\cup \\Iw$: \nAnalogous with $\\DlacCTp(k)$ and $\\DlacCTm(k)$;\n\n\\item[-]\n$\\DlacCFp(c)\\in [0,|\\Lambda^\\inte|],\n\\DlacCFm(c) \\in [0,|\\Lambda^\\inte|],  c\\in [1,\\widetilde{\\tC}]$: \n$\\DlacCFp(c)=\\DlacCFm(c) =0$ (resp., \n $\\DlacCFp(c)=\\alpha(u)$ and $\\DlacCFm(c)=\\alpha(v)$) \n $\\Leftrightarrow$ \nedge  $\\eCF_{c,i}=(u,v)\\in \\ECF$  \n   for some $i\\in [1,\\tF]$ is used in $\\C$ (resp., otherwise);  \n\\item[-]\n  $\\DlacTFp(i)\\in [0,|\\Lambda^\\inte|],\n  \\DlacTFm(i)\\in [0,|\\Lambda^\\inte|],  i\\in [1,\\tT]$:  \n  Analogous with $\\DlacCFp(c)$ and $\\DlacCFm(c)$;\n\\end{enumerate}\n \n  \n\\smallskip\\noindent\n{\\bf constraints: }\n\n\\begin{align} \n \\acC([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacC , \\notag \\\\\n \\acT([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacT , \\notag \\\\\n \\acF([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacF , \\notag \\\\\n\n %\n \\acCT([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacCT , \\notag \\\\\n \\acTC([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacTC , \\notag \\\\\n \\acCF([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacCF , \\notag \\\\\n \\acTF([\\nu]^\\inte) =0,  &&  \\nu \\in \\Gac^\\inte\\setminus \\tGacTF , \\notag \\\\\n \\label{eq:AC_first} \n\\end{align}    \n\n\\begin{align} \n \\sum_{(\\ta, \\tb,m)=\\nu\\in  \\Gac^\\inte}\\acC([\\nu]^\\inte) \n      =\\sum_{i\\in [\\widetilde{\\kC}+1,\\mC]}\\delbC(i,m),  &&   m\\in [1,3]   , \\notag \\\\\n \\sum_{(\\ta, \\tb,m)=\\nu\\in  \\Gac^\\inte}\\acT([\\nu]^\\inte) \n      =\\sum_{i\\in [2,\\tT]}\\delbT(i,m) ,  &&   m\\in [1,3] , \\notag \\\\\n \\sum_{(\\ta, \\tb,m)=\\nu\\in \\Gac^\\inte}\\acF([\\nu]^\\inte)\n      =\\sum_{i\\in [2,\\tF]}\\delbF(i,m) ,  &&   m\\in [1,3]  , \\notag \\\\\n %\n \\sum_{(\\ta, \\tb,m)=\\nu\\in \\Gac^\\inte}\\acCT([\\nu]^\\inte)\n     =\\sum_{k\\in [1, \\kC]} \\delbCT(k,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\sum_{(\\ta, \\tb,m)=\\nu\\in \\Gac^\\inte}\\acTC([\\nu]^\\inte)\n    =\\sum_{k\\in [1, \\kC]} \\delbTC(k,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\sum_{(\\ta, \\tb,m)=\\nu\\in \\Gac^\\inte}\\acCF([\\nu]^\\inte)\n    =\\sum_{c\\in [1,\\widetilde{\\tC}]} \\delbsF(c,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\sum_{(\\ta, \\tb,m)=\\nu\\in \\Gac^\\inte}\\acTF([\\nu]^\\inte) \n    =\\sum_{c\\in [\\widetilde{\\tC}+1, \\cF]} \\delbsF(c,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\label{eq:AC_first2} \n\\end{align}    \n\n\\begin{align} \n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacC }\\!\\!\\!\\! m\\cdot \\dlacC(i, [\\nu]^\\inte) \n=\\bC(i), && \\notag  \\\\  \n\\DlacCp(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacC }\\!\\!\\!\\! [\\ta]^\\inte\\dlacC(i, [\\nu]^\\inte) \n=\\aC(\\tail(i)),  && \\notag  \\\\\n\\DlacCm(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacC }\\!\\!\\!\\! [\\tb]^\\inte\\dlacC(i, [\\nu]^\\inte) \n=\\aC(\\hd(i)),  &&\\notag  \\\\\n\\DlacCp(i)+\\DlacCm(i) \\leq 2|\\Lambda^\\inte|(1 - \\eC(i)),\n&& i\\in [\\widetilde{\\kC}+1,\\mC],  \\notag  \\\\\n\\sum_{i\\in [\\widetilde{\\kC}+1,\\mC]}\\!\\!\\!\\! \\dlacC(i, [\\nu]^\\inte) =\\acC([\\nu]^\\inte),  \n&&  \\nu \\in \\tGacC ,   \\label{eq:AC1}   \n\\end{align}    \n \n \n\n\\begin{align} \n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacT }\\!\\!\\!\\! m\\cdot \\dlacT(i, [\\nu]^\\inte) \n=\\bT(i), && \\notag  \\\\  \n\\DlacTp(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacT }\\!\\!\\!\\! [\\ta]^\\inte\\dlacT(i, [\\nu]^\\inte) \n=\\aT(i-1),  && \\notag  \\\\\n\\DlacTm(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacT }\\!\\!\\!\\! [\\tb]^\\inte\\dlacT(i, [\\nu]^\\inte) \n=\\aT(i),  &&\\notag  \\\\\n\\DlacTp(i)+\\DlacTm(i) \\leq 2|\\Lambda^\\inte|(1 - \\eT(i)),\n&& i\\in [2,\\tT],   \\notag  \\\\\n \\sum_{ i\\in [2,\\tT]} \\!\\! \\dlacT(i, [\\nu]^\\inte) =\\acT([\\nu]^\\inte),    \n&& \\nu \\in \\tGacT ,    \\label{eq:AC2} \n\\end{align}    \n\n\\begin{align} \n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacF }\\!\\!\\!\\! m\\cdot \\dlacF(i, [\\nu]^\\inte) \n=\\bF(i), && \\notag  \\\\  \n\\DlacFp(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacF }\\!\\!\\!\\! [\\ta]^\\inte\\dlacF(i, [\\nu]^\\inte) \n=\\aF(i-1),  && \\notag  \\\\\n\\DlacFm(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacF }\\!\\!\\!\\! [\\tb]^\\inte\\dlacF(i, [\\nu]^\\inte) \n=\\aF(i),  &&\\notag  \\\\\n\\DlacFp(i)+\\DlacFm(i) \\leq 2|\\Lambda^\\ex|(1 - \\eF(i)),\n&& i\\in [2,\\tF],   \\notag  \\\\\n  \\sum_{ i\\in [2,\\tF]} \\!\\! \\dlacF(i, [\\nu]^\\inte) =\\acF([\\nu]^\\inte),  \n &&  \\nu \\in \\tGacF ,    \\label{eq:AC3} \n\\end{align}    \n  \n\n\n\\begin{align} \n \\aT(i)+|\\Lambda^\\inte|(1-\\chiT(i,k)+\\eT(i))\\geq \\aCT(k),  && \\notag  \\\\  \n\\aCT(k)\\geq \\aT(i)- |\\Lambda^\\inte|(1-\\chiT(i,k)+\\eT(i)), \n&&  i\\in [1,\\tT],     \\notag  \\\\  \n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacCT }\\!\\!\\!\\! m\\cdot \\dlacCT(k, [\\nu]^\\inte) \n=\\bCT(k), && \\notag  \\\\  \n\\DlacCTp(k) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacCT }\\!\\!\\!\\! [\\ta]^\\inte\\dlacCT(k, [\\nu]^\\inte) \n=\\aC(\\tail(k)),  && \\notag  \\\\\n\\DlacCTm(k) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacCT }\\!\\!\\!\\! [\\tb]^\\inte\\dlacCT(k, [\\nu]^\\inte) \n=  \\aCT(k),  &&\\notag  \\\\\n\\DlacCTp(k)+\\DlacCTm(k) \\leq 2|\\Lambda^\\inte|(1 - \\dclrT(k)),\n&& k\\in [1, \\kC],  \\notag  \\\\\n\\sum_{k\\in [1, \\kC]}\\!\\! \\dlacCT(k, [\\nu]^\\inte) =\\acCT([\\nu]^\\inte),  \n && \\nu \\in  \\tGacCT ,  \\label{eq:AC5} \n\\end{align}    \n\n \n\\begin{align} \n \\aT(i)+|\\Lambda^\\inte|(1-\\chiT(i,k)+\\eT(i+1))\\geq \\aTC(k),   && \\notag  \\\\  \n\\aTC(k)\\geq \\aT(i)- |\\Lambda^\\inte|(1-\\chiT(i,k)+\\eT(i+1)), \n&&  i\\in [1,\\tT],    \\notag  \\\\  \n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacTC }\\!\\!\\!\\! m\\cdot \\dlacTC(k, [\\nu]^\\inte) \n=\\bTC(k), && \\notag  \\\\  \n\\DlacTCp(k) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacTC }\\!\\!\\!\\! [\\ta]^\\inte\\dlacTC(k, [\\nu]^\\inte) \n=  \\aTC(k),  &&\\notag  \\\\\n\\DlacTCm(k) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacTC }\\!\\!\\!\\! [\\tb]^\\inte\\dlacTC(k, [\\nu]^\\inte) \n=\\aC(\\hd(k)),  &&   \\notag  \\\\\n\\DlacTCp(k)+\\DlacTCm(k) \\leq 2|\\Lambda^\\inte|(1 - \\dclrT(k)), && k\\in [1, \\kC],  \\notag  \\\\  \n\\sum_{k\\in [1, \\kC]}\\!\\! \\dlacTC(k, [\\nu]^\\inte) =\\acTC([\\nu]^\\inte),  \n && \\nu \\in  \\tGacTC ,  \\label{eq:AC5} \n\\end{align}    \n \n\n\\begin{align} \n\\aF(i)+|\\Lambda^\\inte|(1-\\chiF(i,c)+\\eF(i))\\geq \\aCF(c),  && \\notag  \\\\  \n\\aCF(c)\\geq \\aF(i)- |\\Lambda^\\inte|(1-\\chiF(i,c)+\\eF(i)), \n   &&   i\\in [1,\\tF], \\notag  \\\\   \n  %\n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacCF }\\!\\!\\!\\! m\\cdot \\dlacCF(c, [\\nu]^\\inte) \n=\\bsF(c), && \\notag  \\\\  \n\\DlacCFp(c) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacCF }\\!\\!\\!\\! [\\ta]^\\inte\\dlacCF(c, [\\nu]^\\inte) \n= \\aC(\\hd(c)),  &&\\notag  \\\\\n\\DlacCFm(c) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacCF }\\!\\!\\!\\! [\\tb]^\\inte\\dlacCF(c, [\\nu]^\\inte) \n=\\aCF(c) ,   &&\\notag  \\\\\n\\DlacCFp(c)+\\DlacCFm(c) \\leq 2\\max\\{|\\Lambda^\\inte|,|\\Lambda^\\inte|\\}(1 - \\dclrF(c)), \n&& c\\in [1,\\widetilde{\\tC}],  \\notag  \\\\\n\\sum_{c\\in [1,\\widetilde{\\tC}]}\\!\\! \\dlacCF(c, [\\nu]^\\inte) =\\acCF([\\nu]^\\inte),  \n && \\nu \\in  \\tGacCF ,  \\label{eq:AC6} \n\\end{align}    \n\n\\begin{align} \n\\aF(j)+|\\Lambda^\\inte|(1-\\chiF(j,i+\\widetilde{\\tC})+\\eF(j))\\geq \\aTF(i), \n   && \\notag  \\\\  \n\\aTF(i)\\geq \\aF(j)- |\\Lambda^\\inte|(1-\\chiF(j,i+\\widetilde{\\tC})+\\eF(j)), \n  &&  j\\in [1,\\tF],  \\notag  \\\\   \n  %\n\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacTF }\\!\\!\\!\\! m\\cdot \\dlacTF(i, [\\nu]^\\inte) \n=\\bsF(i+\\widetilde{\\tC}), && \\notag  \\\\  \n\\DlacTFp(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacTF }\\!\\!\\!\\! [\\ta]^\\inte\\dlacTF(i, [\\nu]^\\inte) \n= \\aT(i),  &&\\notag  \\\\\n\\DlacTFm(i) +\\sum_{\\nu=(\\ta,\\tb,m) \\in \\tGacTF }\\!\\!\\!\\! [\\tb]^\\inte\\dlacTF(i, [\\nu]^\\inte) \n=\\aTF(i) ,   \n&& \\notag  \\\\\n\\DlacTFp(i)+\\DlacTFm(i) \\leq 2\\max\\{|\\Lambda^\\inte|,|\\Lambda^\\inte|\\}\n(1 - \\dclrF(i+\\widetilde{\\tC})), \n&& i\\in [1,\\tT],  \\notag  \\\\\n\\sum_{i\\in [1,\\tT]}\\!\\! \\dlacTF(i, [\\nu]^\\inte) =\\acTF([\\nu]^\\inte),  \n && \\nu \\in  \\tGacTF ,  \\label{eq:AC5} \n\\end{align}    \n\n\n  \n\\begin{align} \n\\sum_{\\mathrm{X}\\in\\{\\mathrm{C,T,F,CT,TC,CF,TF}\\}}\\!\\!\\!\\!\\!\\!\n   (\\acX([\\nu]^\\inte)+\\acX([\\overline{\\nu}]^\\inte)) \n =\\ac^\\inte([\\nu]^\\inte) , && \\nu\\in \\Gacs^\\inte,  \\notag \\\\  \n\\sum_{\\mathrm{X}\\in\\{\\mathrm{C,T,F,CT,TC,CF,TF}\\}} \\!\\!\\!\\!\\!\\!\n    \\acX([\\nu]^\\inte)\n =\\ac^\\inte([\\nu]^\\inte) , && \\nu\\in \\Gace^\\inte.    \n   \\label{eq:AC_last} \n\\end{align}    \n\n\n \n  \n\\subsection{Descriptor for the Number of Chemical Symbols}  \n\\label{sec:CS\n\nWe include constraints for computing\n the frequency of each chemical symbol in $\\Ldg$.\n Let $\\cs(v)$ denote the chemical symbol of an interior-vertex $v$ in \n a chemical graph $\\C$ to be inferred; i.e.,\n $\\cs(v)=\\mu=\\ta d\\in \\Ldg$ such that $\\alpha(v)=\\ta$ and\n  $\\deg_{\\anC}(v)=\\deg_H(v)-\\deghyd_\\C(v)=d$ in $\\C=(H,\\alpha,\\beta)$. \n\n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate}[leftmargin=*\n\\item[-] A set  $\\Ldg^\\inte$  of chemical symbols;\n \n\\item[-]\n Prepare a coding of each of the two sets \n$\\Ldg^\\inte$    and let $[\\mu]^\\inte$  denote  \nthe coded integer of  an element $\\mu \\in \\Ldg^\\inte$; \n\n\\item[-]\nChoose subsets  $\\tLdgC, \\tLdgT, \\tLdgF  \\subseteq \\Ldg^\\inte$:   \n To compute the frequency of chemical symbols exactly,  set\n  $\\tLdgC:= \\tLdgT :=  \\tLdgF :=\\Ldg^\\inte$;  \n  \n\\end{enumerate}\n\n\n\\smallskip\\noindent\n{\\bf variables: }\n\\begin{enumerate}[leftmargin=*\n\\item[-] $\\ns^\\inte([\\mu]^\\inte )\\in[0,\\nint_\\UB]$,  $\\mu\\in \\Ldg^\\inte$: \n      the number of interior-vertices $v$  with $\\cs(v)=\\mu$;   \n\\item[-] \n   $\\dlnsX(i,[\\mu]^\\inte)\\in [0,1]$, $ i\\in [1,\\tX],\\mu\\in \\Ldg^\\inte$, \n   $\\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$;\n\\end{enumerate}\n\n\\smallskip\\noindent\n{\\bf constraints: } \n\\begin{align}  \n   \\sum_{\\mu\\in \\tLdgX\\cup\\{\\epsilon\\} } \\dlnsX(i,[\\mu]^\\inte)=1, ~~ \n   \\sum_{\\mu=\\ta d\\in \\tLdgX }[\\ta]^\\inte\\cdot\\dlnsX(i,[\\mu]^\\inte)=\\aX(i), \n   \\notag \\\\\n   \\sum_{\\mu=\\ta d\\in \\tLdgX }d\\cdot\\dlnsX(i,[\\mu]^\\inte)\n   =\\degX(i),\n    \\hspace{2cm}\\notag \\\\ \n  ~~   i\\in [1,\\tX],   \n  \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\},  \n  \\label{eq:CS_first} \n\\end{align}    \n \n\n\\begin{align}  \n   \\sum_{i\\in [1,\\tC]} \\dlnsC(i,[\\mu]^\\inte)\n   +    \\sum_{i\\in [1,\\tT]} \\dlnsT(i,[\\mu]^\\inte) \n + \\sum_{i\\in [1,\\tF]} \\dlnsF(i,[\\mu]^\\inte)=\\ns^\\inte([\\mu]^\\inte),\n    && \\mu\\in \\Ldg^\\inte.\n  \\label{eq:CS_last} \n\\end{align}    \n\n\n\n  \n\\subsection{Descriptor for the Number of Edge-configurations}  \n\\label{sec:EC\n\n\nWe include constraints to compute the frequency of each edge-configuration\nin an inferred chemical graph $\\C$. \n \n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate}[leftmargin=*\n\\item[-] A set  $\\Gamma^\\inte$ of edge-configurations $\\gamma=(\\mu,\\mu',m)$\nwith $\\mu\\leq \\mu'$;\n\n\\item[-]  Let $\\Gamma_{<}^\\inte=\\{(\\mu,\\mu',m)\\in  \\Gamma^\\inte\\mid \\mu < \\mu' \\}$, \n$\\Gamma_{=}^\\inte=\\{(\\mu,\\mu',m)\\in  \\Gamma^\\inte\\mid \\mu= \\mu' \\}$\nand   $\\Gamma_{>}^\\inte=\\{(\\mu',\\mu,m)\\mid \n    (\\mu,\\mu',m)\\in  \\Gamma_{<}^\\inte  \\}$;\n    \n\\item[-] \n Prepare a coding of  the set \n$\\Gamma^\\inte \\cup \\Gamma_{>}^\\inte$  and let \n$[\\gamma]^\\inte$   denote  \nthe coded integer of  an element $\\gamma$ in $\\Gamma^\\inte \\cup \\Gamma_{>}^\\inte$; \n  \n\\item[-] \nChoose subsets   $\\tGecC,\\tGecT,\\tGecCT,\\tGecTC,\\tGecF, \\tGecCF , \\tGecTF\n \\subseteq \\Gamma^\\inte\\cup\\Gamma_{>}^\\inte$;  \n To compute the frequency  of edge-configurations exactly,  set \n  $\\tGecC:= \\tGecT :=\\tGecCT:=  \\tGecTC :=\\tGecF:=  \\tGecCF := \\tGecTF := \\Gamma^\\inte\\cup\\Gamma_{>}^\\inte$;  \n  \n\\item[-] $\\ec_\\LB^\\inte(\\gamma),  \\ec_\\UB^\\inte(\\gamma) \\in [0,2\\nint_\\UB ], \n\\gamma=(\\mu,\\mu',m)\\in \\Gamma^\\inte$: \nlower and upper bounds on the number  of interior-edges  $e=uv$ \nwith $\\cs(u)=\\mu$,  $\\cs(v)=\\mu'$ and $\\beta(e)=m$; \n\\end{enumerate}\n\n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n\\item[-]\n$\\ec^\\inte([\\gamma]^\\inte) \\in [\\ec_\\LB^\\inte(\\gamma),\\ec_\\UB^\\inte(\\gamma)], \\gamma\\in \\Gamma^\\inte$: \nthe number of interior-edges     with  edge-configuration $\\gamma$;   \n\\item[-]\n$\\ecC([\\gamma]^\\inte)\\in [0,\\mC],  \\gamma\\in \\tGecC$, \n$\\ecT([\\gamma]^\\inte)\\in [0,\\tT],   \\gamma\\in \\tGecT$, \n$\\ecF([\\gamma]^\\inte)\\in [0,\\tF], \\gamma\\in \\tGecF$:  \nthe number of  edges $\\eC\\in \\EC$ (resp.,   edges $\\eT\\in \\ET$\nand  edges $\\eF\\in \\EF$)   with  edge-configuration $\\gamma$;  \n\\item[-]\n$\\ecCT([\\gamma]^\\inte)\\in [0, \\min\\{\\kC,\\tT\\} ], \\gamma\\in \\tGecCT$,\n$\\ecTC([\\gamma]^\\inte)\\in [0,\\min\\{\\kC,\\tT\\} ], \\gamma\\in \\tGecCT$, \n$\\ecCF([\\gamma]^\\inte)\\in [0,\\widetilde{\\tC}],  \\gamma\\in \\tGecCF$,\n$\\ecTF([\\gamma]^\\inte)\\in [0,\\tT], \\gamma\\in \\tGecTF$:\nthe number of  edges $\\eCT\\in \\ECT$  \n(resp.,   edges $\\eTC\\in \\ETC$\nand   edges $\\eCF\\in \\ECF$ and $\\eTF\\in \\ETF$)  with  edge-configuration $\\gamma$;  \n\\item[-]\n$\\dlecC(i,[\\gamma]^\\inte)\\in [0,1], \n  i\\in [\\widetilde{\\kC}+1,\\mC]=\\Iw\\cup \\Iz\\cup\\Iew, \\gamma\\in \\tGecC$,  \n$\\dlecT(i,[\\gamma]^\\inte)\\in [0,1],  i\\in [2,\\tT],   \\gamma\\in \\tGecT$,  \n$\\dlecF(i,[\\gamma]^\\inte)\\in [0,1] , i\\in [2,\\tF],\\gamma\\in \\tGecF$: \n$\\dlecX(i,[\\gamma]^\\typ)=1$  $\\Leftrightarrow$\nedge  $\\eX_i$ has  edge-configuration $\\gamma$;  \n\\item[-]\n$\\dlecCTC(k,[\\gamma]^\\inte),\\dlecTCC(k,[\\gamma]^\\inte)\\in [0,1],\nk\\in [1, \\kC]=\\It\\cup \\Iw,  \\gamma\\in \\tGecCT$: \n$\\dlecCTC(k,[\\gamma]^\\inte)=1$   (resp., $\\dlecTCC(k,[\\gamma]^\\inte)=1$)\n  $\\Leftrightarrow$\nedge  $\\eCT_{\\tail(k),j}$ (resp.,  $\\eTC_{\\hd(k),j}$) \n   for some $j\\in [1,\\tT]$ has  edge-configuration $\\gamma$; \n   \n\\item[-]\n$\\dlecCFC(c,[\\gamma]^\\inte)\\in [0,1],  c\\in [1,\\widetilde{\\tC}],\\gamma\\in \\tGecCF$:\n$\\dlecCFC(c,[\\gamma]^\\inte)=1$    $\\Leftrightarrow$\nedge   $\\eCF_{c,i}$  for some $i\\in [1,\\tF]$ has  edge-configuration $\\gamma$; \n\n\\item[-]\n  $\\dlecTFT(i,[\\gamma]^\\inte)\\in [0,1],  i\\in [1,\\tT],  \\gamma\\in \\tGecTF$:\n    $\\dlecTFT(i,[\\gamma]^\\inte)=1$  $\\Leftrightarrow$\nedge     $\\eTF_{i,j}$ for some $j\\in [1,\\tF]$ has  edge-configuration $\\gamma$; \n\n\\item[-]\n$\\degCTT(k),\\degTCT(k)\\in [0, 4],   k\\in [1, \\kC]$: \n$\\deg_{\\anC}(v)$  of an end-vertex $v\\in \\VT$ of  \nthe edge $(\\vC_{\\mathrm{tail}(k)},v)\\in \\ECT$  \n (resp., $(v,\\vC_{\\mathrm{head}(k)})\\in \\ETC$)  if any; \n \n\\item[-]\n$\\degCFF(c)\\in [0, 4], c\\in [1,\\widetilde{\\tC}]$: \n $\\deg_{\\anC}(v)$  of an end-vertex $v\\in \\VF$ of  \n the edge $(\\vC_{c},v)\\in \\ECF$   if any; \n \n\\item[-]\n$\\degTFF(i)\\in [0, 4], i\\in [1,\\tT]$: \n $\\deg_{\\anC}(v)$   of an end-vertex $v\\in \\VF$ of  \n  the edge $(\\vT_{i},v)\\in \\ETF$   if any;  \n\\item[-]\n$\\DlecCp(i),  \\DlecCm(i), \\in [0,4], \n  i\\in [\\widetilde{\\kC}+1,\\mC]$,  \n$\\DlecTp(i),\\DlecTm(i)\\in [0,4],  i\\in [2,\\tT]$,  \n$\\DlecFp(i),\\DlecFm(i)\\in [0,4] , i\\in [2,\\tF]$: \n$\\DlecXp(i)=\\DlecXm(i)=0$ (resp., \n $\\DlecXp(i)=\\deg_{\\anC}(u)$\n  and $\\DlecXm(i)=\\deg_{\\anC}(v)$) $\\Leftrightarrow$  \nedge  $\\eX_i=(u,v)\\in \\EX$  is used in ${\\anC}$ (resp., $\\eX_i\\not\\in E({\\anC})$);  \n\\item[-]\n$\\DlecCTp(k),\\DlecCTm(k)\\in [0,4],\nk\\in [1, \\kC]=\\It\\cup \\Iw$: \n$\\DlecCTp(k)=\\DlecCTm(k) =0$ \n(resp.,  $\\DlecCTp(k)=\\deg_{\\anC}(u)$\n and $\\DlecCTm(k)=\\deg_{\\anC}(v)$) \n $\\Leftrightarrow$  \nedge  $\\eCT_{\\tail(k),j}=(u,v)\\in \\ECT$   \n  for some $j\\in [1,\\tT]$ is used in ${\\anC}$ (resp., otherwise); \n  \n\\item[-]\n$\\DlecTCp(k),\\DlecTCm(k)\\in [0,4],\nk\\in [1, \\kC]=\\It\\cup \\Iw$: \nAnalogous with $\\DlecCTp(k)$ and $\\DlecCTm(k)$;\n\n\\item[-]\n$\\DlacCFp(c), \\DlecCFm(c) \\in [0,4],  c\\in [1,\\widetilde{\\tC}]$: \n$\\DlecCFp(c)=\\DlecCFm(c) =0$ (resp., \n $\\DlecCFp(c)=\\deg_{\\anC}(u)$ \n and $\\DlecCFm(c)=\\deg_{\\anC}(v)$) \n $\\Leftrightarrow$  \nedge  $\\eCF_{c,j}=(u,v)\\in \\ECF$  \n   for some $j\\in [1,\\tF]$ is used in ${\\anC}$ (resp., otherwise);  \n\\item[-]\n  $\\DlecTFp(i),  \\DlecTFm(i)\\in [0,4],  i\\in [1,\\tT]$:\n  Analogous with $\\DlecCFp(c)$ and $\\DlecCFm(c)$;\n\\end{enumerate}\n  \n\\smallskip\\noindent\n{\\bf constraints: }\n\n\\begin{align} \n \\ecC([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecC , \\notag \\\\\n \\ecT([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecT , \\notag \\\\\n \\ecF([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecF , \\notag \\\\\n %\n %\n \\ecCT([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecCT , \\notag \\\\\n \\ecTC([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecTC , \\notag \\\\\n \\ecCF([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecCF , \\notag \\\\\n \\ecTF([\\gamma]^\\inte) =0,  &&  \\gamma \\in \\Gamma^\\inte\\setminus \\tGecTF , \\notag \\\\\n \\label{eq:EC_first} \n\\end{align}    \n\n\n\\begin{align} \n \\sum_{(\\mu, \\mu',m)=\\gamma\\in  \\Gamma^\\inte}\\ecC([\\gamma]^\\inte) \n      =\\sum_{i\\in [\\widetilde{\\kC}+1,\\mC]}\\delbC(i,m),  &&   m\\in [1,3]   , \\notag \\\\\n \\sum_{(\\mu, \\mu',m)=\\gamma\\in  \\Gamma^\\inte}\\ecT([\\gamma]^\\inte) \n      =\\sum_{i\\in [2,\\tT]}\\delbT(i,m) ,  &&   m\\in [1,3] , \\notag \\\\\n \\sum_{(\\mu, \\mu',m)=\\gamma\\in \\Gamma^\\inte}\\ecF([\\gamma]^\\inte)\n      =\\sum_{i\\in [2,\\tF]}\\delbF(i,m) ,  &&   m\\in [1,3]  , \\notag \\\\\n %\n \\sum_{(\\mu, \\mu',m)=\\gamma\\in \\Gamma^\\inte}\\ecCT([\\gamma]^\\inte)\n     =\\sum_{k\\in [1, \\kC]} \\delbCT(k,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\sum_{(\\mu, \\mu',m)=\\gamma\\in \\Gamma^\\inte}\\ecTC([\\gamma]^\\inte)\n    =\\sum_{k\\in [1, \\kC]} \\delbTC(k,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\sum_{(\\mu, \\mu',m)=\\gamma\\in \\Gamma^\\inte}\\ecCF([\\gamma]^\\inte)\n    =\\sum_{c\\in [1,\\widetilde{\\tC}]} \\delbsF(c,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\sum_{(\\mu, \\mu',m)=\\gamma\\in \\Gamma^\\inte}\\ecTF([\\gamma]^\\inte) \n    =\\sum_{c\\in [\\widetilde{\\tC}+1, \\cF]} \\delbsF(c,m),  &&   m\\in [1,3]  , \\notag \\\\\n \\label{eq:EC_first2} \n\\end{align}    \n\n\n\\begin{align}  \n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecC }\\!\\!\\!\\! [(\\ta,\\tb,m)]^\\inte\\cdot \\dlecC(i, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacC } [\\nu]^\\inte\\cdot \\dlacC(i, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecCp(i) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecC }\\!\\!\\!\\! \n  d\\cdot \\dlecC(i, [\\gamma]^\\inte) \n=\\degC(\\tail(i)),  && \\notag  \\\\\n\\DlecCm(i) +\\sum_{\\gamma=(\\mu,\\tb d,m) \\in \\tGecC }\\!\\!\\!\\!\n  d\\cdot\\dlecC(i, [\\gamma]^\\inte) \n= \\degC(\\hd(i)),  &&\\notag  \\\\\n\\DlecCp(i)+\\DlecCm(i) \\leq 8(1 - \\eC(i)),\n&& i\\in [\\widetilde{\\kC}+1,\\mC],  \\notag  \\\\\n\\sum_{i\\in [\\widetilde{\\kC}+1,\\mC]}\\!\\!\\!\\! \\dlecC(i, [\\gamma]^\\inte) =\\ecC([\\gamma]^\\inte),  \n&&  \\gamma \\in \\tGecC ,   \\label{eq:EC1}   \n\\end{align}    \n\n\n\\begin{align} \n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecT }\\!\\!\\!\\! [(\\ta,\\tb,m)]^\\inte\\cdot \\dlecT(i, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacT} [\\nu]^\\inte\\cdot \\dlacT(i, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecTp(i) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecT }\\!\\!\\!\\!\n  d\\cdot  \\dlecT(i, [\\gamma]^\\inte) \n   =\\degT(i-1 ),  && \\notag  \\\\\n\\DlecTm(i) +\\sum_{\\gamma=(\\mu,\\tb d,m) \\in \\tGecT }\\!\\!\\!\\! \n d\\cdot \\dlecT(i, [\\gamma]^\\inte) \n =\\degT(i),     &&\\notag  \\\\\n\\DlecTp(i)+\\DlecTm(i) \\leq 8(1 - \\eT(i)),\n&& i\\in [2,\\tT],   \\notag  \\\\\n \\sum_{ i\\in [2,\\tT]} \\!\\! \\dlecT(i, [\\gamma]^\\inte) =\\ecT([\\gamma]^\\inte),    \n&& \\gamma \\in \\tGecT ,    \\label{eq:EC2} \n\\end{align}     \n\n\\begin{align} \n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecF }\\!\\!\\!\\! \n[(\\ta,\\tb,m)]^\\inte\\cdot \\dlecF(i, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacF } [\\nu]^\\inte\\cdot \\dlacF(i, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecFp(i) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecF }\\!\\!\\!\\! \n  d\\cdot \\dlecF(i, [\\gamma]^\\inte) \n=\\degF(i-1 ),    && \\notag  \\\\\n\\DlecFm(i) +\\sum_{\\gamma=(\\mu,\\tb d, m) \\in \\tGecF }\\!\\!\\!\\! \n d\\cdot  \\dlecF(i, [\\gamma]^\\inte) \n=\\degF(i,0 ),    &&\\notag  \\\\\n\\DlecFp(i)+\\DlecFm(i) \\leq 8(1 - \\eF(i)),\n&& i\\in [2,\\tF],   \\notag  \\\\\n  \\sum_{ i\\in [2,\\tF]} \\!\\! \\dlecF(i, [\\gamma]^\\inte) =\\ecF([\\gamma]^\\inte),  \n &&  \\gamma \\in \\tGecF ,    \\label{eq:EC3} \n\\end{align}    \n  \n\n\n\\begin{align} \n\\degT(i)+4(1-\\chiT(i,k)+\\eT(i))\\geq \\degCTT(k),  \n && \\notag  \\\\  \n\\degCTT(k)\\geq \\degT(i)- 4(1-\\chiT(i,k)+\\eT(i)), &&  i\\in [1,\\tT],    \\notag  \\\\  \n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecCT }\\!\\!\\!\\! \n[(\\ta,\\tb,m)]^\\inte\\cdot \\dlecCTC(k, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacCT} [\\nu]^\\inte\\cdot \\dlacCT(k, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecCTp(k) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecCT }\\!\\!\\!\\! \n  d\\cdot \\dlecCTC(k, [\\gamma]^\\inte) \n=\\degC(\\tail(k)),    && \\notag  \\\\\n\\DlecCTm(k) +\\sum_{\\gamma=(\\mu,\\tb d, m) \\in \\tGecCT }\\!\\!\\!\\! \n d\\cdot  \\dlecCTC(k, [\\gamma]^\\inte) \n= \\degCTT(k),     &&\\notag  \\\\\n\\DlecCTp(k)+\\DlecCTm(k) \\leq 8(1 - \\dclrT(k)),\n&& k\\in [1, \\kC],  \\notag  \\\\\n\\sum_{k\\in [1, \\kC]}\\!\\! \\dlecCTC(k, [\\gamma]^\\inte) =\\ecCT([\\gamma]^\\inte),  \n && \\gamma \\in  \\tGecCT ,  \\label{eq:EC5} \n\\end{align}    \n\n\n\\begin{align} \n \\degT(i)+4(1-\\chiT(i,k)+\\eT(i+1))\\geq \\degTCT(k),  \n   && \\notag  \\\\  \n\\degTCT(k)\\geq \\degT(i)- 4(1-\\chiT(i,k)+\\eT(i+1)), \n&&  i\\in [1,\\tT],    \\notag  \\\\  \n  %\n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecTC }\\!\\!\\!\\! \n[(\\ta,\\tb,m)]^\\inte\\cdot \\dlecTCC(k, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacTC} [\\nu]^\\inte\\cdot \\dlacTC(k, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecTCp(k) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecTC }\\!\\!\\!\\! \n  d\\cdot \\dlecTCC(k, [\\gamma]^\\inte) \n= \\degTCT(k),     &&\\notag  \\\\\n\\DlecTCm(k) +\\sum_{\\gamma=(\\mu,\\tb d, m) \\in \\tGecTC }\\!\\!\\!\\! \n d\\cdot  \\dlecTCC(k, [\\gamma]^\\inte) \n=\\degC(\\hd(k)),    && \\notag  \\\\\n\\DlecTCp(k)+\\DlecTCm(k) \\leq 8(1 - \\dclrT(k)),\n&& k\\in [1, \\kC],  \\notag  \\\\\n\\sum_{k\\in [1, \\kC]}\\!\\! \\dlecTCC(k, [\\gamma]^\\inte) =\\ecTC([\\gamma]^\\inte),  \n && \\gamma \\in  \\tGecTC ,  \\label{eq:EC5} \n\\end{align}    \n \n\n\\begin{align} \n\\degF(i)+4(1-\\chiF(i,c)+\\eF(i))\\geq \\degCFF(c), \n   && \\notag  \\\\  \n\\degCFF(c)\\geq \\degF(i)- 4(1-\\chiF(i,c)+\\eF(i)), \n && i\\in [1,\\tF],   \\notag  \\\\   \n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecCF }\\!\\!\\!\\! \n[(\\ta,\\tb,m)]^\\inte\\cdot \\dlecCFC(c, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacCF} [\\nu]^\\inte\\cdot \\dlacCF(c, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecCFp(c) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecCF }\\!\\!\\!\\! \n  d\\cdot \\dlecCFC(c, [\\gamma]^\\inte) \n=\\degC(c),    && \\notag  \\\\\n\\DlecCFm(c) +\\sum_{\\gamma=(\\mu,\\tb d, m) \\in \\tGecCF }\\!\\!\\!\\! \n d\\cdot  \\dlecCFC(c, [\\gamma]^\\inte) \n= \\degCFF(c),     &&\\notag  \\\\\n\\DlecCFp(c)+\\DlecCFm(c) \\leq 8(1 - \\dclrF(c)),\n&& c\\in [1,\\widetilde{\\tC}],  \\notag  \\\\\n\\sum_{c\\in [1,\\widetilde{\\tC}]}\\!\\! \\dlecCFC(c, [\\gamma]^\\inte) =\\ecCF([\\gamma]^\\inte),  \n && \\gamma \\in  \\tGecCF ,  \\label{eq:EC6} \n\\end{align}    \n\n\n\\begin{align} \n\\degF(j)+4(1-\\chiF(j,i+\\widetilde{\\tC})+\\eF(j))\\geq \\degTFF(i), \n  && \\notag  \\\\  \n\\degTFF(i)\\geq \\degF(j)- 4(1-\\chiF(j,i+\\widetilde{\\tC})+\\eF(j)), \n && j\\in [1,\\tF],   \\notag  \\\\ \n\\sum_{\\gamma=(\\ta d,\\tb d',m) \\in \\tGecTF }\\!\\!\\!\\! \n[(\\ta,\\tb,m)]^\\inte\\cdot \\dlecTFT(i, [\\gamma]^\\inte) \n= \\sum_{\\nu \\in \\tGacTF} [\\nu]^\\inte\\cdot \\dlacTF(i, [\\nu]^\\inte) , && \\notag  \\\\  \n\\DlecTFp(i) +\\sum_{\\gamma=(\\ta d,\\mu',m) \\in \\tGecTF }\\!\\!\\!\\! \n  d\\cdot \\dlecTFT(i, [\\gamma]^\\inte) \n=\\degT(i),    && \\notag  \\\\\n\\DlecTFm(i) +\\sum_{\\gamma=(\\mu,\\tb d, m) \\in \\tGecTF }\\!\\!\\!\\! \n d\\cdot  \\dlecTFT(i, [\\gamma]^\\inte) \n= \\degTFF(i),     &&\\notag  \\\\\n\\DlecTFp(i)+\\DlecTFm(i) \\leq 8(1 - \\dclrF(i+\\widetilde{\\tC})),\n&& i\\in [1,\\tT],  \\notag  \\\\\n\\sum_{i\\in [1,\\tT]}\\!\\! \\dlecTFT(i, [\\gamma]^\\inte) =\\ecTF([\\gamma]^\\inte),  \n && \\gamma \\in  \\tGecTF ,  \\label{eq:EC6} \n\\end{align}    \n\n\n\n\\begin{align} \n\\sum_{\\mathrm{X}\\in\\{\\mathrm{C,T,F,CT,TC,CF,TF}\\}}(\\ecX([\\gamma]^\\inte)\n+\\ecX([\\overline{\\gamma}]^\\inte)) \n =\\ec^\\inte([\\gamma]^\\inte) , &&  \\gamma\\in \\Gamma_{<}^\\inte,  \\notag \\\\  \n\\sum_{\\mathrm{X}\\in\\{\\mathrm{C,T,F,CT,TC,CF,TF}\\}} \\ecX([\\gamma]^\\inte) \n =\\ec^\\inte([\\gamma]^\\inte) , && \\gamma\\in \\Gamma_{=}^\\inte. \n   \\label{eq:EC_last} \n\\end{align}    \n\n\n \n\\section{All Constraints in an MILP Formulation for Chemical Graphs}\\label{sec:full_milp\n\n   \n\n \nWe define a standard encoding of a finite set $A$ of elements\nto be a bijection $\\sigma: A \\to [1, |A|]$, \nwhere we denote by $[A]$   the set $[1, |A|]$ of integers\nand by $[{\\tt e}]$ the encoded element $\\sigma({\\tt e})$.\nLet $\\epsilon$ denote {\\em null}, a fictitious chemical element \nthat does not belong to any set of chemical elements,\nchemical symbols, adjacency-configurations and\nedge-configurations in the following formulation.\nGiven a finite set $A$, let $A_\\epsilon$ denote the set $A\\cup\\{\\epsilon\\}$\nand define a standard encoding of $A_\\epsilon$\n  to be a bijection $\\sigma: A \\to [0, |A|]$ such that\n$\\sigma(\\epsilon)=0$, \nwhere we denote by $[A_\\epsilon]$   the set $[0, |A|]$ of integers\nand by $[{\\tt e}]$ the encoded element $\\sigma({\\tt e})$,\nwhere $[\\epsilon]=0$.\n\n\n \\bigskip \n Let $\\sigma=(\\GC,\\sint,\\sce)$ be a target specification,\n ${\\rho}$ denote  the branch-parameter in the specification $\\sigma$\n and  $\\C$ denote a chemical   graph in $\\mathcal{G}(\\GC, \\sint,\\sce)$. \n\n \\subsection{Selecting  a Cyclical-base} \n\\label{sec:co\n \nRecall that  \n\\[ \\begin{array}{ll}\n   \\Eew = \\{e\\in \\EC\\mid \\ell_\\LB(e)=\\ell_\\UB(e)=1 \\}; &\n   \\Ez =\\{e\\in \\EC\\mid \\ell_\\LB(e)=0, \\ell_\\UB(e)=1 \\}; \\\\\n  \\Ew=\\{e\\in \\EC\\mid \\ell_\\LB(e)=1,  \\ell_\\UB(e)\\geq 2 \\}; &\n  \\Et= \\{e\\in \\EC\\mid \\ell_\\LB(e)\\geq 2 \\}; \\end{array} \\]\n\\begin{enumerate} [leftmargin=*\n\\item[-]\nEvery edge $a_i\\in \\Eew$ is  included in  $\\anC$;\n\n\\item[-]\nEach edge $a_i\\in \\Ez$ is   included in $\\anC$ if necessary;\n \n\\item[-]\nFor each edge  $a_i  \\in \\Et$, edge $a_i$ is not included in $\\anC$\nand instead a path \n\\[P_i=(\\vC_{\\tail(i)}, \\vT_{j-1},\\vT_{j},\\ldots,\n    \\vT_{j+t}, \\vC_{\\hd(i)})\\]\n     of length at least 2\n  from vertex $\\vC_{\\tail(i)}$ to vertex $\\vC_{\\hd(i)}$ \n  visiting some  vertices in $\\VT$ is constructed in $\\anC$; and  \n \n\\item[-]\nFor each edge $a_i  \\in \\Ew$, either  edge $a_i$   is directly used in $\\anC$ or\nthe above path $P_i$ of length at least 2   is constructed in $\\anC$.  \n \\end{enumerate}\n \nLet  $\\tC\\triangleq |\\VC|$ and denote $\\VC$ by \n$\\{\\vC_{i}\\mid i\\in [1,\\tC]\\}$.\nRegard the seed graph $\\GC$ as a digraph such that\neach edge $a_i$ with end-vertices $\\vC_{j}$ and $\\vC_{j'}$\nis directed from  $\\vC_{j}$ to $\\vC_{j'}$ when $j<j'$.\n For each directed edge $a_i  \\in \\EC $,\n let $\\hd(i)$ and $\\tail(i)$ denote the head and tail of $\\eC(i)$;\n i.e., $a_i=(\\vC_{\\tail(i)}, \\vC_{\\hd(i)})$. \n  \nDefine \n \\[ \\kC \\triangleq  |\\Et\\cup \\Ew| , ~~ \\widetilde{\\kC} \\triangleq  |\\Et| ,\\]\n and denote   $\\EC=\\{a_i\\mid i\\in[1,\\mC]\\}$,\n$\\Et=\\{a_k\\mid k\\in[1,\\widetilde{\\kC}]\\}$,\n$\\Ew=\\{a_k\\mid k\\in[\\widetilde{\\kC}+1,\\kC]\\}$,\n$\\Ez=\\{a_i\\mid i\\in[\\kC+1,\\kC+|\\Ez|]\\}$ and \n$\\Eew=\\{a_i\\mid i\\in[\\kC+|\\Ez|+1,\\mC]\\}$.\nLet $\\Iew$ denote the set of indices $i$ of edges $a_i\\in \\Eew$.\nSimilarly for $\\Iz$, $\\Iw$  and $\\It$.\n\n\nTo control the construction of such a path $P_i$\n for each edge  $a_k\\in  \\Et\\cup \\Ew $,\nwe regard the index $k\\in [1,\\kC]$ of each edge $a_k\\in  \\Et\\cup \\Ew$\nas the ``color'' of the edge.\nTo introduce necessary linear constraints \nthat can construct such a path $P_k$ properly   in our MILP,\nwe assign the color $k$ to the vertices $\\vT_{j-1},\\vT_{j},\\ldots,$ \n$\\vT_{j+t}$ in $\\VT$\nwhen the above path  $P_k$ is used in $\\anC$.\n \nFor each index $s\\in [1,\\tC]$, let  \n$\\IC(s)$ denote the set of edges $e\\in \\EC$ incident to vertex $\\vC_{s}$,\nand \n $\\Eew^+(s)$ (resp., $\\Eew^-(s)$) denote the set of \n edges $a_i\\in \\Eew$ such that \nthe tail (resp., head) of $a_i$ is vertex $\\vC_{s}$.\nSimilarly for \n$\\Ez^+(s)$,  $\\Ez^-(s)$, $\\Ew^+(s)$,  $\\Ew^-(s)$,\n$\\Et^+(s)$ and $\\Et^-(s)$.\nLet $\\IC(s)$ denote the set of indices $i$ of edges $a_i\\in \\IC(s)$.\nSimilarly for   \n$\\Iew^+(s)$,  $\\Iew^-(s)$,\n$\\Iz^+(s)$,  $\\Iz^-(s)$, \n$\\Iw^+(s)$,  $\\Iw^-(s)$,\n$\\It^+(s)$ and $\\It^-(s)$.\nNote that $[1, \\kC]=\\It\\cup \\Iw$ and \n$[\\widetilde{\\kC}+1,\\mC]=\\Iw\\cup \\Iz\\cup\\Iew$.\n\n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate} [leftmargin=*\n\\item[-] $\\tC=|\\VC|$, $\\widetilde{\\kC}=  |\\Et|$, $\\kC= |\\Et\\cup \\Ew|$,\n      $\\tT=\\nint_\\UB-|\\VC|$, $\\mC=|\\EC|$.\n      Note that \n      $a_i\\in \\EC\\setminus (\\Et\\cup \\Ew)$ holds $i\\in [\\kC+1,\\mC]$;   \n\n\\item[-] $\\ell_\\LB(k), \\ell_\\UB(k)\\in [1, \\tT]$, $k\\in [1,\\kC]$: \nlower and upper bounds on the length of path $P_k$;  \n       \n\\item[-] $r_{\\GC}\\in[1,\\mC]$: the rank $\\mathrm{r}(\\GC)$ of\nseed graph $\\GC$;   \\newone\n\\end{enumerate}\n\n\n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n\\item[-] $\\eC(i)\\in[0,1]$,  $i\\in [1, \\mC]$: \n$\\eC(i)$ represents edge $a_i\\in \\EC$, $i\\in [1,\\mC]$  \n ($\\eC(i)=1$, $i\\in \\Iew$;  $\\eC(i)=0$, $i\\in \\It$)     \n  ($\\eC(i)=1$ $\\Leftrightarrow $   edge $a_i$ is  used in  $\\anC$);    \n\\item[-]  $\\vT(i)\\in[0,1]$,   $i\\in [1,\\tT]$:  \n  $\\vT(i)=1$ $\\Leftrightarrow $ vertex $\\vT_{i}$ is used in  $\\anC$;   \n\\item[-]  $\\eT(i)\\in[0,1]$, $i\\in [1,\\tT+1]$:  $\\eT(i)$ represents edge \n$\\eT_{i}=(\\vT_{i-1}, \\vT_{i})\\in \\ET$,  \nwhere $\\eT_{1}$ and $\\eT_{\\tT+1}$ are fictitious edges\n  ($\\eT(i)=1$ $\\Leftrightarrow $   edge $\\eT_{i}$ is  used in  $\\anC$);    \n  \n\\item[-]  $\\chiT(i)\\in [0,\\kC]$, $i\\in [1,\\tT]$: $\\chiT(i)$ represents\n the color assigned to vertex $\\vT_{i}$ \n  ($\\chiT(i)=k>0$\n   $\\Leftrightarrow $  vertex $\\vT_{i}$ is  assigned color $k$;\n   $\\chiT(i)=0$ means that vertex $\\vT_{i}$ is not used in $\\anC$);    \n   \n\\item[-]  $\\clrT(k)\\in [\\ell_\\LB(k)-1, \\ell_\\UB(k)-1]$, $k\\in [1,\\kC]$, \n$\\clrT(0)\\in [0, \\tT]$: the number of vertices \n$\\vT_{i}\\in \\VT$  with color $c$;\n\\item[-]  $\\dclrT(k)\\in [0,1]$,   $k\\in [0,\\kC]$:\n      $\\dclrT(k)=1$    $\\Leftrightarrow $ $\\chiT(i)=k$ \n      for some $i\\in [1,\\tT]$;\n      \n\\item[-]    $\\chiT(i,k)\\in[0,1]$,  $i\\in [1,\\tT]$, $k\\in [0,\\kC]$  \n  ($\\chiT(i,k)=1$    $\\Leftrightarrow $ $\\chiT(i)=k$);  \n\\item[-]   $\\tldgC^+(i)\\in [0,4]$, $i\\in [1,\\tC]$: \nthe out-degree of vertex $\\vC_{i}$ with the used edges $\\eC$ in $\\EC$; \n\n\\item[-]   $\\tldgC^-(i)\\in [0,4]$, $i\\in [1,\\tC]$: \nthe in-degree of vertex $\\vC_{i}$  with the used edges $\\eC$ in $\\EC$; \n\n\\item[-] $\\mathrm{rank}$:  the rank $\\mathrm{r}(\\C)$ of a target \nchemical graph $\\C$;   \\newone\n\\end{enumerate}\n  \n\\smallskip\\noindent\n{\\bf constraints: }   \n\\begin{align} \n \\mathrm{rank} =  r_{\\GC} -\\sum_{i\\in \\Iz}(1-\\eC(i)), && \\label{eq:co_rank} \\\\\n  \\eC(i)=1,  ~~~  i\\in \\Iew,       &&  \\label{eq:co_first}  \\\\\n  \\eC(i)=0,  ~~ \\clrT(i)\\geq 1,   ~~~  i\\in \\It,     &&   \\label{eq:co_first} \\\\\n  \\eC(i)+ \\clrT(i)\\geq 1,  ~~~~~  \\clrT(i)\\leq \\tT\\cdot (1-\\eC(i) ), \n~~~  i\\in \\Iw,    &&    \\label{eq:co1c} \n\\end{align}   \n  \n\\begin{align}  \n\\sum_{ c\\in \\Iw^-(i)\\cup \\Iz^-(i)\\cup \\Iew^-(i) }\\!\\!\\!\\!\\!\\! \\eC(c) \n = \\tldgC^-(i),  ~~ \n\\sum_{ c\\in \\Iw^+(i)\\cup \\Iz^+(i)\\cup \\Iew^+(i) }\\!\\!\\!\\!\\!\\! \\eC(c) \n = \\tldgC^+(i),  &&   i\\in [1,\\tC],   \\label{eq:co_5}\n\\end{align}   \n\\begin{align} \n\\chiT(i,0)=1 -\\vT(i), ~~~\n\\sum_{k\\in [0,\\kC]} \\chiT(i,k)=1,  ~~~ \n\\sum_{k\\in [0,\\kC]}k\\cdot \\chiT(i,k)=\\chiT(i),  && i\\in[1,\\tT],  \\label{eq:co2} \n\\end{align}   \n\n\\begin{align}  \n\\sum_{i\\in[1,\\tT]} \\chiT(i,k)=\\clrT(k), ~~\n\\tT\\cdot \\dclrT(k)\\geq  \\sum_{i\\in [1,\\tT]} \\chiT(i,k)\n\\geq \\dclrT(k), &&  k\\in [0,\\kC],    \\label{eq:co3}   \n\\end{align}     \n \n\\begin{align}  \n\\vT(i-1)\\geq \\vT(i), && \\notag \\\\\n \\kC\\cdot (\\vT(i-1)-\\eT(i )) \\geq \\chiT(i-1)-\\chiT(i )\n  \\geq \\vT(i-1) - \\eT(i ), && i\\in[2,\\tT]. \\label{eq:co_last} \n \\end{align}    \n \n \n \n \n\\subsection{Constraints for Including Leaf Paths} \n\\label{sec:int\n\nLet\n$\\widetilde{\\tC}$  denote the number of vertices $u\\in \\VC$ such that \n$\\bl_\\UB(u)=1$ and assume that \n$\\VC=\\{u_1,u_2,\\ldots, u_p\\}$ so that \n\\[ \\mbox{ \n$\\bl_\\UB(u_i)=1$, $i\\in [1,\\widetilde{\\tC}]$ and \n$\\bl_\\UB(u_i)=0$, $i\\in[\\widetilde{\\tC}+1, \\tC]$. }\\]\nDefine the set of colors for the vertex set \n$\\{u_i\\mid i\\in [1,\\widetilde{\\tC}] \\}\\cup \\VT$\n to be $[1,\\cF]$ with \n\\[ \\cF \\triangleq \\widetilde{\\tC} + \\tT \n=|\\{u_i\\mid i\\in[1,\\widetilde{\\tC}]\\}\\cup \\VT|. \\]\nLet each  vertex   $\\vC_{i}$, $i\\in[1,\\widetilde{\\tC}]$ \n(resp., $\\vT_{i}\\in \\VT$)\n  correspond to \na color $i\\in [1,\\cF]$ (resp., $i+\\widetilde{\\tC} \\in [1,\\cF]$). \nWhen a path $P=(u, \\vF_{j}, \\vF_{j+1},\\ldots, \\vF_{j+t})$ \nfrom a vertex $u\\in \\VC\\cup \\VT$ \n  is used in $\\anC$, we assign the color $i\\in [1,\\cF]$ of the vertex $u$\nto the vertices $\\vF_{j}, \\vF_{j+1},\\ldots, \\vF_{j+t}\\in \\VF$.\n\n\n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate}[leftmargin=*\n\\item[-] $\\cF$: the maximum number of different colors \nassigned to the vertices in $\\VF$;  \n\n\\item[-]   $n^*$: an upper bound  \non the number $n(\\C)$ of non-hydrogen atoms in $\\C$;  \n\n\\item[-] $\\nint_\\LB, \\nint_\\UB \\in [2,n^* ]$:\n lower and upper bounds on\nthe number of interior-vertices in $\\C$; \n\n\\item[-] $\\bl_\\LB(i) \\in [0,1]$,  $i\\in [1, \\widetilde{\\tC}]$: \na lower   bound  on the number of leaf ${\\rho}$-branches  in\nthe leaf path rooted  at a vertex $\\vC_{i}$; \n\n\\item[-]  $\\bl_\\LB(k),\\bl_\\UB(k)\\in [0,\\ell_\\UB(k)-1]$, \n $k\\in[1,\\kC]=\\It\\cup\\Iw$: \nlower and upper bounds on the number of \nleaf ${\\rho}$-branches in the trees rooted at internal vertices \nof a pure path $P_k$  for an edge $a_k\\in \\Ew\\cup \\Et$; \n\\end{enumerate}\n\n\n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n  \n\\item[-]   $\\nint_G\\in [\\nint_\\LB, \\nint_\\UB]$: \nthe number of interior-vertices in $\\C$; \n\n\\item[-]  $\\vF(i)\\in[0,1]$,   $i\\in [1,\\tF]$:\n $\\vF(i)=1$ $\\Leftrightarrow $ vertex $\\vF_{i}$ is used in  $\\C$;   \n \n\\item[-]  $\\eF(i)\\in[0,1]$, $i\\in [1,\\tF+1]$:  $\\eF(i)$ represents edge \n$\\eF_{i}=\\vF_{i-1} \\vF_{i}$,  \nwhere $\\eF_{1}$ and $\\eF_{\\tF+1}$ are fictitious edges\n  ($\\eF(i)=1$ $\\Leftrightarrow $   edge $\\eF_{i}$ is  used in  $\\C$);    \n\\item[-]  $\\chiF(i)\\in [0,\\cF]$, $i\\in [1,\\tF]$: $\\chiF(i)$ represents\n the color assigned to  vertex $\\vF_{i}$  \n  ($\\chiF(i)=c$ $\\Leftrightarrow $  vertex $\\vF_{i}$ is  assigned color $c$);   \n  \n\\item[-]  $\\clrF(c)\\in [0, \\tF]$, $c\\in [0,\\cF]$: the number of vertices $\\vF_{i}$\n with color $c$;  \n\\item[-]  $\\dclrF(c)\\in [\\bl_\\LB(c), 1]$,  $c\\in [1, \\widetilde{\\tC}]$:\n      $\\dclrF(c)=1$    $\\Leftrightarrow $ $\\chiF(i)=c$ for some $i\\in [1,\\tF]$;  \n\\item[-]  $\\dclrF(c)\\in[0,1]$,  $c\\in [\\widetilde{\\tC}+1,\\cF]$:\n      $\\dclrF(c)=1$    $\\Leftrightarrow $ $\\chiF(i)=c$ for some $i\\in [1,\\tF]$;  \n\\item[-]    $\\chiF(i,c)\\in[0,1]$,\n $i\\in [1,\\tF]$, $c\\in [0,\\cF]$:  \n   $\\chiF(i,c)=1$    $\\Leftrightarrow $ $\\chiF(i)=c$;     \n\\item[-]  $\\bl(k,i)\\in [0,1]$, $k\\in[1,\\kC]= \\It\\cup\\Iw$,  $i\\in[1,\\tT]$: \n    $\\bl(k,i)=1$ $\\Leftrightarrow$ path $P_k$ contains vertex $\\vT_{i}$ \n    as an internal vertex\n    and the ${\\rho}$-fringe-tree rooted at $\\vT_{i}$ contains a leaf ${\\rho}$-branch;\n\\end{enumerate}\n  \n\\smallskip\\noindent\n{\\bf constraints: }   \n\\begin{align} \n\\chiF(i,0)=1 -\\vF(i), ~~~\n\\sum_{c\\in [0,\\cF]} \\chiF(i,c)=1,  ~~~ \n\\sum_{c\\in [0,\\cF]}c\\cdot \\chiF(i,c)=\\chiF(i),  &&  i\\in[1,\\tF],  \\label{eq:int_first} \n\\end{align}   \n\n\\begin{align}  \n\\sum_{i\\in[1,\\tF]} \\chiF(i,c)=\\clrF(c), ~~~ \\tF\\cdot \\dclrF(c)\\geq\n\\sum_{i\\in [1,\\tF]} \\chiF(i,c)\\geq \\dclrF(c), &&  c\\in [0,\\cF],    \\label{eq:int3}   \n\\end{align}   \n \n\\begin{align}  \n \\eF(1)=\\eF(\\tF+1)=0,  && \\label{eq:int4} \n \\end{align}   \n \n\\begin{align}  \n\\vF(i-1)\\geq \\vF(i), && \\notag \\\\\n \\cF\\cdot (\\vF(i-1)-\\eF(i)) \\geq \\chiF(i-1)-\\chiF(i) \n \\geq \\vF(i-1)- \\eF(i), && i\\in[2,\\tF], \\label{eq:int6} \n \\end{align}    \n \n \n\n\n\\begin{align}  \n \\bl(k,i)\\geq  \\dclrF(\\widetilde{\\tC} + i)+\\chiT(i,k)-1 , ~~~\n ~~~~~   k \\in[1,\\kC],   i\\in[1,\\tT], &&   \n \\end{align}   \n \n\\begin{align}  \n \\sum_{k \\in[1,\\kC],  i\\in[1,\\tT]} \\bl(k,i)\n \\leq \\sum_{i\\in[1,\\tT]}\\dclrF( \\widetilde{\\tC} +i),   &&   \n  \\label{eq:int12} \n \\end{align}   \n  \n  \\begin{align}  \n \\bl_\\LB(k)\\leq  \\sum_{ i\\in[1,\\tT]} \\bl(k,i) \\leq  \\bl_\\UB(k) , ~~~~~~\n     k \\in[1,\\kC], &&      \n       \\label{eq:int_last} \n \\end{align} \n \n  \n\\begin{align}  \n \\tC +\\sum_{i\\in [1,\\tT]} \\vT(i) + \\sum_{i\\in [1,\\tF]} \\vF(i) =\\nint_G.  &&  \n  \\label{eq:int_last} \n \\end{align}   \n \n  \n  \n \n\\subsection{Constraints for Including Fringe-trees} \\label{sec:ex\n \n Recall that   $\\mathcal{F}(D_\\pi)$ denotes the set of \nchemical rooted trees $\\psi$  \nr-isomorphic to a chemical rooted tree in $\\mathcal{T}(\\C)$\n  over all chemical graphs $\\C\\in D_\\pi$,\n  where possibly a chemical rooted tree $\\psi\\in \\mathcal{F}(D_\\pi)$\n  consists of a single chemical element $\\ta\\in \\Lambda\\setminus \\{{\\tt H}\\}$.\n\nTo express the condition that\nthe ${\\rho}$-fringe-tree is chosen from a rooted tree $C_i$, $T_i$  or  $F_i$, \nwe introduce the following set of variables and constraints.  \n  \n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate}[leftmargin=*\n\\item[-]   $n_\\LB$: a lower bound  \non the number $n(\\C)$ of non-hydrogen atoms in $\\C$,\nwhere $n_\\LB, n^*\\geq \\nint_\\LB$;  \n\n\\item[-]   $\\ch_{\\LB}(i),\\ch_{\\UB}(i)\\in [0,n^* ]$, $i\\in [1,\\tT]$: \nlower and upper bounds on $\\h(\\langle T_i\\rangle)$ of the tree $T_i$ rooted \nat a vertex $\\vC_{i}$; \n\n\\item[-]   $\\ch_{\\LB}(k),\\ch_{\\UB}(k)\\in [0,n^* ]$, $k \\in[1,\\kC]= \\It\\cup\\Iw$: \nlower and upper bounds on the maximum  \n height $\\h(\\langle T \\rangle)$ of the tree $T\\in \\F(P_k)$ rooted at \nan internal vertex of a path $P_k$   for an edge $a_k\\in \\Ew\\cup \\Et$;  \n\n\n\\item[-]  Prepare a coding of the set  $\\mathcal{F}(D_\\pi)$ and let \n    $[\\psi]$ denote  the coded integer of  \n     an element $\\psi$ in $\\mathcal{F}(D_\\pi)$;  \n\n\\item[-]   Sets  $\\mathcal{F}(v) \\subseteq \\mathcal{F}(D_\\pi), v\\in \\VC$\nand $\\mathcal{F}_E \\subseteq \\mathcal{F}(D_\\pi)$ \n of chemical rooted trees $T$ with $\\h(T)\\in [1,{\\rho}]$;  \n \n\\item[-]  Define\n$\\mathcal{F}^*:=\\bigcup_{v\\in \\VC}\\mathcal{F}(v)\\cup \\mathcal{F}_E$, \n $\\FrC_i:= \\mathcal{F}(\\vC_i)$, $i\\in[1,\\tC]$,\n$\\FrT_i:= \\mathcal{F}_E$, $i\\in[1,\\tT]$  and \n$\\FrF_i:= \\mathcal{F}_E$, $i\\in[1,\\tF]$;\n\n\\item[-]    \n $\\fc_\\LB(\\psi),\\fc_\\UB(\\psi)\\in[0,n^*], \\psi\\in \\mathcal{F}^*$:\nlower and upper bound functions  \n  on the number of   interior-vertices $v$ \n  such that $\\C[v]$ is r-isomorphic to $\\psi $  in $\\C$;\n  \n\\item[-]  \n$\\FrX_i[p], p\\in [1,{\\rho}], \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\}$:\nthe set of  chemical rooted trees  $T\\in  \\FrX_i$\nwith   $\\h(\\langle T\\rangle)= p$;  \n\n\\item[-]  \n$n_{\\oH}([\\psi])\\in [0, 3^{\\rho}], \\psi\\in \\mathcal{F}^*$: \nthe number $n(\\langle \\psi\\rangle)$ \nof non-root hydrogen vertices in a chemical rooted tree  $\\psi$; \n \n\\item[-]  \n$\\h_{\\oH}([\\psi])\\in [0,{\\rho}], \\psi\\in \\mathcal{F}^*$: \n the height $\\h(\\langle \\psi\\rangle)$  of the\n hydrogen-suppressed chemical rooted tree  $\\langle \\psi\\rangle$; \n\n\\item[-]  \n$\\deg_\\mathrm{r}^{\\oH}([\\psi])\\in [0,3], \\psi\\in \\mathcal{F}^*$: \nthe number $\\deg_\\mathrm{r}(\\anpsi)$ of non-hydrogen children of the root $r$\n of a chemical rooted tree  $\\psi$; \n \n\\item[-]  \n$\\deghyd_\\mathrm{r}([\\psi])\\in [0,3], \\psi\\in \\mathcal{F}^*$: \nthe number $\\deg_\\mathrm{r}(\\psi)-\\deg_\\mathrm{r}(\\anpsi)$ \nof hydrogen children of the root $r$ of a chemical rooted tree  $\\psi$; \n \n\\item[-] \n$\\vion(\\psi)\\in [-3,+3], \\psi\\in \\mathcal{F}^*$: \n  the ion-valence of the root in  $\\psi$; \n \n  \n\\item[-] \n  $\\ac^\\lf_\\nu(\\psi), \\nu\\in \\Gac^\\lf$:\nthe frequency of leaf-edges with adjacency-configuration $\\nu$ in $\\psi$;\n \n  \n \\item[-] \n$\\ac^\\lf_\\LB,\\ac^\\lf_\\UB: \\Gac^\\lf \\to  [0,n^*]$:\nlower and upper bound functions    on the number of  leaf-edges $uv$ in $\\acC$\n  with adjacency-configuration $\\nu$; \n\\end{enumerate}\n\n\\smallskip\\noindent\n{\\bf variables: }  \n\\begin{enumerate}[leftmargin=*\n\\item[-]\n  $n_G\\in [n_\\LB, n^*]$: the number $n(\\C)$ of non-hydrogen atoms in $\\C$;  \n %\n\\item[-] $\\vX(i)\\in[0,1], i\\in [1,\\tX]$,   $\\mathrm{X}\\in\\{\\mathrm{T,F}\\}$: \n $\\vX(i)=1$ $\\Leftrightarrow $ vertex $\\vX_{i}$ is used in $\\C$; \n       \n\\item[-] \n$\\dlfrX(i,[\\psi])\\in [0,1], \n    i\\in[1,\\tX], \\psi\\in \\FrX_i, \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$:  \n$\\dlfrX(i,[\\psi])=1$  $\\Leftrightarrow $\n $\\psi$ is  the ${\\rho}$-fringe-tree rooted at vertex $\\vX_i$ in $\\C$;  \n\n\\item[-]    \n$\\fc([\\psi])\\in [\\fc_\\LB(\\psi),\\fc_\\UB(\\psi)], \\psi\\in \\mathcal{F}^*$:\n  the number of   interior-vertices $v$ \n  such that $\\C[v]$ is r-isomorphic to $\\psi$  in $\\C$;  \n  \n\\item[-]    \n$\\ac^\\lf([\\nu])\\in [\\ac^\\lf_\\LB(\\nu),\\ac^\\lf_\\UB(\\nu)], \\nu\\in \\Gac^\\lf$: \n  the number of leaf-edge with adjacency-configuration $\\nu$  in $\\C$;  \n  \n\\item[-]\n $\\degXex(i)\\in [0,3],  i\\in [1,\\tX],     \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\}$:\nthe number of non-hydrogen children of the root\n of  the ${\\rho}$-fringe-tree rooted at vertex $\\vX_i$ in $\\C$;  \n  \n\\item[-]  $\\hyddegX(i)\\in [0,4]$,  $i\\in [1,\\tX]$, \n $\\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$: \n the number of  hydrogen atoms adjacent to  vertex $\\vX_{i}$\n (i.e.,  $\\hyddeg(\\vX_{i})$) in $\\C=(H,\\alpha,\\beta)$; \n \n\\item[-] \n $\\eledegX(i)\\in [-3,+3]$,  $i\\in [1,\\tX]$, \n $\\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$: \n the  ion-valence $\\vion(\\psi)$ of vertex $\\vX_{i}$\n (i.e.,  $\\eledegX(i)=\\vion(\\psi)$ \n for the ${\\rho}$-fringe-tree $\\psi$ rooted at $\\vX_{i}$) in $\\C=(H,\\alpha,\\beta)$; \n\n \n\\item[-] $\\hX(i)\\in [0,{\\rho}]$, $i\\in [1,\\tX]$,\n$\\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$: the height $\\h(\\langle T\\rangle)$ of\nthe hydrogen-suppressed chemical rooted tree $\\langle T\\rangle$ of  \n the ${\\rho}$-fringe-tree $T$ rooted at vertex $\\vX_i$ in $\\C$;  \n\\item[-] $\\sigma(k,i)\\in[0,1]$, $k \\in[1,\\kC]=\\It\\cup\\Iw,  i\\in [1,\\tT]$: \n    $\\sigma(k,i)=1$ $\\Leftrightarrow$ \n    the ${\\rho}$-fringe-tree $T_v$ rooted at  vertex $v=\\vT_{i}$ \n      with color $k$  has the largest height $\\h(\\langle \\T_v \\rangle)$ among such trees\n      $T_v, v\\in \\VT$;\n\\end{enumerate}\n\n\\smallskip\\noindent\n{\\bf constraints: }    \n\\begin{align}    \n\\sum_{\\psi\\in \\FrC_i}\\!\\!\\dlfrC(i,[\\psi]) =1, &&  i\\in [1,\\tC], \\notag \\\\\n\\sum_{\\psi\\in \\FrX_i }\\!\\!\\dlfrX(i,[\\psi]) =\\vX(i),  \n&&   i\\in [1,\\tX],  \\mathrm{X}\\in\\{\\mathrm{T,F}\\},  \\label{eq:ex_first}   \n\\end{align}     \n \n\\begin{align}    \n\\sum_{\\psi\\in \\FrX_i }\\!\\! \\deg_\\mathrm{r}^{\\oH}([\\psi]) \\cdot \\dlfrX(i,[\\psi]) \n = \\degXex(i),  &&  \\notag \\\\\n\\sum_{\\psi\\in \\FrX_i }\\!\\!   \\deghyd_\\mathrm{r}([\\psi]) \\cdot \\dlfrX(i,[\\psi])\n = \\hyddegX(i),   &&  \\notag \\\\\n\\sum_{\\psi\\in \\FrX_i }\\!\\! \\vion([\\psi]) \\cdot \\dlfrX(i,[\\psi]) \n = \\eledegX(i),\n&&   i\\in [1,\\tX],  \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\},  \\label{eq:ex_1}  \n\\end{align}    \n\n  \n \n\\begin{align}    \n\\sum_{\\psi\\in \\FrF_i[{\\rho}] }\\dlfrF(i,[\\psi]) \n\\geq \\vF(i) - \\eF(i+1),\n      && i\\in [1,\\tF]~(\\eF(\\tF+1)=0), \\label{eq:ex3}  \n\\end{align}   \n \n\\begin{align}   \n\\sum_{\\psi\\in \\FrX_i } \\h_{\\oH}([\\psi]) \\cdot \\dlfrX(i,[\\psi]) =  \\hX(i), && \n   i\\in[1,\\tX],  \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}, \\label{eq:ex5}  \n\\end{align}   \n\n\\begin{align}     \n\\sum\\limits_{\\substack{  \\psi\\in \\FrX_i \\\\\n              i\\in [1,\\tX],    \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}  }}\\!\\!\n                 n_{\\oH}([\\psi])  \\cdot \\dlfrX(i,[\\psi])    \n +    \\sum_{  i\\in [1,\\tX], \\mathrm{X}\\in \\{\\mathrm{T,F}\\} } \\vX(i)\n   +\\tC\n  = n_G, ~~\n &&\n  \\label{eq:ex2} \n\\end{align}   \n\n\\begin{align} \n\\sum_{ i\\in [1,\\tX], \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\}} \n\\dlfrX(i,[\\psi]) =  \\fc([\\psi]), &&  \\psi\\in \\mathcal{F}^*, \n \\label{eq:ex3} \n\\end{align}    \n\n\\begin{align}    \n\\sum_{\\psi\\in \\FrX_i, i\\in[1,\\tX], \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}}\n\\ac^\\lf_\\nu(\\psi)\\cdot \\dlfrX(i,[\\psi]) = \\ac^\\lf([\\nu]), &&\n\\nu\\in \\Gac^\\lf, \\label{eq:ex4}  \n\\end{align}   \n\n\n\n\\begin{align}  \n\\hC(i)    \\geq \\ch_\\LB(i)- n^* \\cdot \\dclrF(i),  ~~\n\\clrF(i)+{\\rho} \\geq \\ch_\\LB(i) , ~~~~~~~~~~~~~~   &&\\notag \\\\\n\\hC(i)          \\leq \\ch_\\UB(i) ,  ~~\n\\clrF(i)+{\\rho} \\leq \\ch_\\UB(i)+ n^* \\cdot (1-\\dclrF(i)),  ~ \n        &&   i\\in [1,\\widetilde{\\tC}],         \n            \\label{eq:int14} \n \\end{align}   \n \n\\begin{align}  \n \\ch_\\LB(i) \\leq  \\hC(i)   \\leq  \\ch_\\UB(i) ,  ~~ \n        &&    i\\in [\\widetilde{\\tC}+1,\\tC],       \n             \\label{eq:int14} \n \\end{align}   \n \n\\begin{align}   \n \\hT(i)    \\leq \\ch_\\UB(k) \n  + n^*\\cdot (\\dclrF( \\widetilde{\\tC}+ i)+1-\\chiT(i,k)),  &&\\notag \\\\\n\\clrF(\\widetilde{\\tC}+i)+{\\rho}\n \\leq \\ch_\\UB(k)+ n^*\\cdot (2-\\dclrF( \\widetilde{\\tC}+ i)-\\chiT(i,k)),    \n&& k \\in[1,\\kC],  i\\in [1,\\tT],      \n    \\label{eq:int15} \n \\end{align}   \n \n\\begin{align}   \n \\sum_{i\\in[1,\\tT]}\\sigma(k,i) =\\dclrT(k),   &&   k \\in[1,\\kC],      \n \\label{eq:int16} \n \\end{align}   \n \n\\begin{align}  \n \\chiT(i,k)\\geq \\sigma(k,i), && \\notag\\\\\n\n \\hT(i)    \\geq \\ch_\\LB(k) - n^*\\cdot (\\dclrF( \\widetilde{\\tC}+ i)+1-\\sigma(k,i) ),\n  && \\notag\\\\ \n\\clrF(\\widetilde{\\tC}+i)+{\\rho}\n \\geq \\ch_\\LB(k) - n^* \\cdot (2-\\dclrF( \\widetilde{\\tC}+ i)-\\sigma(k,i)),   \n&&  k \\in[1,\\kC],  i\\in [1,\\tT]. \n    \\label{eq:ex_last} \n \\end{align}   \n\n \n\\subsection{Descriptor for the  Number of Specified Degree} \n\\label{sec:Deg\n\nWe include constraints to compute descriptors for degrees in $\\C$. \\\\\n \n\n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n\\item[-]  $\\degX(i)\\in [0,4]$,  $i\\in [1,\\tX]$, \n $\\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$: \n the number of non-hydrogen atoms adjacent to  vertex $v=\\vX_{i}$\n (i.e.,  $\\deg_{\\anC}(v)=\\deg_H(v)-\\hyddeg_{\\C}(v)$) in $\\C=(H,\\alpha,\\beta)$; \n\n\\item[-] $\\degCT(i)\\in [0,4]$,  $i\\in [1, \\tC]$: the number of edges\nfrom vertex $\\vC_{i}$ to vertices $\\vT_{j}$, $j\\in [1,\\tT]$;  \n\\item[-]  $\\degTC(i)\\in [0,4]$,  $i\\in [1, \\tC]$: the number of edges\nfrom  vertices $\\vT_{j}$, $j\\in [1,\\tT]$ to vertex $\\vC_{i}$;    \n\\item[-]    $\\ddgC(i,d)\\in[0,1]$,  $i\\in [1,\\tC]$, $d\\in [1,4]$, \n  $\\ddgX(i,d)\\in[0,1]$,  $i\\in [1,\\tX]$,\n $d\\in [0,4]$,  $\\mathrm{X}\\in \\{\\mathrm{T,F}\\}$: \n        $\\ddgX(i,d)=1$ $\\Leftrightarrow$   $\\degX(i)+\\hyddegX(i)=d$;  \n       \n\\item[-]   $\\dg(d)\\in[\\dg_\\LB(d),\\dg_\\UB(d)]$,  $d \\in[1,4]$:\n    the number  of interior-vertices $v$ with \n       $\\mathrm{deg}_H(\\vX_{i})=d$   in $\\C=(H,\\alpha,\\beta)$; \n       \n       \n\\item[-] $\\degCint(i)\\in [1,4]$,  $i\\in [1, \\tC]$, \n $\\degXint(i)\\in [0,4]$,  $i\\in [1, \\tX], \\mathrm{X}\\in \\{\\mathrm{T,F}\\}$: \nthe interior-degree $\\deg_{H^\\inte}(\\vX_i)$ \n  in the interior $H^\\inte=(V^\\inte(\\C),E^\\inte(\\C))$ of  $\\C$; i.e., \nthe number of interior-edges incident to vertex $\\vX_{i}$;\n\n\\item[-]    $\\ddgCint(i,d)\\in[0,1]$,  $i\\in [1,\\tC]$,  $d\\in [1,4]$,  \n  $\\ddgXint(i,d)\\in[0,1]$,  $i\\in [1,\\tX]$,\n $d\\in [0,4]$,  $\\mathrm{X}\\in \\{\\mathrm{T,F}\\}$: \n       $\\ddgXint(i,d)=1$ $\\Leftrightarrow$   $\\degXint(i)=d$;  \n       \n\\item[-]   $\\dg^\\inte(d)\\in[\\dg_\\LB(d),\\dg_\\UB(d)]$,  $d \\in[1,4]$:\n    the number  of interior-vertices $v$ with\n    the interior-degree  $\\deg_{H^\\inte}(v)=d$\n  in the interior $H^\\inte=(V^\\inte(\\C),E^\\inte(\\C))$ of  $\\C=(H,\\alpha,\\beta)$.\n  \n\\end{enumerate}\n   \n\\smallskip\\noindent\n{\\bf constraints: }   \n\\begin{align}   \n\\sum_{   k\\in \\It^+(i)\\cup \\Iw^+(i)} \\dclrT(k) = \\degCT(i), ~~\n \\sum_{   k\\in \\It^-(i)\\cup \\Iw^-(i)} \\dclrT(k) = \\degTC(i), \n    &&    i\\in [1, \\tC],     \\label{eq:Deg_first}  \n\\end{align}   \n  \n\n\\begin{align}   \n\\tldgC^-(i)+\\tldgC^+(i)   + \\degCT(i)  + \\degTC(i) + \\dclrF(i) = \\degCint(i),  \n    &&    i\\in [1, \\widetilde{\\tC}],     \\label{eq:Deg2}  \n\\end{align}   \n\n\\begin{align}      \n\\tldgC^-(i)+\\tldgC^+(i)  + \\degCT(i)  + \\degTC(i)   = \\degCint(i),  \n     &&   i\\in [\\widetilde{\\tC}+1,\\tC],     \\label{eq:Deg2b}  \n\\end{align}   \n\n\\begin{align}      \n  \\degCint(i)+ \\degCex(i) = \\degC(i),  \n    &&    i\\in [1, \\tC],     \\label{eq:Deg2c}  \n\\end{align}   \n\n\\begin{align}    \n\\sum_{\\psi\\in \\FrC_i[{\\rho}] }\\dlfrC(i,[\\psi]) \\geq 2-\\degCint(i)\n      &&  i\\in [1, \\tC],    \\label{eq:Deg2d} \n\\end{align}   \n\n\\begin{align}   \n  2\\vT(i)   + \\dclrF(\\widetilde{\\tC}+i)    =\\degTint(i),   && \\notag \\\\\n \\degTint(i)+ \\degTex(i)  =\\degT(i),   && \n  i\\in [1,\\tT]~(\\eT(1)=\\eT(\\tT+1)=0), \\label{eq:Deg3}  \n %\n\\end{align}   \n\n\\begin{align}\n   \\vF(i) +\\eF(i+1)  =\\degFint(i),  && \\notag \\\\\n   \\degFint(i)  +\\degFex(i)   = \\degF(i),   && \n  i\\in [1,\\tF] ~(\\eF(1)=\\eF(\\tF+1)=0),   \\label{eq:Deg4}  \n\\end{align} \n\n\\begin{align}   \n\\sum_{d\\in [0,4]}\\ddgX(i,d)=1, ~\n\\sum_{d\\in [1,4]}d\\cdot\\ddgX(i,d)=\\degX(i)+\\hyddegX(i), && \\notag \\\\\n\\sum_{d\\in [0,4]}\\ddgXint(i,d)=1, ~\n\\sum_{d\\in [1,4]}d\\cdot\\ddgXint(i,d)=\\degXint(i), && \n i\\in [1,\\tX],  \\mathrm{X}\\in \\{\\mathrm{T, C, F}\\}, \\label{eq:Deg5}  \n\\end{align}   \n \n\\begin{align}   \n\\sum_{ i\\in [1,\\tC]} \\ddgC(i,d) + \\sum_{ i\\in [1,\\tT]} \\ddgT(i,d) \n + \\sum_{ i\\in [1,\\tF] }  \\ddgF(i,d) = \\dg(d),   &&   \\notag \\\\\n\\sum_{ i\\in [1,\\tC]} \\ddgCint(i,d) + \\sum_{ i\\in [1,\\tT]} \\ddgTint(i,d) \n + \\sum_{ i\\in [1,\\tF] }  \\ddgFint(i,d) = \\dg^\\inte(d),     \n    && d\\in [1,4].  \n  \\label{eq:Deg_last}  \n\\end{align}   \n\n \n\\subsection{Assigning Multiplicity}\n\\label{sec:beta\n\n We prepare an integer variable $\\beta(e)$  \n for each edge $e$ in the scheme graph $\\mathrm{SG}$ \n to denote the bond-multiplicity of $e$ in a selected graph $H$ and\n include necessary constraints for the variables to satisfy in $H$. \n \n\\smallskip\\noindent\n{\\bf constants: }\n\\begin{enumerate}[leftmargin=*\n\\item[-]\n$\\betar([\\psi])$: the sum $\\beta_\\psi(r)$ of bond-multiplicities of edges\nincident to  the root $r$ of a chemical rooted tree $\\psi\\in \\mathcal{F}^*$; \n\\end{enumerate}\n\n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n\\item[-] $\\bX(i)\\in [0,3]$,   $i\\in [2,\\tX]$, $\\mathrm{X}\\in \\{\\mathrm{T,F}\\}$:   \n the bond-multiplicity of edge  $\\eX_{i}$ in $\\C$;  \n \n\\item[-] $\\bC(i)\\in [0,3]$,     $i\\in [\\widetilde{\\kC}+1,\\mC]= \\Iw\\cup \\Iz\\cup\\Iew$:    \n     the bond-multiplicity of \n     edge  $a_{i}\\in \\Ew\\cup \\Ez\\cup\\Eew$ in $\\C$;        \n\\item[-]\n   $\\bCT(k), \\bTC(k)\\in [0,3]$, $k\\in [1, \\kC]=\\It\\cup \\Iw$: \n   the bond-multiplicity of the first (resp., last) edge of the pure path $P_k$ in $\\C$;    \n   \n\\item[-]\n   $\\bsF(c)\\in [0,3], c\\in [1,\\cF=\\widetilde{\\tC} + \\tT ]$:  \n   the bond-multiplicity of the first edge of the leaf path $Q_c$\n   rooted at vertex $\\vC_{c}, c\\leq\\widetilde{\\tC} $\n    or $\\vT_{c-\\widetilde{\\tC}}, c>\\widetilde{\\tC} $  in $\\C$;   \n    \n\\item[-] $\\bXex(i)\\in [0,4],  i\\in [1,\\tX],   \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\}$:\nthe sum $\\beta_{\\C[v]}(v)$ of bond-multiplicities of edges in the ${\\rho}$-fringe-tree\n$\\C[v]$ rooted at  interior-vertex $v=\\vX_{i}$;  \n\n\\item[-] $\\delbX(i,m)\\in [0,1]$, $i\\in [2,\\tX]$,   $m\\in[0,3]$, \n       $\\mathrm{X}\\in \\{\\mathrm{T,F}\\}$:  \n  $\\delbX(i,m)=1$  $\\Leftrightarrow$  $\\bX(i)=m$; \n\\item[-] $\\delbC(i,m)\\in [0,1]$,  \n   $i\\in [\\widetilde{\\kC},\\mC]=\\Iw\\cup \\Iz\\cup\\Iew$,  $m\\in[0,3]$:  \n   $\\delbC(i,m)=1$  $\\Leftrightarrow$  $\\bC(i)=m$; \n %\n\\item[-]\n   $\\delbCT(k,m), \\delbTC(k,m)\\in [0,1]$, $k\\in [1, \\kC]=\\It\\cup \\Iw$,  $m\\in[0,3]$:\n     $\\delbCT(k,m)=1$   (resp., $\\delbTC(k,m)=1$)    $\\Leftrightarrow$  \n           $\\bCT(k)=m$ (resp., $\\bTC(k)=m$); \n           \n\\item[-]\n   $\\delbsF(c,m)\\in [0,1]$, $c\\in [1,\\cF]$,  \n    $m\\in[0,3], \\mathrm{X}\\in \\{\\mathrm{C,T}\\}$: \n     $\\delbsF(c,m)=1$ $\\Leftrightarrow$  $\\bsF(c)=m$;  \n\\item[-] $\\bd^\\inte(m)\\in[0, 2\\nint_\\UB]$, $m\\in[1,3]$:\n      the number of interior-edges with bond-multiplicity  $m$ in $\\C$;  \n      \n\\item[-] $\\bdX(m)\\in [0,2\\nint_\\UB],  \\mathrm{X}\\in \\{\\mathrm{C,T,CT,TC}\\}$,\n      $\\bdX(m)\\in [0,2\\nint_\\UB], \\mathrm{X}\\in \\{\\mathrm{F,CF,TF}\\}$, $m\\in[1,3]$:  \n the number of interior-edges $e\\in \\EX$ with bond-multiplicity  $m$ in  $\\C$; \n\\end{enumerate}\n \n\\smallskip\\noindent\n{\\bf constraints: } \n\\begin{align}    \n\\eC(i)\\leq \\bC(i)\\leq 3\\eC(i), \n  i\\in [\\widetilde{\\kC}+1,\\mC]=\\Iw\\cup \\Iz\\cup\\Iew, \\label{eq:beta_first} \n\\end{align}   \n\n\\begin{align}   \n  \\eX(i)\\leq \\bX(i)\\leq 3 \\eX(i), \n  &&    i\\in [2,\\tX],   \\mathrm{X}\\in \\{\\mathrm{T, F}\\},    \\label{eq:beta1}  \n\\end{align}   \n\n\\begin{align}   \n\\dclrT(k)\\leq \\bCT(k)\\leq 3 \\dclrT(k), ~~~ \n\\dclrT(k)\\leq \\bTC(k)\\leq 3 \\dclrT(k), &&  k\\in [1, \\kC], \\label{eq:beta8} \\\\ \n\\dclrF(c)\\leq \\bXF(c)\\leq 3 \\dclrF(c), &&   c\\in [1,\\cF], \\label{eq:beta8}  \n\\end{align}\n\n\\begin{align} \n\\sum_{m\\in[0,3]} \\delbX(i,m)=1,  ~~\n\\sum_{m\\in[0,3]}m\\cdot \\delbX(i,m)=\\bX(i), &&   i\\in [2,\\tX],  \n  \\mathrm{X}\\in \\{\\mathrm{T,F}\\},  \\label{eq:beta10}    \n\\end{align}   \n\n\\begin{align} \n\\sum_{m\\in[0,3]} \\delbC(i,m)=1,  ~~\n\\sum_{m\\in[0,3]}m\\cdot \\delbC(i,m)=\\bC(i), &&   i\\in [\\widetilde{\\kC}+1,\\mC],   \\label{eq:beta11}    \n\\end{align}   \n \n\\begin{align}   \n\\sum_{m\\in[0,3]} \\delbCT(k,m)=1,    ~~ \n\\sum_{m\\in[0,3]}m\\cdot\\delbCT(k,m)=\\bCT(k),\n&&     k\\in [1, \\kC],  \\notag \\\\      \n\\sum_{m\\in[0,3]} \\delbTC(k,m)=1,  ~~ \n\\sum_{m\\in[0,3]} m\\cdot\\delbTC(k,m)=\\bTC(k),  &&\n      k\\in [1, \\kC], \\notag \\\\\n\\sum_{m\\in[0,3]} \\delbsF(c,m)=1,  ~~ \n\\sum_{m\\in[0,3]} m\\cdot\\delbsF(c,m)=\\bsF(c),  &&    c\\in [1,\\cF],\n   \\label{eq:beta15}    \n\\end{align}    \n\n \\begin{align}           \n\\sum_{\\psi\\in \\FrX_i } \\betar([\\psi]) \\cdot  \\dlfrX(i,[\\psi]) = \\bXex(i),     \n &&  i\\in [1,\\tX],     \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\}, \n \\label{eq:beta16a}   \n\\end{align}    \n  \n\n\\begin{align} \n \\sum_{i\\in [\\widetilde{\\kC}+1,\\mC]} \\delbC(i,m) =\\bdC(m), ~~\n  \\sum_{i\\in [2,\\tT]} \\delbT(i,m)    =\\bdT(m),   \\notag \\\\ \n   \\sum_{k\\in [1, \\kC]}\\delbCT(k,m)=\\bdCT(m), ~~\n   \\sum_{k\\in [1, \\kC]}\\delbTC(k,m)=\\bdTC(m),    \\notag \\\\ \n %\n\\sum_{i\\in [2,\\tF]}\\!\\!\\! \\delbF(i,m) =\\bdF(m), ~~\n \\sum_{c\\in [1,\\widetilde{\\tC}]} \\delbsF(c,m)  =\\bdCF(m),   \\notag \\\\ \n  \\sum_{c\\in [\\widetilde{\\tC}+1,\\cF]}  \\delbsF(c,m) =\\bdTF(m),  \n   \\notag \\\\ \n \\bdC(m)+\\bdT(m) + \\bdF(m)\n +\\bdCT(m)+\\bdTC(m) +\\bdTF(m)+\\bdCF(m) = \\bd^\\inte(m),   \\notag \\\\ \n  m\\in [1,3].       \\label{eq:beta_last} \n\\end{align}   \n\n\n \n\\subsection{Assigning Chemical Elements and  Valence Condition}\n\\label{sec:alpha} \n\nWe include constraints so that each vertex $v$ in a selected graph $H$\nsatisfies the valence condition; i.e., \n$\\beta_\\C(v)=  \\val(\\alpha(v)) +\\eledeg_\\C(v)$, \nwhere $\\eledeg_\\C(v)=\\vion(\\psi)$ for the ${\\rho}$-fringe-tree $\\C[v]$\nr-isomorphic to $\\psi$. \nWith these constraints, a chemical graph\n   $\\C=(H,\\alpha,\\beta)$ on a selected subgraph $H$\nwill be constructed. \n \n\\smallskip\\noindent\n{\\bf constants: }\n \\begin{enumerate}[leftmargin=*\n\\item[-] Subsets\n $\\Lambda^\\inte \\subseteq \\Lambda\\setminus\\{{\\tt H}\\}, \n \\Lambda^\\ex \\subseteq \\Lambda$ of chemical elements,\n where we denote by $[{\\tt e}]$ (resp., $[{\\tt e}]^\\inte$ and $[{\\tt e}]^\\ex$)  \n of a standard encoding of an element ${\\tt e}$ in the set $\\Lambda$ \n (resp.,    $\\Lambda^\\inte_\\epsilon$ and  $\\Lambda^\\ex_\\epsilon$);  \n\\item[-]  A valence function: $\\val: \\Lambda \\to [1,6]$;  \n\n\\item[-]  A function $\\mathrm{mass}^*:\\Lambda\\to \\mathbb{Z}$ \n(we let $\\mathrm{mass}(\\ta)$ denote  the observed mass of a chemical element  \n$\\ta\\in \\Lambda$, and define \n   $\\mathrm{mass}^*(\\ta)\\triangleq\n    \\lfloor 10\\cdot \\mathrm{mass}(\\ta)\\rfloor$);  \n        \n\\item[-]  \n Subsets $\\Lambda^*(i)\\subseteq \\Lambda^\\inte$, $i\\in[1,\\tC]$; \n \n\\item[-] \n $\\na_\\LB(\\ta),\\na_\\UB(\\ta)\\in [0,n^* ]$,  $\\ta\\in  \\Lambda$:\nlower and upper bounds on the number of vertices  $v$    with $\\alpha(v)=\\ta$;  \n\\item[-] \n  $\\na_\\LB^\\inte(\\ta),\\na_\\UB^\\inte(\\ta)\\in [0,n^* ]$,\n $\\ta\\in  \\Lambda^\\inte$:\nlower and upper bounds on the number  of interior-vertices  \n $v$ with $\\alpha(v)=\\ta$; \n\n\\item[-] \n$\\alpha_\\mathrm{r}([\\psi])\\in [\\Lambda^\\ex], \\in \\mathcal{F}^*$:\n the chemical element $\\alpha(r)$  of the root $r$ of  $\\psi$;\n\n\\item[-]   $\\na_\\ta^\\ex([\\psi])\\in [0,n^*]$,  \n$\\ta\\in \\Lambda^\\ex, \\psi\\in \\mathcal{F}^*$: \nthe frequency of chemical element $\\ta$ in the set of  \nnon-rooted vertices   in   $\\psi$, where possibly $\\ta={\\tt H}$;\n\n\n\\item[-]  A positive integer $\\mathrm{M}\\in \\Z_+$: \nan upper bound for the average $\\overline{\\mathrm{ms}}(\\C)$ of mass$^*$ \nover all atoms in $\\C$;\n\n\\end{enumerate}\n      \n\\smallskip\\noindent\n{\\bf variables: } \n\\begin{enumerate}[leftmargin=*\n\\item[-]\n   $\\bCT(i),\\bTC(i)\\in [0,3], i\\in [1,\\tT]$:\nthe bond-multiplicity of edge $\\eCT_{j,i}$ (resp., $\\eTC_{j,i}$)\nif one exists;  \n\n\\item[-] \n $\\bCF(i), \\bTF(i)\\in [0,3], i\\in [1,\\tF]$:\nthe bond-multiplicity of $\\eCF_{j,i}$ (resp., $\\eTF_{j,i}$)\nif one exists;  \n\n\\item[-]  $\\aX(i)\\in [\\Lambda^\\inte_\\epsilon ],\n       \\delaX(i,[\\ta]^\\inte)\\in [0,1],  \\ta\\in \\Lambda^\\inte_\\epsilon, i\\in [1,\\tX],\n         \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$:  \n$\\aX(i)= [\\ta]^\\inte\\geq 1$  (resp., $\\aX(i)=0$)\n  $\\Leftrightarrow$ $\\delaX(i,[\\ta]^\\inte)=1$ (resp., $\\delaX(i,0)=0$)  \n  $\\Leftrightarrow$ $\\alpha(\\vX_{i})= \\ta\\in \\Lambda$ \n(resp., vertex $\\vX_{i}$ is not used in $\\C$); \n\n\\item[-] \n$\\delaX(i,[\\ta]^\\inte)\\in [0,1], i\\in [1,\\tX],  \n\\ta  \\in \\Lambda^\\inte,   \\mathrm{X}\\in\\{\\mathrm{C,T,F}\\}$:   \n   $\\delaX(i,[\\ta]^\\typ)=1$   $\\Leftrightarrow$   $\\alpha(\\vX_{i})=\\ta$;  \n\n\\item[-]  $\\mathrm{Mass}\\in \\mathbb{Z}_+$: \n $\\sum_{v\\in V(H)} \\mathrm{mass}^*(\\alpha(v))$;  \n \n\\item[-]  $\\overline{\\mathrm{ms}}\\in \\mathbb{R}_+$: \n $\\sum_{v\\in V(H)} \\mathrm{mass}^*(\\alpha(v)) \/ |V(H)|$;  \n \\item[-] \n$\\delta_{\\mathrm{atm}}(i)\\in [0,1], i\\in [n_\\LB + \\na_\\LB({\\tt H}), \nn^* + \\na_\\UB({\\tt H})]$:   \n   $\\delta_{\\mathrm{atm}}(i)=1$   $\\Leftrightarrow$   $|V(H)| = i$;  \n\n\\item[-]   $\\na([\\ta])\\in[\\na_\\LB(\\ta),\\na_\\UB(\\ta)]$,\n $\\ta \\in \\Lambda$:\n    the number  of vertices $v\\in V(H)$\n     with $\\alpha(v)=\\ta$, where possibly $\\ta={\\tt H}$; \n    \n\\item[-]   $\\na^{\\inte}([\\ta]^\\inte) \\in[\\na_\\LB^\\inte(\\ta),\\na_\\UB^\\inte(\\ta)]$,\n $\\ta \\in \\Lambda, \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$:\n    the number  of interior-vertices  $v\\in V(\\C)$ \n    with $\\alpha(v)=\\ta$; \n    \n\\item[-]   $\\naX^\\ex([\\ta]^\\ex) , \\na ^\\ex([\\ta]^\\ex) \\in [0,\\na_\\UB(\\ta)]$,\n $\\ta \\in \\Lambda$,   $\\mathrm{X}\\in \\{\\mathrm{C,T,F}\\}$: \n    the number    of   exterior-vertices rooted at vertices $v\\in\\VX$ \n  and  the number    of   exterior-vertices $v$\n     such that  $\\alpha(v)=\\ta$;\n     \n\n\\end{enumerate} \n    \n\\smallskip\\noindent\n{\\bf constraints: } \n\\begin{align}    \n \\bCT(k)-3(\\eT(i)-\\chiT(i,k)+1) \\leq \n\\bCT(i)\\leq \\bCT(k)+3(\\eT(i)-\\chiT(i,k)+1),  i\\in [1,\\tT], && \\notag\\\\   \n  \\bTC(k)-3(\\eT(i+1)-\\chiT(i,k)+1) \\leq \n\\bTC(i)\\leq \\bTC(k)+3(\\eT(i+1)-\\chiT(i,k)+1),  i\\in [1,\\tT], && \\notag\\\\ \n   k\\in [1, \\kC],   &&  \\label{eq:alpha_first}  \n\\end{align}\n\n\\begin{align}   \n \\bsF(c)-3(\\eF(i)-\\chiF(i,c)+1) \\leq \n\\bCF(i)\\leq \\bsF(c)+3(\\eF(i)-\\chiF(i,c)+1),   i\\in [1,\\tF], \n&&  c\\in[1,\\widetilde{\\tC}] ,    \\notag\\\\  \n  \\bsF(c)-3(\\eF(i)-\\chiF(i,c)+1) \\leq \n\\bTF(i)\\leq \\bsF(c)+3(\\eF(i)-\\chiF(i,c)+1),    i\\in [1,\\tF], \n&& c\\in[\\widetilde{\\tC}+1,\\cF] ,    \\notag\\\\  \n  \\label{eq:alpha2}  \n\\end{align}\n\n\\begin{align}  \n   \\sum_{\\ta\\in \\Lambda^\\inte} \\delaC(i,[\\ta]^\\inte)=1, ~~ \n   \\sum_{\\ta\\in \\Lambda^\\inte} [\\ta]^\\inte\\cdot\\delaX(i,[\\ta]^\\inte)=\\aC(i),   \n     &&  i\\in [1,\\tC], \\notag  \\\\\n  \\sum_{\\ta\\in \\Lambda^\\inte } \\delaX(i,[\\ta]^\\inte)=\\vX(i), ~~ \n   \\sum_{\\ta\\in \\Lambda^\\inte} [\\ta]^\\inte\\cdot\\delaX(i,[\\ta]^\\inte)=\\aX(i), \n    &&  i\\in [1,\\tX],  \\mathrm{X}\\in \\{\\mathrm{T,F}\\},  \n  \\label{eq:alpha_first} \n\\end{align}    \n\n\n\\begin{align}  \n\\sum_{\\psi\\in \\FrX_i } \n \\alpha_\\mathrm{r}([\\psi])\\cdot \\dlfrX(i,[\\psi]) = \\aX(i),   \n &&   i\\in [1,\\tX],   \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\},  \n  \\label{eq:alpha_1} \n\\end{align}    \n  \n\n\n\\begin{align}  \n\\sum_{j\\in \\IC(i)}\\bC(j)  \n+ \\sum_{  k\\in \\It^+(i)\\cup \\Iw^+(i)} \\bCT(k)\n+ \\sum_{  k\\in \\It^-(i)\\cup \\Iw^-(i)} \\bTC(k)   &&  \\notag\\\\\n     + \\bsF(i)  +\\bCex(i) -\\eledegC(i) \n\n     =\n     \\sum_{\\ta\\in \\Lambda^\\inte}\\val(\\ta)\\delaC(i,[\\ta]^\\inte),  \n &&  i\\in [1,\\widetilde{\\tC}],  \\label{eq:alpha3} \n\\end{align}\n\n\\begin{align}   \n\\sum_{j\\in \\IC(i)}\\bC(j)  \n+ \\sum_{  k\\in \\It^+(i)\\cup \\Iw^+(i)} \\bCT(k)\n+ \\sum_{  k\\in \\It^-(i)\\cup \\Iw^-(i)} \\bTC(k)   &&  \\notag\\\\\n+\\bCex(i) -\\eledegC(i) \n\n   =\n      \\sum_{\\ta\\in \\Lambda^\\inte}\\val(\\ta)\\delaC(i,[\\ta]^\\inte),  \n  &&  i\\in [\\widetilde{\\tC}+1,\\tC],   \\label{eq:alpha3b} \n\\end{align} \n\n\\begin{align}  \n \\bT(i)+\\bT(i\\!+\\!1)   +   \\bTex(i)  \n  + \\bCT(i) + \\bTC(i) \\hspace{1cm} \\notag\\\\\n  + \\bsF(\\widetilde{\\tC}+i) -\\eledegT(i) \n\n   =\n      \\sum_{\\ta\\in \\Lambda^\\inte}\\val(\\ta)\\delaT(i,[\\ta]^\\inte), \n   \\notag \\\\\n  i\\in [1,\\tT]~  (\\bT(1)=\\bT(\\tT+1)=0),   \\label{eq:alpha4} \n\\end{align}\n \n\\begin{align} \n \\bF(i)+\\bF(i\\!+\\!1) +\\bCF(i) +\\bTF(i)  \\hspace{1cm}    \\notag\\\\\n  +\\bFex(i)  -\\eledegF(i) \n \n   =\n      \\sum_{\\ta\\in \\Lambda^\\inte}\\val(\\ta)\\delaF(i,[\\ta]^\\inte),  \n   \\notag \\\\\n  i\\in [1,\\tF] ~  (\\bF(1)=\\bF(\\tF+1)=0),   \\label{eq:alpha5} \n\\end{align}   \n   \n  \n\\begin{align}  \n \\sum_{i\\in [1,\\tX] } \\delaX(i,[\\ta]^\\inte) = \\naX([\\ta]^\\inte) ,  \n &&  \\ta\\in \\Lambda^\\inte, \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\},  \n     \\label{eq:alpha6} \n\\end{align}    \n\n\\begin{align}   \n\\sum_{\\psi\\in \\FrX_i , i\\in [1,\\tX] } \\na_\\ta^\\ex([\\psi])\\cdot  \\dlfrX(i,[\\psi])  \n   = \\naX^\\ex([\\ta]^\\ex),     && \n \\ta\\in \\Lambda^\\ex,  \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\},  \\label{eq:alpha6} \n\\end{align}    \n       \n\\begin{align}  \n \\naC([\\ta]^\\inte)+ \\naT([\\ta]^\\inte)+ \\naF([\\ta]^\\inte) =  \\na^\\inte([\\ta]^\\inte),  \n    &&   \\ta\\in \\Lambda^\\inte,       \\notag \\\\\n  \\sum_{  \\mathrm{X}\\in \\{\\mathrm{C,T,F}\\} }\n   \\naX^\\ex([\\ta]^\\ex)       =\\na^\\ex([\\ta]^\\ex),  \n  &&  \\ta\\in \\Lambda^\\ex,    \\notag \\\\\n  \\na^\\inte([\\ta]^\\inte) + \\na^\\ex([\\ta]^\\ex)=\\na([\\ta]),  \n &&   \\ta\\in \\Lambda^\\inte\\cap \\Lambda^\\ex,     \\notag \\\\\n  \\na^\\inte([\\ta]^\\inte)  =\\na([\\ta]),  \n &&      \\ta\\in \\Lambda^\\inte \\setminus \\Lambda^\\ex,     \\notag \\\\ \n   \\na^\\ex([\\ta]^\\ex)  =\\na([\\ta]),  \n &&      \\ta\\in \\Lambda^\\ex \\setminus \\Lambda^\\inte,        \n     \\label{eq:alpha6} \n\\end{align}    \n  \n \n \\begin{align}    \n \\sum_{\\ta\\in \\Lambda^*(i)} \\delaC(i,[\\ta]^\\inte) = 1,  \n  &&  i\\in [1,\\tC],   \\label{eq:alpha8} \n\\end{align}   \n\n\n\\begin{align}   \n\\sum_{ \\ta\\in\\Lambda }\\mathrm{mass}^*(\\ta )\\cdot \\na([\\ta])\n =\\mathrm{Mass}, &&    \\label{eq:alpha7} \n\\end{align}  \n\n\n\n \\begin{align}\n \\sum_{i \\in [n_\\LB + \\na_\\LB({\\tt H}), n^* + \\na_\\UB({\\tt H})]} \\delta_{\\mathrm{atm}}(i) = 1, && \\\\\n \\sum_{i \\in [n_\\LB + \\na_\\LB({\\tt H}), n^* + \\na_\\UB({\\tt H})]} i \\cdot \\delta_{\\mathrm{atm}}(i) \n = n_G + \\na^\\ex([{\\tt H}]^\\ex), &&  \\\\\n \\mathrm{Mass}  - \\mathrm{M}\\cdot (1 - \\delta_{\\mathrm{atm}}(i)) \n  \\leq i \\cdot \\overline{\\mathrm{ms}} \\leq\n   \\mathrm{Mass}  + \\mathrm{M}\\cdot (1 - \\delta_{\\mathrm{atm}}(i)), \n   && i \\in [n_\\LB + \\na_\\LB({\\tt H}), n^* + \\na_\\UB({\\tt H})]. \n  \\label{eq:alpha_last} \n \\end{align}\n \n\n\n  \n\\subsection{Constraints for Bounds on the Number of Bonds}  \n\\label{sec:BDbond\n\nWe include constraints for specification of lower and upper bounds\n$\\bd_\\LB$ and $\\bd_\\UB$. \n\n\\smallskip\\noindent\n{\\bf constants: } \n\\begin{enumerate}[leftmargin=*\n\\item[-]\n$\\bd_{m, \\LB}(i), \\bd_{m, \\UB}(i)\\in [0,\\nint_\\UB]$, \n$i\\in [1,\\mC]$,  $m\\in [2,3]$:  lower and upper bounds \n on the number  of edges $e\\in E(P_i)$ with bond-multiplicity $\\beta(e)=m$\n in the pure path $P_i$ for edge $e_i\\in \\EC$; \n\\end{enumerate}\n\n\n\\smallskip\\noindent\n{\\bf variables : } \n\\begin{enumerate}[leftmargin=*\n\\item[-]\n  $\\bdT(k,i,m)\\in [0,1]$, $k\\in [1, \\kC]$, $i\\in [2,\\tT]$, $m\\in [2,3]$:  \n  $\\bdT(k,i,m)=1$  $\\Leftrightarrow$ the pure path $P_k$ for edge $e_k\\in \\EC$ \n  contains edge $\\eT_i$ with $\\beta(\\eT_i)=m$; \n\\end{enumerate}\n  \n\\smallskip\\noindent\n{\\bf constraints: } \n\\begin{align}    \n\\bd_{m,\\LB}(i)\\leq \\delbC(i,m)\\leq \\bd_{m,\\UB}(i), \n  i\\in \\Iew\\cup \\Iz, m\\in [2,3], &&\n  \\label{eq:BDbond_first}  \n\\end{align}    \n\n    \n\\begin{align}   \n\\bdT(k,i,m)\\geq \\delbT(i,m)+\\chiT(i,k)-1, \n~~~ k \\in [1, \\kC], i\\in [2,\\tT],  m\\in [2,3], &&\n \\label{eq:BDbond2}  \n\\end{align}   \n \n\\begin{align}    \n\\sum_{j\\in[2,\\tT]}\\delbT(j,m) \\geq \n\\sum_{k\\in[1, \\kC], i\\in [2,\\tT]}\\!\\!\\!\\! \\bdT(k,i,m) , \n~~ m\\in [2,3],  \\label{eq:BDbond3}  \n\\end{align}    \n\n\\begin{align}    \n \\bd_{m, \\LB}(k) \\leq \n   \\sum_{i\\in [2,\\tT]}\\bdT(k,i,m) +\\delbCT(k,m)+\\delbTC(k,m)    \n   \\leq   \\bd_{m, \\UB}(k), ~~~~  \\notag \\\\\n    k\\in [1, \\kC],   m\\in [2,3]. ~~\n     \\label{eq:BDbond_last}  \n\\end{align}   \n\n\\subsection{Constraints for Normalization  of Feature Vectors} \n\\label{sec:NSFV\n\n\n\n  By introducing a tolerance $\\varepsilon>0$   in the conversion \n between integers and reals, we include the following constraints for \nnormalizing of a  feature vector   $x=(x(1),x(2),\\ldots,x(K))$: \n\\begin{equation}\\label{eq:normalization2}\n\\frac{(1 - \\varepsilon)(x(j)  - \\min(\\dcp_j;D_\\pi))}\n{ \\max(\\dcp_j;D_\\pi) - \\min(\\dcp_j;D_\\pi) }\n\\leq \\widehat{x}(j)   \\leq \n\\frac{(1 + \\varepsilon) ( x(j) - \\min(\\dcp_j;D_\\pi))}\n{ \\max(\\dcp_j;D_\\pi) - \\min(\\dcp_j;D_\\pi) }, \n~ j\\in [1,K].  \n\\end{equation} \nAn example of  a tolerance is $\\varepsilon=1\\times 10^{-5}$.\n\nWe use the same conversion for descriptor $x_j = \\overline{\\mathrm{ms}}$. \n\n \n  \n\n\n\\section{A Full Description of Descriptors}\\label{sec:descriptor\n\n \nAssociated with the two functions \n$\\alpha$ and $\\beta$ in a chemical graph $\\Co=(H,\\alpha,\\beta)$,\nwe introduce   functions  \n $\\ac: V(E)\\to (\\Lambda\\setminus\\{\\ttH\\})\\times (\\Lambda\\setminus\\{\\ttH\\})\\times [1,3]$, \n $\\cs: V(E)\\to (\\Lambda\\setminus\\{\\ttH\\})\\times [1,6]$ and\n$\\ec: V(E)\\to ((\\Lambda\\setminus\\{\\ttH\\})\\times [1,6])\\times ((\\Lambda\\setminus\\{\\ttH\\})\\times [1,6])\\times [1,3]$\nin the following. \n\n To represent  a feature of the exterior  of  $\\Co$, \n  a  chemical rooted tree in $\\mathcal{T}(\\Co)$ is\n  called a {\\em fringe-configuration} of $\\Co$. \n\nWe also represent leaf-edges in the exterior of $\\Co$.\nFor a leaf-edge $uv\\in E(\\anC)$ with $\\deg_{\\anC}(u)=1$, we define\nthe {\\em adjacency-configuration} of $e$ to be an ordered tuple\n$(\\alpha(u),\\alpha(v),\\beta(uv))$. \nDefine \n\\[ \\Gac^\\lf\\triangleq \\{(\\ta,\\tb,m)\\mid \\ta,\\tb\\in\\Lambda, \nm\\in[1,\\min\\{\\val(\\ta),\\val(\\tb)\\}]\\} \\]\nas a set of possible adjacency-configurations for leaf-edges. \n\nTo  represent a feature of an interior-vertex $v\\in V^\\inte(\\Co)$ such that\n$\\alpha(v)=\\ta$  and  $\\deg_{\\anC}(v)=d$\n(i.e., the number of non-hydrogen atoms adjacent to $v$ is $d$) \n   in a chemical   graph  $\\Co=(H,\\alpha,\\beta)$,\n we use  a pair $(\\ta, d)\\in (\\Lambda\\setminus\\{{\\tt H}\\})\\times [1,4]$,\n which we call the {\\em chemical symbol} $\\cs(v)$ of the vertex $v$.\n We treat $(\\ta, d)$ as a single symbol $\\ta d$,  and  \ndefine $\\Ldg$   to be  the set of all chemical symbols\n$\\mu=\\ta d\\in  (\\Lambda\\setminus\\{{\\tt H}\\})\\times [1,4]$.  \n\nWe define a method for featuring interior-edges  as follows.\nLet $e=uv\\in E^\\inte(\\Co)$  be \n an interior-edge $e=uv\\in E^\\inte(\\Co)$ \n such that $\\alpha(u)=\\ta$, $\\alpha(v)=\\tb$ and $\\beta(e)=m$ \n   in a chemical graph  $\\Co=(H,\\alpha,\\beta)$.\nTo feature this edge $e$, \n we use a tuple $(\\ta,\\tb,m)\\in (\\Lambda\\setminus\\{{\\tt H}\\})\n    \\times (\\Lambda\\setminus\\{{\\tt H}\\})\\times [1,3]$,\n which we call the {\\em adjacency-configuration} $\\ac(e)$ of the edge $e$. \n We introduce a total order $<$ over the elements in $\\Lambda$\n to distinguish  between $(\\ta,\\tb, m)$ and $(\\tb,\\ta, m)$ \n $(\\ta\\neq \\tb)$ notationally.\n For a tuple  $\\nu=(\\ta,\\tb, m)$,\n let $\\overline{\\nu}$ denote the tuple $(\\tb,\\ta, m)$. \n \n\nLet $e=uv\\in E^\\inte(\\Co)$  be \nan  interior-edge $e=uv\\in E^\\inte(\\Co)$ \n such that $\\cs(u)=\\mu$, $\\cs(v)=\\mu'$ and $\\beta(e)=m$ \n   in a chemical  graph  $\\Co=(H,\\alpha,\\beta)$.\nTo feature this edge $e$, \n we use a tuple $(\\mu,\\mu',m)\\in \\Ldg\\times \\Ldg\\times [1,3]$, \n which we call  the {\\em edge-configuration} $\\ec(e)$ of the edge $e$. \n %\n We introduce a total order $<$ over the elements in $\\Ldg$\n to distinguish between $(\\mu,\\mu', m)$ and $(\\mu', \\mu, m)$ \n $(\\mu \\neq \\mu')$ notationally. \n For a tuple  $\\gamma=(\\mu,\\mu', m)$,\n let $\\overline{\\gamma}$ denote the tuple $(\\mu', \\mu, m)$. \n\nLet $\\pi$ be a chemical property for which we will construct\na prediction function $\\eta$ from a feature\nvector $f(\\Co)$ of a chemical graph $\\Co$ \nto a predicted value $y\\in \\mathbb{R}$\nfor the  chemical property of $\\Co$.\n\nWe first choose a set $\\Lambda$ of chemical elements\n and then collect a data set  $D_{\\pi}$ of\n  chemical compounds  $C$ whose \n  chemical elements belong to $\\Lambda$,\n  where we regard  $D_{\\pi}$ as a set of chemical graphs $\\Co$\n  that represent the chemical compounds $C$  in  $D_{\\pi}$.\nTo define the interior\/exterior of \nchemical graphs  $\\Co\\in D_{\\pi}$,\nwe  next choose a branch-parameter ${\\rho}$, where\n we recommend ${\\rho}=2$.  \n \nLet $\\Lambda^\\inte(D_\\pi)\\subseteq \\Lambda$ \n(resp., $\\Lambda^\\ex(D_\\pi)\\subseteq \\Lambda$)\ndenote the set  of chemical elements  used in\nthe set $V^\\inte(\\Co)$ of interior-vertices\n(resp., the set $V^\\ex(\\Co)$ of  exterior-vertices) of $\\Co$\n over all chemical graphs $\\Co\\in D_\\pi$, \nand $\\Gamma^\\inte(D_\\pi)$\ndenote the set of edge-configurations used in\nthe set $E^\\inte(\\Co)$  of interior-edges in $\\Co$\n over all chemical graphs $\\Co\\in D_\\pi$. \nLet $\\mathcal{F}(D_\\pi)$ denote the set of\nchemical rooted trees $\\psi$  \nr-isomorphic to a chemical rooted tree in $\\mathcal{T}(\\Co)$\n  over all chemical graphs $\\Co\\in D_\\pi$,\n  where possibly a chemical rooted tree $\\psi\\in \\mathcal{F}(D_\\pi)$\n  consists of a single chemical element $\\ta\\in \\Lambda\\setminus \\{{\\tt H}\\}$.\n  \nWe define an integer encoding of a finite set $A$ of elements\nto be a bijection $\\sigma: A \\to [1, |A|]$, \nwhere we denote by $[A]$   the set $[1, |A|]$ of integers.\nIntroduce  an integer coding of each of the   sets \n$\\Lambda^\\inte(D_\\pi)$, $\\Lambda^\\ex(D_\\pi)$, \n$\\Gamma^\\inte(D_\\pi)$ and $\\mathcal{F}(D_\\pi)$. \nLet $[\\ta]^\\inte$  \n(resp., $[\\ta]^\\ex$)  denote   \nthe coded integer of  an element $\\ta\\in \\Lambda^\\inte(D_\\pi)$\n(resp., $\\ta\\in \\Lambda^\\ex(D_\\pi)$),  \n$[\\gamma]$   denote  \nthe coded integer of  an element $\\gamma$ in $\\Gamma^\\inte(D_\\pi)$\nand \n$[\\psi]$   denote  an element $\\psi$ in $\\mathcal{F}(D_\\pi)$. \n \n Over 99\\% of  chemical compounds $\\Co$ with up to\n  100 non-hydrogen atoms in  PubChem  have degree at most 4\n in the hydrogen-suppressed graph $\\anC$~\\cite{AZSSSZNA20}. \nWe assume that a chemical graph $\\Co$\n treated in this paper satisfies  $\\deg_{\\anC}(v)\\leq 4$\nin the hydrogen-suppressed graph $\\anC$.\n \nIn our model, we  use an integer \n  $\\mathrm{mass}^*(\\ta)=\\lfloor 10\\cdot \\mathrm{mass}(\\ta)\\rfloor$, \n for each $\\ta\\in \\Lambda$.\n \n  We define the {\\em feature vector} $f(\\C)$ \n  of a  chemical graph $\\C=(H,\\alpha,\\beta)\\in D_{\\pi}$ \n  to be a vector that consists of the following  \nnon-negative integer descriptors $\\dcp_i(\\C)$, $i\\in [1,K]$, where \n$K = 14+ |\\Lambda^\\inte(D_\\pi)|+|\\Lambda^\\ex(D_\\pi)|\n         +|\\Gamma^\\inte(D_\\pi)|+|\\mathcal{F}(D_\\pi)|+|\\Gac^\\lf|$. \n\n\n\\begin{enumerate} \n\\item   \n$\\dcp_1(\\C)$: the number  $|V(H)|-|\\VH|$ of non-hydrogen atoms  in  $\\C$.  \n \n\\item   \n$\\dcp_2(\\C)$: the rank $\\mathrm{r}(\\C)$ of   $\\C$.  \n\n\\item \n$\\dcp_3(\\C)$:  the number $|V^\\inte(\\C)|$ of interior-vertices in  $\\C$.\n  \n\\item \n$\\dcp_4(\\C)$: \nthe average $\\overline{\\mathrm{ms}}(\\C)$ of mass$^*$ \nover all atoms in $\\C$; \\\\\n i.e., $\\overline{\\mathrm{ms}}(\\C)\\triangleq \n \\frac{1}{|V(H)|}\\sum_{v\\in V(H)}\\mathrm{mass}^*(\\alpha(v))$. \n\n\\item \n$\\dcp_i(\\C)$,  $i=4+d,   d\\in [1,4]$: \nthe number $\\dg_d^{\\oH} (\\C)$ \n of non-hydrogen vertices $v\\in V(H)\\setminus \\VH$\n of degree $\\deg_{\\anC}(v)=d$\n in the hydrogen-suppressed chemical graph $\\anC$.  \n \n\\item \n$\\dcp_i(\\C)$,  $i=8+d,   d\\in [1,4]$: \nthe number $\\dg_d^\\inte(\\C)$\n of interior-vertices of interior-degree  $\\deg_{\\C^\\inte}(v)=d$\n  in the interior $\\C^\\inte=(V^\\inte(\\C),E^\\inte(\\C))$ of  $\\C$. \n  \n   \n\\item $\\dcp_i(\\C)$, $i=12+m$,  $m\\in[2,3]$: \nthe number $\\bd_m^\\inte(\\C)$\n of  interior-edges with bond multiplicity $m$ in  $\\C$; \n i.e., $\\bd_m^\\inte(\\C)\\triangleq \\{e\\in E^\\inte(\\C)\\mid \\beta(e)=m\\}$. \n  \n      \n\\item $\\dcp_i(\\C)$, $i=14+[\\ta]^\\inte$, \n $\\ta\\in \\Lambda^\\inte(D_\\pi)$: \n the frequency $\\na_\\ta^\\inte(\\C)=|V_\\ta(\\C)\\cap V^\\inte(\\C) |$ \n of chemical element $\\ta$ in\n the set $V^\\inte(\\C)$ of  interior-vertices in  $\\C$. \n \n\\item $\\dcp_i(\\C)$, \n$i=14+|\\Lambda^\\inte(D_\\pi)|+[\\ta]^\\ex$, \n $\\ta\\in \\Lambda^\\ex(D_\\pi)$: \n the frequency $\\na_\\ta^\\ex(\\C)=|V_\\ta(\\C)\\cap V^\\ex(\\C) |$\n  of chemical element $\\ta$ in\n the set $V^\\ex(\\C)$ of  exterior-vertices in  $\\C$. \n \n\\item $\\dcp_i(\\C)$, \n$i=14+|\\Lambda^\\inte(D_\\pi)|+|\\Lambda^\\ex(D_\\pi)|+ [\\gamma]$, \n$\\gamma \\in \\Gamma^\\inte(D_\\pi)$: \nthe frequency $\\ec_{\\gamma} (\\Co)$ of edge-configuration $\\gamma$\nin the set $E^\\inte(\\C)$ of interior-edges   in  $\\C$.\n\n\\item $\\dcp_i(\\C)$, \n$i= 14+|\\Lambda^\\inte(D_\\pi)|+|\\Lambda^\\ex(D_\\pi)|\n+ |\\Gamma^\\inte(D_\\pi)|+ [\\psi]$,  \n $\\psi \\in \\mathcal{F}(D_\\pi)$: \nthe frequency $\\fc_{\\psi}(\\C)$ of fringe-configuration $\\psi $\nin the set of ${\\rho}$-fringe-trees in  $\\C$. \n\n\\item $\\dcp_i(\\C)$, \n$i= 14+|\\Lambda^\\inte(D_\\pi)|+|\\Lambda^\\ex(D_\\pi)|\n+ |\\Gamma^\\inte(D_\\pi)|+|\\mathcal{F}(D_\\pi)|+ [\\nu]$,  \n $\\nu \\in \\Gac^\\lf$: \nthe frequency $\\ac_{\\nu}^\\lf(\\C)$ of adjacency-configuration $\\nu$\nin the set of leaf-edges in  $\\anC$.\n\\end{enumerate} \n\n\n\n \n\\section{Grid Neighbor Search}\\label{sec:grid_search\n\n\nThis section introduces a procedure of generating solutions\nof an MILP as a new building block of the framework of inferring chemical graphs.\n\nFor a notational convenience, let $(x,1)$ for a vector  $x\\in \\RK$\ndenote the vector $y\\in \\RKw$ such that $y(j)=x(j), j\\in [1,K]$\nand $y(K+1)=1$. \n\nChoose an integer $p_{\\max}\\geq 1$ as the dimension of a search space $\\R^{p_{\\max}}$,\na vector $s^*\\in  \\R^{p_{\\max}}$ as the center of  $\\R^{p_{\\max}}$\nand a vector $\\delta\\in  \\R_+^{p_{\\max}}$ with \n  $\\delta(p)>0, p\\in[1,p_{\\max}]$ as the width of a grid   in the space $\\R^{p_{\\max}}$.\nA {\\em grid} is defined to be  an integer vector  $z\\in \\Z^{p_{\\max}}$\nfor which we define a subspace \n $S(z)\\subseteq  \\R^{p_{\\max}}$  to be \n\\[  S(z)\\triangleq \\{s\\in  \\R^{p_{\\max}}  \\mid  \n           s^*(p)+(z(p)-0.5)\\delta(p)  \\leq   s(p)\\leq s^*(p)+(z(p)+0.5)\\delta(p), p\\in[1,p_{\\max}] \\} .\\]\nWe call a grid  $z\\in \\Z^{p_{\\max}}$ with $z(p)=0, p\\in[1,p_{\\max}]$ the {\\em center grid}.\nA neighbor  $N(r)$ of the center  with a radius vector $r\\in \\Z_+^{p_{\\max}}$ is defined\nto be a set  of grids such that \n\\[ N(r)\\triangleq \n\\{  z\\in \\Z^{p_{\\max}} \\mid -r(p) \\leq z(p)\\leq r(p), p\\in[1,p_{\\max}] \\}. \\]\nLet us introduce a partial order $\\preceq$ over the set  $\\Z^{p_{\\max}}$ of grids.\nFor two grids $z,z'\\in \\Z^{p_{\\max}}$, $z'\\preceq z $ if\n$0\\leq z'(p)\\leq z(p)$ or $0\\geq z'(p)\\geq z(p)$ \nfor each $p\\in[1,p_{\\max}]$, where we let $z'\\prec z$ mean that\n$z'\\preceq z$ and $z'\\neq z$.\n\n \nWe introduce a linear function $\\theta_p: \\RK\\to \\R$ for each $p\\in[1,p_{\\max}]$,\ncalled a {\\em projection function}  such that \n\\[ \\theta_p(x) \\triangleq w_p\\cdot (x,1)=\\sum_{j\\in [1,K]}w_p(j)x(j) + w_p(K+1) \\]\nby choosing a vector $w_p\\in \\RKw$. \nFor a notational convenience, let $\\theta(x), x\\in \\RK$ denote the vector \n$(\\theta_1(x), \\theta_2(x), \\ldots, \\theta_{p_{\\max}}(x)) \\in \\R^{p_{\\max}}$. \n\n\\bigskip \nIn the framework of inferring chemical graphs, \nwe formulate an MILP $\\mathcal{M}(g, x,y;\\mathcal{C}_1,\\mathcal{C}_2)$\nthat consists of   two MILPs \n$\\mathcal{M}(x,y;\\mathcal{C}_1)$ and $\\mathcal{M}(g,x;\\mathcal{C}_2)$,\nwhere the former simulates \nthe computation process of  a prediction function $\\eta$ for a chemical property $\\pi$ and \nthe latter  simulates the computation process of   a feature function $f$\nand describes \nconstruction of a chemical graph that satisfies a given topological specification $\\sigma$.\nBy solving the MILP for lower and upper bounds on\na  target value  of $\\pi$, we obtain a desired chemical graph $\\C^\\dagger$ \nwhen the MILP instance is feasible (or we detect that there is no such chemical graph\nwhen the instance is infeasible).\n\nWe design a procedure for finding other solutions of the MILP by\nsearching the neighbor of the output chemical graph  $\\C^\\dagger$.\nSuppose that we have chosen\n a dimension $p_{\\max}$, \n a width vector  $\\delta\\in \\R_+^{p_{\\max}}$ with \n  $\\delta(p)>0, p\\in[1,p_{\\max}]$, \n a radius vector $r\\in \\Z_+^{p_{\\max}}$ and \n projection functions $\\theta_p, p\\in[1,p_{\\max}]$. \n\n\\begin{enumerate}\n\\item[1.] \nLet  $\\underline{y}^*$ and $\\overline{y}^*$ be lower and upper bounds on\nthe value of property $\\pi$ of a chemical graph to be inferred.\nWe first solve $\\mathcal{M}(g, x, y;\\mathcal{C}_1,\\mathcal{C}_2)$ with\n the input values $(\\underline{y}^*,\\overline{y}^*)$ \n to find a desired chemical graph  $\\C^\\dagger$.\n (When the MILP instance is infeasible, we halt.)\nLet $x^*:=f(\\C^\\dagger)\\in \\RK$. \n\n\\item[2.]  Set the center of the space $\\R^{p_{\\max}}$ to be\n$s^*:=\\theta(x^*)$. \nFor each grid $z \\in N(r)$, we solve\nan MILP $\\mathcal{M}(g, x,y;\\mathcal{C}_1,\\mathcal{C}_2)$ \nwith an additional linear constraint $\\theta(x)\\in S(z)$ \nfor $(\\underline{y}^*,\\overline{y}^*)$,\nwhere we call a grid $z$ {\\em feasible} (resp., {\\em infeasible}) \nif the augmented MILP instance is feasible  (resp., infeasible). \n \n\\item[3.]\nFor each feasible grid $z \\in N(r)$, output a feasible solution $\\C^\\dagger_z$\nof  the augmented MILP instance.\nWe check the feasibility of grids  $z \\in N(r)$ in a non-decreasing order of\n$\\max_{p\\in [1,p_{\\max}]}|z(p)|$, and discard any grid $z \\in N(r)$\nwithout testing the feasibility of $z$ if we find an infeasible grid $z'\\in N(r)$\nwith $z'\\prec z$.\nNote that $\\C^\\dagger_z$ and $\\C^\\dagger_{z'}$ are not isomorphic unless\n$\\theta(f(\\C^\\dagger_z))$ and $\\theta(f(\\C^\\dagger_{z'}))$ happen to belong to\nthe common boundary $S(z) \\cap S(z')$.\n\\end{enumerate}\n\nIn the above method, we can choose arbitrarily many grids in the space of  $\\R^{p_{\\max}}$\naround the center $s^*$ by choosing small   $\\delta(p)$ and large  $r(p),  p\\in[1,p_{\\max}]$,\nwhere  each $\\delta(p)$ needs to be large enough to avoid a possible numerical error. \n\nWe can also choose arbitrary linear functions as projection functions  $\\theta_p,  p\\in[1,p_{\\max}]$.\nWhen we have constructed a prediction function $\\eta_\\tau$ as a linear function\nwith linear regression for several chemical properties $\\tau$ other than the current\ntarget property $\\pi$, we can use such functions as projection functions.\nFor example, if  a linear prediction function $\\eta_\\tau$ is available \nfor a chemical property $\\tau$ such as \nsolubility ({\\sc Sl}) and lipophilicity ({\\sc Lp}), \nthen we can infer chemical graphs $\\C^\\dagger_z$ with slightly different values\nof these properties  $\\tau_1= $ {\\sc Sl}   and $\\tau_2=$ {\\sc Lp} \nby setting $\\theta_p:=\\eta_{\\tau_p}, p=1,2$. \n\n\n\\section{Introduction}\\label{sec:introduction}\n\n\n\\noindent {\\bf Background~}\nAnalysis of chemical compounds is one of the important applications of\nintelligent computing.\nIndeed, various machine learning methods have been applied to\nthe prediction of chemical activities from their structural data,\nwhere such a problem is often referred to as\n\\emph{quantitative structure activity relationship} (QSAR)\n\\cite{Lo18,Tetko20}.\nRecently, neural networks and deep-learning technologies have extensively\nbeen applied to QSAR \\cite{Ghasemi18}.\n\nIn addition to QSAR, extensive studies have been done on\ninverse quantitative structure activity relationship\n(inverse QSAR), which seeks for chemical structures having\ndesired chemical activities under some constraints.\nSince it is difficult to directly handle chemical structures\nin both QSAR and inverse QSAR,\nchemical compounds are usually represented \nas vectors of real or integer numbers,\nwhich are often called \\emph{descriptors} in chemoinformatics and\ncorrespond to \\emph{feature vectors} in machine learning.\nOne major approach in inverse QSAR is \nto infer feature vectors from given chemical activities and constraints\nand then reconstruct chemical structures from these feature\nvectors~\\cite{Miyao16,Ikebata17,Rupakheti15},\nwhere chemical structures are usually treated as undirected graphs.\nHowever, the  reconstruction itself is a challenging task\nbecause the number of possible chemical graphs is huge.\nFor example, \nchemical graphs with up to 30 atoms (vertices)\n{\\tt C}, {\\tt N}, {\\tt O}, and  {\\tt S}\nmay  exceed~$10^{60}$~\\cite{BMG96}. \nIndeed, it is NP-hard to infer a chemical graph from a given feature vector\nexcept for some simple cases~\\cite{AFJS12}.  \nDue to this inherent difficulty, most existing methods for  inverse QSAR\ndo not guarantee optimal or exact solutions.\n\nAs a new approach,\nextensive studies have recently been done for inverse QSAR using \n\\emph{artificial neural networks} (ANNs),\nespecially using graph convolutional networks~\\cite{Kipf16}.\nFor example, recurrent neural networks~\\cite{Segler18,Yang17}, \nvariational autoencoders~\\cite{Gomez18}, \ngrammar variational autoencoders~\\cite{Kusner17},\ngenerative adversarial networks~\\cite{DeCao18},\nand invertible flow models~\\cite{Madhawa19,Shi20}\nhave been applied.\nHowever, these methods do not yet guarantee optimal or exact solutions.  \n\n\\begin{figure}[!ht]  \\begin{center}\n\\includegraphics[width=.77\\columnwidth]{fig\/framework_interval.eps}\n\\end{center} \\caption{An illustration of a framework for inferring\na set of chemical graphs $\\Co^*$.   } \n\\label{fig:framework}  \\end{figure}    \n\n\\smallskip\n\\noindent {\\bf Framework~}\nAkutsu and Nagamochi~\\cite{AN19} proved that  \nthe computation process of a given ANN can be simulated\nwith a mixed integer linear programming (MILP).\nBased on this,\na novel  framework for inferring chemical graphs has been developed\nand revised~\\cite{ACZSNA20,ZAHZNA21}, \nas illustrated in Figure~\\ref{fig:framework}. \nIt constructs a prediction function in the first phase and\ninfers a chemical graph in the second phase. \nThe first phase of the framework consists of three stages.\nIn Stage~1, we choose a chemical property $\\pi$ and a class $\\mathcal{G}$ \nof graphs, where a property function\n$a$ is defined so that $a(\\Co)$ is the value of $\\pi$  for a compound $\\Co\\in \\mathcal{G}$,\nand collect a data set $D_{\\pi}$ of chemical graphs in  $\\mathcal{G}$ \nsuch that $a(\\Co)$ is available for every $\\Co \\in D_{\\pi}$.\nIn Stage~2, we introduce a feature function $f: \\mathcal{G}\\to \\mathbb{R}^K$ \nfor a positive integer $K$.    \nIn Stage~3, we construct a prediction function $\\eta$ \nwith an ANN $\\mathcal{N}$ that,  \ngiven a   vector  $x\\in \\mathbb{R}^K$, \nreturns a value $y=\\eta(x)\\in \\mathbb{R}$    \nso that $\\eta(f(\\Co))$ serves as a predicted value\nto the real value $a(\\Co)$ of $\\pi$ for each $\\Co\\in D_\\pi$.  \nGiven two reals $\\underline{y}^*$ and $\\overline{y}^*$\n as an interval  for a  target chemical value,\nthe  second phase infers  chemical graphs $\\Co^*$\nwith $\\underline{y}^*\\leq \\eta(f(\\C^*))\\leq \\overline{y}^*$ in the next two stages. \nWe have obtained a  feature function $f$ and a  prediction function $\\eta$\nand call an additional constraint on the substructures of target chemical graphs \na {\\em topological specification}. \nIn Stage~4, we prepare the following two  MILP formulations: \n\\begin{enumerate}[nosep,  leftmargin=*]\n\\item[-]\n MILP $\\mathcal{M}(x,y;\\mathcal{C}_1)$\nwith a set $\\mathcal{C}_1$ of linear constraints on variables $x$ and $y$\n(and some other auxiliary variables) \n  simulates the  process of computing $y:=\\eta(x)$ from a vector $x$; and\n\\item[-]\n MILP $\\mathcal{M}(g,x;\\mathcal{C}_2)$\nwith a set $\\mathcal{C}_2$ of linear constraints on  variable  $x$ and\n a variable vector  $g$ that represents a chemical graph $\\Co$\n(and some other auxiliary variables)  \n  simulates the  process of computing $x:=f(\\Co)$ from a chemical graph $\\Co$\nand chooses a chemical graph $\\Co$ that satisfies the given topological specification\n$\\sigma$. \n\\end{enumerate} \nGiven an interval with boundaries $\\underline{y}^*,\\overline{y}^* \\in \\mathbb{R}$,\n we solve the combined\nMILP $\\mathcal{M}(g,x,y;\\mathcal{C}_1,\\mathcal{C}_2)$\nto find  a feature vector $x^*\\in \\mathbb{R}^K$\n and a chemical graph $\\Co^{\\dagger}$  with the specification\n$\\sigma$ such that $f(\\Co^\\dagger)=x^*$ and  \n$\\underline{y}^*\\leq \\eta(x^*) \\leq \\overline{y}^*$\n(where if the MILP instance is infeasible then this suggests that there \ndoes  not exist\nsuch a desired chemical graph).\nIn Stage~5,  we generate other  chemical graphs $\\Co^*$\nsuch that $\\underline{y}^*\\leq \\eta(f(\\C^*))\\leq \\overline{y}^*$\n based on the output chemical graph $\\Co^\\dagger$.\n\nMILP formulations required in Stage~4 have been designed   \nfor chemical compounds with \ncycle index at most 2~\\cite{ZCSNA20}.\nAfterwards, a modeling of chemical compounds together with \nan MILP formulation has been improved so that\na chemical compound with any graph structure can treated\n(see  Shi et~al.~\\cite{SZAHZNA21}). \nNot only ANNs but also other machine learning methods\nhave been used to construct a prediction function $\\eta$ in Stage~3\nrecently. \nTanaka et~al.~\\cite{TZAHZNA21} \n (resp.,  Zhu~et~al.~\\cite{ZAHZNA21}) used a decision tree\n  (resp.,  linear regression)\nto construct a prediction function $\\eta$ in Stage~3 in the framework\nand derived an MILP $\\mathcal{M}(x,y;\\mathcal{C}_1)$ \nthat simulates the computation process of  a decision tree\n (resp.,  linear regression). \n  \n \n\\smallskip\n\\noindent {\\bf Contribution~}\nIn the current framework,  a chemical graph $\\C^\\dagger$ with a desired \nchemical property $\\pi$ in Stage~4 is constructed \nas a feasible solution of an MILP $\\mathcal{M}(g,x,y;\\mathcal{C}_1,\\mathcal{C}_2)$\nbefore we generate isomers $\\C^*$ of $\\C^\\dagger$ by an algorithm such as \nthe dynamic programming algorithm due to Azam~et~al.~\\cite{AZSSSZNA20}. \nIn this paper, we design  a procedure of generating solutions\nof the same  MILP as a new building block in Stage~4 of the framework of inferring chemical graphs. \nWhen a feasible solution  $\\C^\\dagger$ of the MILP is constructed in Stage~4,\nwe try to find other feasible solutions $\\C^*$ of the same MILP by\nsolving the MILP with additional $p_{\\max}$  linear constraints for some integer $p_{\\max}$.\nFor this,  we first prepare arbitrary linear functions $\\theta_p: \\RK\\to \\R, p\\in [1,p_{\\max}]$\nand consider a neighbor of $\\C^\\dagger$ \n defined by a set of chemical graphs $\\C^*$ that satisfy linear constraints \n $\\delta_1\\leq |\\theta_p(f(\\C^*))-\\theta_p(f(\\C^\\dagger))|\\leq \\delta_2, p\\in [1,p_{\\max}]$ for \n small reals $\\delta_2>\\delta_1>0$.  \nBy changing the  reals $\\delta_2$ and $\\delta_1$ systematically, we can search for new solutions\nof the same MILP other than $\\C^\\dagger$.\nAs a candidate for a function $\\theta_p$, we can use a linear prediction \nfunction $\\eta_\\tau: \\RK\\to \\R$  \nfor a different chemical property $\\tau$ such as a function $\\eta_\\tau$ that has been obtained\nby linear regression. \nWith a linear prediction function $\\theta_p=\\eta_\\tau$, we can \nsearch for other chemical graphs $\\C^*$ by specifying a predicted value of $\\C^*$ for the property $\\tau$.\n\nWe implemented the framework with the new building block \nbased on  the two-layered model and the feature function\nproposed by  Zhu~et~al.~\\cite{ZAHZNA21}. \nWe used  the same MILP $\\mathcal{M}(g,x;\\mathcal{C}_2)$ \nformulation proposed by  Zhu~et~al.~\\cite{ZAHZNA21} \n and omit the details in this paper.  \nFrom the results of our computational experiments,\nwe observe that \n the proposed method can generate an additional number of chemical graphs $\\C^*$ in Stage~4\nwith  up to 50 non-hydrogen atoms. \n\n\nThe paper is organized as follows.  \nSection~\\ref{sec:preliminary} introduces some notions on graphs,\n a modeling of chemical compounds and a choice of descriptors.  \nSection~\\ref{sec:2LM} reviews  the two-layered model.\nSection~\\ref{sec:grid_search} introduces a new method of\ngenerating solutions of an MILP in Stage~4. \nSection~\\ref{sec:experiment} reports the results on  computational \nexperiments conducted for  47 chemical properties such as \nbiological half life and boiling point for monomers\nand  characteristic ratio and refractive index for polymers. \nSection~\\ref{sec:conclude} makes some concluding remarks.   \nSome technical details are given in Appendices:   \n Appendix~\\ref{sec:descriptor} for  all descriptors in our feature function; \n Appendix~\\ref{sec:specification} for a full description of \na topological specification; and \nAppendix~\\ref{sec:test_instances} for the detail of test instances\nused in our computational experiment for Stages~4 and 5.\n  \n\n\n\\section{Preliminary}\\label{sec:preliminary\n\nThis section  introduces some notions and terminologies on graphs,\n  modeling of chemical compounds and our choice of descriptors. \n \nLet $\\mathbb{R}$, $\\mathbb{R}_+$, $\\mathbb{Z}$  and $\\mathbb{Z}_+$ \ndenote the sets of reals,  non-negative reals, \nintegers and non-negative integers, respectively.\nFor two integers $a$ and $b$, let $[a,b]$ denote the set of \nintegers $i$ with $a\\leq i\\leq b$.\nFor a vector $x\\in \\R^p$, the $j$-th entry of $x$ is denoted by $x(j)$.\n\n\\bigskip\\noindent\n{\\bf  Graph} \nGiven a  graph $G$, let $V(G)$ and $E(G)$ denote the sets\nof vertices and edges, respectively.     \nFor a subset $V'\\subseteq V(G)$ (resp., $E'\\subseteq E(G))$ of\na graph $G$, \nlet $G-V'$ (resp., $G-E'$) denote the graph obtained from $G$\nby removing the vertices in $V'$ (resp.,  the edges in $E'$),\nwhere we remove all edges incident to a vertex in $V'$\nin $G-V'$. \nAn edge subset $E'\\subseteq E(G)$ in a connected graph $G$ is called\n{\\em separating} (resp., {\\em non-separating})\nif $G-E'$  becomes disconnected (resp., $G-E'$ remains connected). \nThe {\\em rank}  $\\mathrm{r}(G)$ of a graph $G$  is defined to be \nthe minimum $|F|$ of an edge subset $F\\subseteq E(G)$\nsuch that $G-F$ contains no cycle, where \n$\\mathrm{r}(G)=|E(G)|-|V(G)|+1$ for a connected graph $G$.   \nObserve that   $\\mathrm{r}(G-E')=\\mathrm{r}(G)-|E'|$ holds\nfor any non-separating edge subset $E'\\subseteq E(G)$. \nAn edge $e\\in E(G)$ in a connected graph $G$\n  is called a {\\em bridge} if $\\{e\\}$ is separating.\nFor a connected cyclic graph $G$, an edge $e$ is called a {\\em core-edge} if\nit is in a cycle of $G$ or is a bridge $e=u_1u_2$ such that\neach of the connected graphs $G_i$, $i=1,2$ of $G-e$ contains a cycle. \nA vertex incident to a core-edge is called a {\\em core-vertex} of $G$. \nA path with two end-vertices $u$ and $v$ is called a {\\em $u,v$-path}. \n \n We define a {\\em rooted} graph to be\n a graph with a  designated vertex, called a {\\em root}. \n\n For a graph $G$ possibly with a root,  \n a {\\em leaf-vertex} is defined to be a non-root vertex \n with degree 1. \nWe call  the edge $uv$ incident to a leaf vertex $v$ a {\\em leaf-edge},\n and denote by $\\Vleaf(G)$ and $\\Eleaf(G)$\n  the sets of leaf-vertices and leaf-edges  in $G$, respectively.\n For a graph  or a rooted graph $G$,\n we define graphs $G_i, i\\in \\mathbb{Z}_+$ obtained from $G$\n by removing the set of leaf-vertices $i$ times so that\n\\[ G_0:=G; ~~ G_{i+1}:=G_i - \\Vleaf(G_i), \\]\nwhere we call a vertex $v$ a {\\em tree vertex} if $v\\in \\Vleaf(G_i)$\nfor some $i\\geq 0$. \nDefine the {\\em height} $\\h(v)$ of each tree vertex $v\\in \\Vleaf(G_i)$\nto be $i$; and \n$\\h(v)$ of each non-tree vertex $v$ adjacent to a tree vertex \nto be $\\h(u)+1$ for the maximum $\\h(u)$ of a tree vertex $u$ adjacent to $v$,\nwhere we do not define height of any non-tree vertex not adjacent to any tree vertex. \nWe call a vertex $v$ with $\\h(v)=k$ a {\\em leaf $k$-branch}.\nThe {\\em height} $\\h(T)$ of a rooted tree $T$ is defined\nto be the maximum of $\\h(v)$ of a vertex $v\\in V(T)$. \nFor an integer $k\\geq 0$, we call a  rooted tree $T$\n {\\em $k$-lean} if $T$ has at most one leaf $k$-branch.\nFor an unrooted cyclic graph $G$, we regard that\nthe set of non-core-edges in $G$ induces\na collection $\\mathcal{T}$ of trees each of which is rooted at a core-vertex,\nwhere we call $G$  {\\em $k$-lean} if each of the rooted trees in $\\mathcal{T}$ \nis $k$-lean. \n \n\\subsection{Modeling of Chemical Compounds}\\label{sec:chemical_model}\n\nWe review a modeling of chemical compounds introduced \nby  Zhu~et~al.~\\cite{ZAHZNA21}. \n\nTo represent a chemical compound, \nwe introduce a set  of   chemical elements such as \n  {\\tt H} (hydrogen),   \n {\\tt C} (carbon), {\\tt O} (oxygen), {\\tt N} (nitrogen)  and so on.\n To distinguish a chemical element $\\ta$ with multiple valences such as {\\tt S} (sulfur),\n we denote a chemical element $\\ta$ with a valence $i$ by $\\ta_{(i)}$,\n where we do not use such a suffix $(i)$ \n for a chemical element $\\ta$ with a unique valence. \nLet $\\Lambda$ be a set of chemical elements $\\ta_{(i)}$.\nFor example,  $\\Lambda=\\{\\ttH,  \\ttC, \\ttO, \\ttN, \\ttP, \\ttS_{(2)}, \\ttS_{(4)}, \\ttS_{(6)}\\}$. \nLet $\\val: \\Lambda\\to [1,6]$ be a valence function.\nFor example, $\\val(\\ttH)=1$, $\\val(\\ttC)=4$, $\\val(\\ttO)=2$, $\\val(\\ttP)=5$,\n$\\val(\\ttS_{(2)})=2$, $\\val(\\ttS_{(4)})=4$ and $\\val(\\ttS_{(6)})=6$.\n For each  chemical element $\\ta\\in \\Lambda$, \nlet $\\mathrm{mass}(\\ta)$  denote the mass   of  $\\ta$.\n\nA chemical compound  is represented by a {\\em chemical graph} defined to be\na tuple $\\Co=(H,\\alpha,\\beta)$  of\n  a simple, connected undirected graph $H$ and  \n    functions   $\\alpha:V(H)\\to \\Lambda$  and  $\\beta: E(H)\\to [1,3]$.\nThe set of atoms and the set of bonds in the compound \nare represented by the vertex set $V(H)$ and the edge set $E(H)$, respectively.\nThe chemical element assigned to a vertex $v\\in V(H)$\nis represented by $\\alpha(v)$ and \n the bond-multiplicity  between two adjacent vertices  $u,v\\in V(H)$\nis represented by $\\beta(e)$ of the edge $e=uv\\in E(H)$.\nWe say that two tuples $(H_i,\\alpha_i,\\beta_i), i=1,2$ are\n{\\em isomorphic} if they admit an isomorphism $\\phi$,\ni.e.,  a bijection $\\phi: V(H_1)\\to V(H_2)$\nsuch that\n $uv\\in E(H_1), \\alpha_1(u)=\\ta, \\alpha_1(v)=\\tb, \\beta_1(uv)=m$\n $\\leftrightarrow$  \n $\\phi(u)\\phi(v) \\in E(H_2), \\alpha_2(\\phi(u))=\\ta, \n \\alpha_2(\\phi(v))=\\tb, \\beta_2(\\phi(u)\\phi(v))=m$. \n When $H_i$ is rooted at a vertex $r_i,  i=1,2$,\nthese chemical graphs $(H_i,\\alpha_i,\\beta_i),  i=1,2$ are\n{\\em rooted-isomorphic} (r-isomorphic) if \nthey admit  an isomorphism $\\phi$ such that $\\phi(r_1)=r_2$. \n\n For a notational convenience, we  use\n a function $\\beta_\\Co: V(H)\\to [0,12]$ \n for a chemical graph $\\Co=(H,\\alpha,\\beta)$\n  such that $\\beta_\\Co(u)$ means the sum of bond-multiplicities\n of edges incident to a vertex $u$; i.e., \n\\[ \\beta_\\Co(u) \\triangleq \\sum_{uv\\in E(H) }\\beta(uv) \n\\mbox{ for each vertex $u\\in V(H)$.}\\]\nFor each vertex $u\\in V(H)$, \n define the {\\em electron-degree} $\\eledeg_\\Co(u)$  to be \n\\[  \\eledeg_\\Co(u) \\triangleq  \\beta_\\Co(u) - \\val(\\alpha(u)). \\]\nFor each  vertex $u\\in V(H)$, let $\\deg_\\Co(v)$ denote \nthe number of vertices adjacent to $u$ in $\\Co$. \n  \n  For a chemical   graph  $\\Co=(H,\\alpha,\\beta)$, \n  let  $V_{\\ta}(\\Co)$, $\\ta\\in \\Lambda$\n   denote the set of vertices $v\\in V(H)$ such that $\\alpha(v)=\\ta$ in $\\Co$\n  and define the {\\em hydrogen-suppressed chemical graph} $\\anC$ \nto be  the graph obtained from $H$ by\n  removing all the vertices $v\\in \\VH(\\Co)$.\n \n\n\\subsection{Evaluating prediction function}\\label{sec:coefficient_determination} \n\nWe review the definition of coefficient of determination.\n\nLet $D$ be a data set   of chemical graphs $\\C$ with\nan observed value $a(\\C)\\in \\R$,\nwhere we denote by $a_i=a(\\C_i)$ \nfor an indexed graph $\\C_i$. \n\nLet $ f$ be a feature function that maps a chemical graph $\\C$\nto a vector $ f(\\C)\\in \\RK$\nwhere we denote by $\\x_i= f(\\C_i)$ \nfor an indexed graph $\\C_i$. \nFor  a prediction function $\\eta: \\RK\\to \\R$, \ndefine an error function \n\\[ \\mathrm{Err}(\\eta;D)  \\triangleq \n\\sum_{\\C_i\\in D}(a_i - \\eta(f(\\C_i)))^2=\\sum_{\\C_i\\in D}(a_i - \\eta(\\x_i))^2, \\]\nand define the {\\em coefficient of determination}\n $\\mathrm{R}^2(\\eta,D)$ \n  to be \n\\[ \\displaystyle{ \\mathrm{R}^2(\\eta,D)\\triangleq \n  1- \\frac{\\mathrm{Err}(\\eta;D) } \n  {\\sum_{ \\C_i\\in D  } (a_i-\\widetilde{a})^2}  \n  \\mbox{   for  }\n   \\widetilde{a}= \\frac{1}{|D |}\\sum_{ \\C\\in D }a(\\C).  } \\] \n\n\n\\section{Results}\\label{sec:experiment\n\nWe implemented our method of Stages~1 to 5 \nfor inferring chemical graphs under a given topological specification and\nconducted experiments  to evaluate the computational efficiency. \nWe executed the experiments on a PC with \n Processor:  Core i7-9700 (3.0GHz; 4.7 GHz at the maximum) and \nMemory: 16 GB RAM DDR4. \n\nTo construct an ANN, \nwe used {\\tt scikit-learn} version 0.23.2  with Python 3.8.5, \nMLPRegressor and ReLU activation function. \n\n\\medskip \\noindent\n{\\bf Results on Phase~1.  }\nWe implemented Stages~1, 2 and 3 in Phase~1 as follows.\n\n\nWe have conducted experiments of Lasso linear regression and\n for 37 chemical properties of monomers\n (resp., ten chemical properties of polymers)  using the same feature function in this paper \n and  we found that \n the test  coefficient of determination ${\\rm R}^2$ \n exceeds 0.927 for the following 11 properties of monomers:  \n octanol\/water partition coefficient ({\\sc Kow}), \nheat of combustion ({\\sc Hc}), \nvapor density ({\\sc Vd}),  \nelectron density on the most positive atom ({\\sc EDPA}), \nheat of atomization ({\\sc Ha}), \nheat of formation ({\\sc Hf}), \ninternal energy at 0K ({\\sc U0}),  \nisotropic polarizability ({\\sc Alpha}), \nheat capacity at 298.15K ({\\sc Cv}),  \nisobaric heat capacities in liquid phase ({\\sc IhcLiq}) and \nisobaric heat capacities in solid phase ({\\sc IhcSol}) \n(see \\cite{ZAHZNA21} for the details)\nand\nthat the test  coefficient of determination ${\\rm R}^2$ \n exceeds 0.9  for the following five properties of polymers:  \nexperimental amorphous density~({\\sc AmD}), \nheat capacity liquid ({\\sc HcL}), \nheat capacity solid ({\\sc HcS}), \nmol volume ({\\sc MlV}) and \nglass transition~({\\sc Tg})\n(see \\cite{ICZAHZNA21}  for the details).\nWe excluded the above properties in our experiment of \nconstructing prediction functions with ANNs. \n \n \nWe have conducted experiments of ANNs \n for the rest of 26 chemical properties of monomers\n (resp., five chemical properties of polymers) among which \nwe report the following 12 properties of monomers\n(resp., two properties of polymers) to which the test  \n coefficient of determination ${\\rm R}^2$ \nby ANNs is better than that by Lasso linear regression:\nbiological half life ({\\sc BHL}),\nboiling point ({\\sc Bp}), \ncritical pressure ({\\sc Cp}),\ndissociation constants  ({\\sc Dc}),\nflash point   ({\\sc Fp}),\nKovats retention index   ({\\sc Kov}),\nlipophilicity ({\\sc Lp}),\nenergy of lowest unoccupied molecular orbital ({\\sc Lumo}),\noptical rotation ({\\sc OptR}),\nsolubility ({\\sc Sl}),\nsurface tension ({\\sc SfT}) and \nviscosity ({\\sc Vis}) \n(resp.,  \ncharacteristic ratio ({\\sc ChaR}) and \nrefractive index ({\\sc RfId})).\nWe explain the data set and the results for these 14 properties \nin detail below.\n\n\nWe used data sets of monomers  provided  by HSDB from PubChem~\\cite{pubchem}\n for {\\sc BHL},  {\\sc Cp},  {\\sc Dc},   {\\sc Fp}   and  {\\sc OptR},\n M.~Jalali-Heravi and M.~Fatemi~\\cite{JF01} for {\\sc Kov}, \n   Roy and Saha~\\cite{RS03} for {\\sc Bp},\n MoleculeNet~\\cite{moleculenet} for  {\\sc Lumo}, \nGoussardet al.~\\cite{GFPDNA17}  for {\\sc SfT}, \n Goussard et al.~\\cite{GFPDNA20} for   {\\sc Vis} and\n %\n Figshare~\\cite{figshare}  for {\\sc Lp}.\nProperty  {\\sc  Lumo} has the original data set $D^*$\nwith  more than 130,000 compounds, and \nwe used  a set $D_\\pi$ of 1,000 graphs randomly selected from $D^*$  \nas a   data set of   property  {\\sc  Lumo}\nin this experiment. \n \nWe used data sets of polymers  provided  by   \nBicerano~\\cite{Bicerano},\nwhere we did not include any polymer whose chemical formula\ncould not be found by its name in the book. \nFor property {\\sc ChaR} (resp.,  {\\sc RfId}), we remove the following   polymer \nas an outlier  from the original data set: \\\\\n ethyleneTerephthalate,                             \n oxy(2-methyl-6-phenyl-1$\\underline{~}$4-phenylene)  and\n N-vinylCarbazole                                     \n(resp.,  2-decyl-1$\\underline{~}$4-butadiene).\n\n\\medskip \\noindent\n{\\bf Stage~1.  }\nWe set  a graph class   $ \\mathcal{G}$   to be\nthe set of all chemical graphs with any graph structure, \nand set a branch-parameter ${\\rho}$ to be 2. \n\nFor each of the properties,  \n we first select a set $\\Lambda$ of chemical elements \n and then collect  a  data set  $D_{\\pi}$ on chemical graphs\n over the set $\\Lambda$ of chemical elements.  \n To construct the data set $D_{\\pi}$,\n  we eliminated  chemical compounds that do not satisfy \n  one of the following: the graph is connected,\n  the number of carbon atoms is at least four,\n  and   the number of non-hydrogen neighbors of each atom is\n  at most 4.   \n   \n\n\\medskip \\noindent\n{\\bf Stage~2.  }\nWe used  the  new  feature function  defined \nin our chemical model without suppressing hydrogen \n(see Appendix~\\ref{sec:descriptor} for the detail).\nWe normalize the range of each descriptor and\n the range $\\{t\\in \\R \\mid \\underline{a}\\leq t\\leq \\overline{a}\\}$ \n of property values   $a(\\Co), \\Co\\in D_\\pi$.\n\n Table~\\ref{table:phase1a}    shows\n  the size and range of data sets   that \n we prepared for each chemical property in Stages~1 and 2,\n where  we denote the following:  \n\\begin{enumerate}[nosep,  leftmargin=*]\n\\item[-] \n  $\\Lambda$: the set of elements used in the data set $D_{\\pi}$; \n  $\\Lambda$ is one of the following nine sets: \n  $\\Lambda_1=\\{\\ttH,\\ttC,\\ttO \\}$; \n   $\\Lambda_2=\\{\\ttH,\\ttC,\\ttO, \\ttN \\}$;\n \n   $\\Lambda_3=\\{\\ttH,\\ttC,\\ttO, \\ttSi_{(4)} \\}$;  \n  $\\Lambda_4=\\{\\ttH,\\ttC,\\ttO, \\ttN,\\ttS_{(2)},\\ttF \\}$;\n   $\\Lambda_5=\\{\\ttH,\\ttC,\\ttO, \\ttN,  \\ttCl, \\ttPb \\}$;\n   $\\Lambda_6=\\{\\ttH,\\ttC,\\ttO, \\ttN,\\ttS_{(2)},\\ttS_{(6)},\\ttCl \\}$;    \n   $\\Lambda_7=\\{\\ttH,\\ttC,\\ttO, \\ttN,\\ttS_{(2)},\\ttS_{(4)},\\ttS_{(6)},\\ttCl \\}$;  \n    $\\Lambda_{8}=\\{\\ttH, \\ttC_{(2)},\\ttC_{(3)},\\ttC_{(4)},\\ttC_{(5)},\\ttO,\n   \\ttN_{(1)}, \\ttN_{(2)},$ $ \\ttN_{(3)}, \\ttF \\}$;\n  $\\Lambda_{9}=\\{\\ttH,\\ttC, \\ttO_{(1)}, \\ttO_{(2)}, \\ttN \\}$;\n $\\Lambda_{10}=\\{\\ttH,\\ttC,\\ttO, \\ttN,\\ttSi_{(4)},\\ttCl,\\ttBr \\}$; \n  $\\Lambda_{11}=\\{\\ttH,\\ttC,\\ttO_{(1)}, \\ttO_{(2)},\\ttN,\\ttSi_{(4)},\\ttCl,\\ttF \\}$; \n   and\n  $\\Lambda_{12}=\\{\\ttH,\\ttC,\\ttO_{(1)}, \\ttO_{(2)}, \\ttN,\\ttSi_{(4)},\\ttCl,\\ttF,\n                            \\ttS_{(2)}, \\ttS_{(6)},\\ttBr \\}$,\n where ${\\tt a}_{(i)}$ for a chemical element ${\\tt a}$ and an integer $i\\geq 1$ \n means that  a chemical element ${\\tt a}$ with valence~$i$. \n \n\\item[-] \n $|D_{\\pi}|$:  the size of data set $D_{\\pi}$ over $\\Lambda$\n  for the property $\\pi$.\n  \n\\item[-]   $ \\underline{n},~\\overline{n} $:  \n  the minimum and maximum  values of the number \n  $n(\\Co)$ of non-hydrogen atoms in \n  the   compounds $\\Co$ in $D_{\\pi}$.\n\\item[-] $ \\underline{a},~\\overline{a} $:  the minimum and maximum values\nof $a(\\Co)$ for $\\pi$ over   the   compounds $\\Co$ in  $D_{\\pi}$.\n  %\n\\item[-]    $|\\Gamma|$: \nthe number of different edge-configurations\nof interior-edges over the compounds in~$D_{\\pi}$. \n %\n\\item[-]  $|\\mathcal{F}|$: the number of non-isomorphic chemical rooted trees\n in the set of all 2-fringe-trees in  the   compounds in $D_{\\pi}$.\n \n\\item[-] $K$: the number of descriptors in the original feature vector $f(\\Co)$.  \n\\end{enumerate}\n\n\n\\medskip \\noindent\n{\\bf Stage~3.  }\nFor each chemical property $\\pi$, we conducted a preliminary experiment\nto choose the following:  \n a subset $S_\\pi$ of the original set of $K$ descriptors; \n  an architecture $A_\\pi$ with at most five hidden layers; \n   a nonnegative real $\\rstp_\\pi\\leq 1$; and\n  an integer $\\itstp_\\pi$, where we will use \n  $\\rstp_\\pi$ and $\\itstp_\\pi$ as parameters to execute an early stopping \n  in constructing a prediction function with a training data set. \n Let $f_\\pi$ denote the feature vector that consists of the descriptors in the set  $S_\\pi$.\n \nFor  each  property $\\pi$, we conducted ten 5-fold cross-validations.\nIn a 5-fold cross-validation, \n  we construct  five prediction functions  $\\eta^{(k)}, k\\in[1,5]$ as follows.\n Partition data set $D_{\\pi}$ \n into five subsets $D_{\\pi}^{(k)}$, $k\\in[1,5]$ randomly.\n For each $k\\in[1,5]$, use the  set $\\Dtrain:=D_{\\pi}\\setminus D_{\\pi}^{(k)}$ as a training set\n and  construct an ANN on the selected architecture $A_\\pi$\n with the feature vector  $f_\\pi$ by the MLPRegressor of {\\tt scikit-learn},\n where we stop updating weights\/biases on $A_\\pi$ during an execution of the iterative\n algorithm when  the coefficient of determination \n $\\mathrm{R}^2(\\eta,\\Dtrain)$ of the prediction function\n $\\eta$ by the current weights\/biases  exceeds  $\\rstp_\\pi$\n (where we terminate the execution when the number of iterations exceeds\n $1.5\\times \\itstp_\\pi$ even if \n $\\mathrm{R}^2(\\eta,\\Dtrain)$ does not reach  $\\rstp_\\pi$).\nSet $\\eta^{(k)}$ to be the  prediction function\n $\\eta$ by the resulting weights\/biases on $A_\\pi$.\nWe evaluate the performance of the prediction function  $\\eta^{(k)}$\nwith the coefficient  $\\mathrm{R}^2(\\eta^{(k)},\\Dtest)$ \n  of determination  for the test set $\\Dtest:=D_{\\pi}^{(k)}$.   \nThe running time per trial in a cross-validation was at most 8.4 seconds.  \n  \n\\begin{table}[h!]\\caption{Results of Stages 1 and 2 in Phase 1.} \n  \\begin{center}\n    \\begin{tabular}{@{} c c r c  c  r r r   @{}}\\hline\n      $\\pi$ & $\\Lambda$  &  $|D_{\\pi}|$  &  $ \\underline{n},~\\overline{n} $ &   $\\underline{a},~\\overline{a}$ &\n   $|\\Gamma|$   &  $|\\mathcal{F}|$ &   $K$~ \\\\ \\hline\n      {\\sc BHL} & $\\Lambda_2$  & 300  &  5,\\,36  &  0.03,\\,732.99   & 20   & 70  & 120  \\\\\n      {\\sc BHL} & $\\Lambda_7$  & 514  &  5,\\,36  &  0.03,\\,732.99   & 26  & 101  & 166    \\\\\n      {\\sc Bp} & $\\Lambda_2$  & 370  &  4,\\,67  &  -11.7,\\,470.0   & 22   & 130  & 184      \\\\\n      {\\sc Bp} & $\\Lambda_6$  & 444  &  4,\\,67  &  -11.7,\\,470.0   & 26   & 163 & 230     \\\\\n      {\\sc Cp} & $\\Lambda_2$  & 125 &  4,\\,63  &  $4.7\\!\\times\\!10^{-6}$,\\,5.52   & 8  & 75  & 112     \\\\\n      {\\sc Cp} & $\\Lambda_5$  & 131  &  4,\\,63  &  $4.7\\!\\times\\! 10^{-6}$,\\,5.52  & 8   & 79  & 119       \\\\ \n      {\\sc Dc} & $\\Lambda_2$  & 141  &  5,\\,44  &  0.5,\\,17.11   & 20   & 62  & 111     \\\\ \n      {\\sc Dc} & $\\Lambda_6$  & 161  &  5,\\,44  &  0.5,\\,17.11   & 25   & 69  & 130     \\\\ \n      {\\sc Fp} & $\\Lambda_2$  & 368  &  4,\\,67  &  -82.99,\\,300.0   & 20   & 131  & 183      \\\\ \n      {\\sc Fp} & $\\Lambda_6$  & 424  &  4,\\,67  &  -82.99,\\,300.0   & 25   & 161  & 229       \\\\ \n      {\\sc Kov} & $\\Lambda_1$  & 52  &  11,\\,16  &  1422.0,\\,1919.0   & 9   & 33  & 64      \\\\ \n      {\\sc Lp} & $\\Lambda_2$  & 615  &  6,\\,60  &  -3.62,\\,6.84   & 32   & 116  & 186      \\\\ \n      {\\sc Lp} & $\\Lambda_7$  & 936  &  6,\\,74  &  -3.62,\\,6.84  & 44   & 136  & 231     \\\\ \n      {\\sc Lumo} & $\\Lambda_8$  & 977  &  6,\\,9  &  -0.1144,\\,0.1026   & 59   & 190 & 297   \\\\ \n      {\\sc OptR} & $\\Lambda_2$  & 147  & 5,\\,44  &  -117.0,\\,165.0   & 21   & 55  & 107      \\\\ \n      {\\sc OptR} & $\\Lambda_4$  & 157  &  5,\\,69  &  -117.0,\\,165.0   & 25   & 62  & 123     \\\\ \n      {\\sc Sl} & $\\Lambda_2$  & 673  &  4,\\,55  &  -9.332,\\,1.11   & 27   & 154  & 217     \\\\ \n      {\\sc Sl} & $\\Lambda_7$  & 915 &  4,\\,55  &  -11.6,\\,1.11  & 42   & 207  & 300      \\\\ \n      {\\sc SfT} & $\\Lambda_3$  & 247  &  5,\\,33  & 12.3,\\,45.1  & 11   & 91  & 128      \\\\ \n      {\\sc Vis} & $\\Lambda_3$  & 282  &  5,\\,36  &  -0.64,\\,1.63   & 12   & 88  & 126      \\\\ \n      {\\sc ChaR} & $\\Lambda_2$  & 27  &  4,\\,18  &  5.5,\\,13.2  & 22   & 17  & 67 \\\\\n      {\\sc ChaR} & $\\Lambda_{10}$  & 32  & 4,\\,18  &  5.5,\\,13.2   & 26   & 21  & 82  \\\\   \n      {\\sc RfId} & $\\Lambda_{9}$  & 91  &  4,\\,29  &  1.339,\\,1.683   & 26   & 35  & 96 \\\\\n      {\\sc RfId} & $\\Lambda_{11}$  & 124 & 4,\\,29  &  1.339,\\,1.683  & 32 & 50  & 124 \\\\\n      {\\sc RfId} & $\\Lambda_{12}$  & 134  &  4,\\,29  &  1.339,\\,1.71   & 38   & 56  & 144  \\\\\n      \\hline\n  \\end{tabular}\\end{center}\\label{table:phase1a}\n\\end{table} \n\n\\begin{table}[h!]\\caption{Results of Stage 3 in Phase 1.} \n  \\begin{center}\n    \\begin{tabular}{@{} c  r    c c  c c  @{}}\\hline\n      $\\pi$    &  $|D_{\\pi}|$  &      $A_\\pi$ &  $\\rstp_\\pi$ &  ANN $\\mathrm{R}^2$  &  LLR $\\mathrm{R}^2$ \\\\ \\hline\n      {\\sc BHL}    & 300      & $(108,64,64,64,64,64,1)$ & 0.86  & 0.630 &  0.364  \\\\\n      {\\sc BHL}    & 514 & $(46,10,8,1)$ & 0.71 & 0.622 &  0.483  \\\\\n      {\\sc Bp}   & 370    & $(71,28,22,17,13,10,1)$ & 0.93 & 0.765 & 0.599   \\\\\n      {\\sc Bp}   & 444    & $(225,135,135,135,135,135,1)$ & 0.98 & 0.720 &  0.663  \\\\\n      {\\sc Cp}    & 125    & $(19,10,10,10,1)$ & 0.66 & 0.694 &  0.445  \\\\\n      {\\sc Cp}    & 131    & $(19,10,10,10,1)$ & 0.66 & 0.727 & 0.556   \\\\ \n      {\\sc Dc}   & 141    & $(53,10,8,1)$ & 0.93 & 0.651&  0.489  \\\\ \n      {\\sc Dc}    & 161    & $(109,87,87,87,87,1)$ & 0.94 & 0.622 &  0.574  \\\\\n      {\\sc Fp}    & 368      & $(30,10,10,10,10,10,1)$ & 0.88 & 0.746 &  0.589  \\\\ \n      {\\sc Fp}   & 424     & $(42,16,16,16,16,1)$ & 0.90 & 0.733 & 0.571   \\\\ \n      {\\sc Kov}    & 52      & $(28,10,10,1)$ & 0.92 & 0.727 & 0.677   \\\\ \n      {\\sc Lp}   & 615      & $(186,74,74,74,74,74,1)$ & 0.98 & 0.867 &  0.856    \\\\\n      {\\sc Lp}   & 936    & $(197,157,157,157,157,157,1)$ & 0.81 & 0.859 &  0.840   \\\\\n      {\\sc Lumo}    & 977   & $(241,192,192,192,192,192,1)$ & 0.99 & 0.860 &   0.841   \\\\\n      {\\sc OptR}    & 147    & $(107,64,51,40,32,1)$ & 0.97 & 0.919 &  0.823  \\\\ \n      {\\sc OptR}    & 157      & $(114,20,10,1)$ & 0.96 & 0.894 &  0.825  \\\\ \n      {\\sc Sl}    & 673   & $(205,10,5,1)$ & 0.94 & 0.819 &  0.772  \\\\ \n      {\\sc Sl}    & 915    & $(126,25,25,25,1)$ & 0.95 & 0.822 &  0.808  \\\\ \n      {\\sc SfT}     & 247     & $(19,15,12,9,1)$ & 0.91 & 0.834 &  0.804  \\\\ \n      {\\sc Vis}    & 282     & $(19,11,8,6,5,1)$ & 0.97 & 0.929 &  0.893  \\\\ \n      {\\sc ChaR}    & 27   & $(38,22,22,22,22,22,1)$ & 0.98 & 0.641 &  0.431 \\\\\n      {\\sc ChaR}    & 32   & $(38,15,15,15,1)$ & 0.90 & 0.622 &  0.235  \\\\\n      {\\sc RfId}    & 91  & $(31,18,14,11,8,6,1)$ & 0.95 & 0.871&  0.852  \\\\\n      {\\sc RfId}    & 124   & $(60,48,38,30,24,1)$ & 0.90 & 0.891 &  0.832  \\\\\n      {\\sc RfId}    & 134  & $(53,42,33,26,1)$ & 0.94 & 0.866 &  0.832  \\\\\n      \\hline\n  \\end{tabular}\\end{center}\\label{table:phase1b}\n\\end{table}\n\n\n\n Table~\\ref{table:phase1b}   shows the results on Stage~3,\n where  we denote the following:     \n\\begin{enumerate}[nosep,  leftmargin=*]  \n\\item[-]  \n$A_\\pi$: an architecture $A_\\pi$  used to construct a prediction function for property $\\pi$,\n where  $(K',p_1,p_2,$ $\\ldots,p_\\ell, 1)$ means an architecture\n with   an input layer with $K'$ nodes, \n $\\ell$ hidden layers  with $p_i, i\\in[1,\\ell]$ nodes \nand an output layer with a single node, where $K'$ is\nthe size $|S_\\pi|$  of the set $S_\\pi$ of selected  descriptors from the original set of $K$ descriptors.\n  \n\\item[-]\n$\\rstp_\\pi$:  \n  a nonnegative real with $0\\leq \\rstp_\\pi\\leq 1$ by which we execute an early stopping \n  in constructing a prediction function with a training data set. \n  \n\\item[-]\nANN $\\mathrm{R}^2$: the median of test $\\mathrm{R}^2$ over all 50 trials\n  in ten 5-fold cross-validations for prediction functions constructed\n  with ANNs in this paper.   \n  \n\\item[-]\nLLR $\\mathrm{R}^2$: the median of test $\\mathrm{R}^2$ over all 50 trials\n  in ten 5-fold cross-validations for prediction functions constructed\n  with Lasso linear regression~\\cite{ZAHZNA21}.  \n\\end{enumerate}\n  \n\n\n\n\\section{Specifying Target Chemical Graphs}\\label{sec:specification} \n\nGiven a prediction function $\\eta$ and \na target value $y^*\\in \\mathbb{R}$, \nwe call a chemical graph $\\C^*$ such that $\\eta(x^*)=y^*$\nfor the feature vector $x^*=f(\\C^*)$ a {\\em target chemical graph}.\nThis section  presents a set of rules for \n specifying  topological substructure\n  of a target chemical graph in a flexible way in Stage~4. \n \n\nWe first describe how to reduce a chemical graph $\\C=(H,\\alpha,\\beta)$ into\nan abstract form based on which our specification rules will be defined.\nTo illustrate the reduction process,\nwe use the chemical graph $\\C=(H,\\alpha,\\beta)$\nsuch that $\\anC$ is given in Figure~\\ref{fig:example_chemical_graph}.\n \n \\begin{enumerate\n \\item[R1] {\\bf Removal of all ${\\rho}$-fringe-trees: } \nThe interior $H^\\inte=(V^\\inte(\\C),E^\\inte(\\C))$ of $\\C$ \nis obtained by removing the non-root vertices of \neach ${\\rho}$-fringe-trees $\\C[u]\\in\\mathcal{T}(\\C), u\\in V^\\inte(\\C)$. \nFigure~\\ref{fig:specification_example_interior} illustrates\nthe interior $H^\\inte$ of \nchemical graph $\\C$ with ${\\rho}=2$\n  in Figure~\\ref{fig:example_chemical_graph}. \n  \n \\item[R2] {\\bf Removal of some leaf paths: } \n We call a $u,v$-path $Q$ in $H^\\inte$  a {\\em leaf path} if \n  vertex $v$ is a leaf-vertex of $H^\\inte$\n  and the degree of each internal vertex of $Q$  in $H^\\inte$  is 2,\n  where we regard that $Q$ is rooted at vertex $u$. \nA connected subgraph $S$ of the interior $H^\\inte$ of $\\C$  \nis called a {\\em cyclical-base}\nif $S$ is obtained from $H$\nby removing the vertices in $V(Q_u)\\setminus \\{u \\}, u\\in X$ \nfor a subset $X$ of interior-vertices  and a set  $\\{Q_u \\mid u\\in X\\}$ of leaf \n $u,v$-paths $Q_u$  such that    \n no two paths $Q_u$ and $Q_{u'}$ share a vertex.\nFigure~\\ref{fig:specification_example_R2_3}(a) illustrates\na cyclical-base  $S=H^\\inte- \\bigcup_{u\\in X}(V(Q_u)\\setminus \\{u\\})$\nof the interior  $H^\\inte$  \nfor a set \n$\\{Q_{u_5}=(u_5,u_{24}), \n     Q_{u_{18}}=(u_{18},u_{25},u_{26},u_{27}),\n     Q_{u_{22}}=(u_{22},u_{28})\\}$ of leaf  paths \nin Figure~\\ref{fig:specification_example_interior}.  \n\n \\item[R3] {\\bf Contraction of some pure paths: } \n A path in $S$ is called {\\em pure} \n if  each internal vertex of the path  is of degree 2. \n Choose a set $\\mathcal{P}$ of several pure paths in $S$ \n so that no two paths share  vertices except for their end-vertices. \n A graph $S'$ is called a {\\em contraction} of a graph $S$\n  (with respect to $\\mathcal{P}$) \n if $S'$ is obtained from $S$ by replacing \n each pure $u,v$-path  with a single edge $a=uv$,\n where $S'$ may contain multiple edges between the same pair of adjacent vertices.\nFigure~\\ref{fig:specification_example_R2_3}(b) illustrates\na contraction $S'$ obtained from \nthe chemical graph  $S$\nby contracting each $uv$-path $P_a\\in  \\mathcal{P}$ into a new edge $a=uv$,\nwhere $a_1=u_1 u_{2},  a_2=u_1 u_{3},  a_3=u_4 u_{7}, a_4=u_{10}u_{11}$\nand $a_5=u_{11}u_{12}$ and \n %\n $\\mathcal{P}=\\{\n P_{a_1}=(u_1,u_{13},u_{2}), \n P_{a_2}=(u_{1},u_{14},u_{3}),\n P_{a_3}=(u_{4},u_{15},u_{16},u_{7}), \n P_{a_4}=(u_{10},u_{17},u_{18},u_{19},u_{11}),\n P_{a_5}=(u_{11},u_{20},u_{21},u_{22},u_{12})\\}$ of pure paths \nin Figure~\\ref{fig:specification_example_R2_3}(a). \n\\end{enumerate}\n  \n  \n\\begin{figure}[h!] \\begin{center}\n\\includegraphics[width=.65\\columnwidth]{fig\/specification_example_interior.eps}\n\\end{center} \\caption{The interior $H^\\inte$ of\nchemical graph $\\C$ with $\\anC$ \n  in Figure~\\ref{fig:example_chemical_graph} for ${\\rho}=2$.\n}\n\\label{fig:specification_example_interior} \\end{figure} \n\n\n\\begin{figure}[h!] \\begin{center}\n\\includegraphics[width=.98\\columnwidth]{fig\/specification_example_R2_3.eps}\n\\end{center} \\caption{\n(a) A cyclical-base  \n$S=H^\\inte- \\bigcup_{u\\in \\{u_5,u_{18},u_{22}\\}}(V(Q_u)\\setminus \\{u\\})$\nof the interior  $H^\\inte$ in Figure~\\ref{fig:specification_example_interior};\n(b) A contraction $S'$ of  $S$ for a pure path set \n $\\mathcal{P}=\\{P_{a_1},P_{a_2},\\ldots,P_{a_5}\\}$ \nin (a),\nwhere a new edge obtained by contracting a pure path is depicted\nwith a thick line.   \n}\n\\label{fig:specification_example_R2_3} \\end{figure} \n  \nWe will define a set of rules so that \na chemical graph can be obtained \nfrom a graph (called a seed graph in the next section)\nby applying processes R3 to R1 in a reverse way. \nWe specify topological substructures of a target chemical graph\nwith a tuple  $(\\GC,\\sint,\\sce)$  called  a {\\em target specification}\ndefined under the set of the following rules. \n\n\\subsection*{Seed Graph\n\nA  {\\em seed graph} $\\GC=(\\VC,\\EC)$ is defined\nto be a graph (possibly with multiple edges) such that \nthe edge set $\\EC$ consists of four sets \n$\\Et$, $\\Ew$, $\\Ez$ and $\\Eew$, \nwhere each of them can be empty.\nA seed graph plays a role of the most abstract form $S'$ in R3.  \nFigure~\\ref{fig:specification_example_1}(a) illustrates an example of a seed graph\n$\\GC$ with $\\mathrm{r}(\\GC)=5$,   \nwhere $\\VC=\\{u_1,u_2,\\ldots,u_{12},u_{23}\\}$, \n$\\Et=\\{a_1,a_2,\\ldots,a_5\\}$, \n$\\Ew=\\{a_6\\}$,\n$\\Ez=\\{a_7\\}$ and \n$\\Eew=\\{a_8,a_9,\\ldots,a_{16}\\}$.\n\n A {\\em subdivision} $S$ of $\\GC$  \nis a graph constructed from a seed graph $\\GC$ \naccording to the following rules:\n\\begin{enumerate}[leftmargin=*]\n\\item[-]\nEach edge $e=uv\\in \\Et$ is replaced\nwith a $u,v$-path $P_e$ of length at least 2;\n\n\\item[-] \nEach edge $e=uv\\in \\Ew$ is replaced\nwith a $u,v$-path $P_e$ of length at least 1\n(equivalently $e$ is directly used or replaced with\na $u,v$-path $P_e$ of length at least 2);\n\n\\item[-] \nEach edge $e\\in \\Ez$ is either used or discarded, where \n $\\Ez$ is required to be chosen as a non-separating edge subset of\n $E(\\GC)$ since otherwise the connectivity of a final chemical graph $\\Co$\n is not guaranteed; \n$\\mathrm{r}(\\Co)= \\mathrm{r}(\\GC)-|E'|$ holds\nfor a subset $E'\\subseteq \\Ez$ of edges discarded \nin a  final chemical graph $\\Co$; \nand \n\n\\item[-]\nEach edge $e\\in \\Eew$ is always used directly. \n\\end{enumerate}\n\nWe allow a possible elimination of edges in $\\Ez$ as an optional rule\nin constructing a target chemical graph from a seed graph, \neven though such an operation has \nnot been included in the process R3. \nA subdivision  $S$ plays a role of a cyclical-base   in R2. \nA target chemical graph $\\C=(H,\\alpha,\\beta)$ will contain  $S$  as a subgraph\nof the interior $H^\\inte$ of $\\C$.\n\n\n\\subsection*{Interior-specification\n\nA graph $H^*$ that serves as the interior $H^\\inte$ of\na target chemical graph $\\C$ will be constructed as follows.\nFirst construct a subdivision  $S$ of a seed graph $\\GC$ \nby replacing each edge $e=u u'\\in \\Et\\cup\\Ew$\nwith a pure $u,u'$-path $P_e$.\nNext construct a supergraph $H^*$ of $S$ by \nattaching a leaf path $Q_v$ at each vertex $v\\in \\VC$ or\nat an internal vertex $v\\in V(P_e)\\setminus\\{u,u'\\}$ \nof each pure $u,u'$-path $P_e$ for some edge $e=uu'\\in \\Et\\cup\\Ew$,\nwhere possibly $Q_v=(v), E(Q_v)=\\emptyset$ \n(i.e., we do not attach any new edges to $v$).\nWe introduce the following rules for specifying\n the size of $H^*$, the length $|E(P_e)|$  of\na pure path  $P_e$,  the length $|E(Q_v)|$ of\na   leaf path $Q_v$, the number of  leaf paths $Q_v$\nand a bond-multiplicity of each interior-edge,\nwhere we call the set of prescribed constants  \n an  {\\em interior-specification}   $\\sint$: \n %\n\\begin{enumerate}[leftmargin=*]\n \\item[-]\n  Lower and upper bounds $\\nint_\\LB, \\nint_\\UB\\in \\mathbb{Z}_+$ \n  on   the number of interior-vertices \nof a target chemical graph~$\\C$. \n  \n\\item[-] \nFor each edge $e=u u'\\in \\Et\\cup\\Ew$, \n\\begin{description} \n\\item[]\n a lower bound $\\ell_{\\LB}(e)$ and \n an upper bound $\\ell_{\\UB}(e)$  on the length $|E(P_e)|$ of\n a pure $u,u'$-path $P_e$. \n(For a notational convenience, set \n$\\ell_\\LB(e):=0$, $\\ell_\\UB(e):=1$, $e\\in \\Ez$ and\n$\\ell_\\LB(e):=1$, $\\ell_\\UB(e):=1$, $e\\in \\Eew$.)\n   \n\\item[]  \n a lower bound $\\bl_{\\LB}(e)$ and \n an upper bound $\\bl_{\\UB}(e)$ on the number of leaf paths $Q_v$ attached \n at  internal vertices $v$ of a pure $u,u'$-path $P_e$.   \n\n\\item[] \n a lower bound $\\ch_{\\LB}(e)$ and \n an upper bound $\\ch_{\\UB}(e)$  on the maximum \n length  $|E(Q_v)|$ of a leaf path $Q_v$ attached  \n at an internal vertex $v\\in V(P_e)\\setminus\\{u,u'\\}$ \n of a pure $u,u'$-path $P_e$.   \n\\end{description} \n\n\\item[-]\nFor each vertex $v\\in \\VC$, \n\\begin{description} \n\\item[]\n a lower bound $\\ch_{\\LB}(v)$ and \n an upper bound $\\ch_{\\UB}(v)$  on  \n the number of leaf paths $Q_v$ attached to $v$,\n where $0\\leq \\ch_{\\LB}(v)\\leq \\ch_{\\UB}(v)\\leq 1$.\n \n\\item[]\n a lower bound $\\ch_{\\LB}(v)$ and \n an upper bound $\\ch_{\\UB}(v)$  on the\n length $|E(Q_v)|$ of a leaf path $Q_v$ attached to $v$. \n\\end{description}  \n\n\\item[-] \nFor each edge $e=u u'\\in \\EC$, \na lower bound $\\bd_{m, \\LB}(e)$ \nand an  upper bound $\\bd_{m, \\UB}(e)$  on\nthe number of edges with bond-multiplicity $m\\in [2,3]$ in\n$u,u'$-path $P_e$, where we regard $P_e$, $e  \\in \\Ez\\cup \\Eew$ \nas single edge $e$.\n\\end{enumerate}\n\nWe call a graph $H^*$ that satisfies an interior-specification $\\sint$\na {\\em $\\sint$-extension of $\\GC$}, \nwhere the bond-multiplicity of each edge has been determined. \n \n\nTable~\\ref{table:interior-spec}  shows an example of \nan interior-specification  $\\sint$ to the seed graph  $\\GC$ in \nFigure~\\ref{fig:specification_example_1}. \n\n\\begin{table}[h!]\\caption{Example~1 of an interior-specification  $\\sint$. }\n\\begin{tabular}{ |  c | c |  } \\hline \n$\\nint_\\LB=20$ & $\\nint_\\UB = 28$ \\\\\\hline \n\\end{tabular}\n\n \n \\begin{tabular}{ |  c | c c c c c c |  } \\hline\n                        & $a_1$ &  $a_2$ &   $a_3$ &   $a_4$ &   $a_5$ &   $a_6$   \\\\\\hline\n $\\ell_\\LB(a_i)$&  2 &  2 &  2 & 3 &  2 &  1 \\\\ \\hline\n $\\ell_\\UB(a_i)$&  3 & 4 &  3 & 5 & 4 &  4 \\\\\\hline\n $\\bl_\\LB(a_i)$&  0 &  0 &   0 & 1 &  1 &   0 \\\\ \\hline\n $\\bl_\\UB(a_i)$&  1 & 1 &   0 & 2 & 1 &   0 \\\\\\hline\n $\\ch_\\LB(a_i)$&  0 &  1 & 0 & 4 &  3 &  0 \\\\ \\hline\n $\\ch_\\UB(a_i)$&  3 & 3 &  1 & 6 & 5 &  2 \\\\\\hline\n\\end{tabular} \n\n\\begin{tabular}{ |  c | c c c c c c   c c c c  c c c |  } \\hline\n                        & $u_1$ &  $u_2$ &   $u_3$ &   $u_4$ &   $u_5$ &   $u_6$ \n                       & $u_7$ &   $u_8$ &   $u_9$ &   $u_{10}$ &   $u_{11}$ \n                       &   $u_{12}$ &   $u_{23}$ \\\\\\hline \n $\\bl_\\LB(u_i)$&  0 &  0 &   0 & 0 &  0 &   0\n                       & 0 &   0 &  0 &   0 &  0 &  0 &  0 \\\\ \\hline\n $\\bl_\\UB(u_i)$&  1 & 1 &   1 & 1 & 1 &   0\n                       & 0 &   0 &  0 &   0 &  0 &  0 &  0\\\\\\hline\n $\\ch_\\LB(u_i)$&  0 &  0 &   0 & 0 &  1 &   0\n                       & 0 &   0 &  0 &   0 &  0 &  0 &  0 \\\\ \\hline\n $\\ch_\\UB(u_i)$&  1 & 0 &   0 & 0 & 3 &   0\n                       & 1 &   1 &  0 &   1 &  2 & 4 &  1 \\\\\\hline\n\\end{tabular} \n\n\\begin{tabular}{ |  c | c c c c c c   c c c c c c  c c c c c |  } \\hline\n                               & $a_1$ &  $a_2$ &   $a_3$ &   $a_4$ &   $a_5$ &   $a_6$ \n                               & $a_7$ &  $a_8$ &   $a_9$ &   $a_{10}$ &   $a_{11}$ &   $a_{12}$ \n                               & $a_{13}$ &   $a_{14}$ &   $a_{15}$ &   $a_{16}$ &   $a_{17}$  \\\\\\hline\n $\\bd_{2, \\LB}(a_i)$ &  0    &  0 &   0 & 1 &  0 &   0\n                                &  0   &  0 &  0 & 0 &  0 &   1\n                                &  0    &  0 &   0 & 0     & 0  \\\\ \\hline\n $ \\bd_{2, \\UB}(a_i)$&  1    & 1 &   0 & 2  & 2 &   0  \n                                &  0    & 0&   0 & 0 &  0 &   1\n                                &  0    &  0 &   0 & 0   & 0   \\\\ \\hline\n $\\bd_{3, \\LB}(a_i)$ &  0    &  0 &   0 & 0 &  0 &   0\n                                &  0   &  0 &  0 & 0 &  0 &   0\n                                &  0    &  0 &   0 & 0   & 0   \\\\ \\hline\n $ \\bd_{3, \\UB}(a_i)$&  0    & 0 &   0 & 0  & 1 &   0 \n                                &  0    &  0 &   0 & 0 &  0 &   0\n                                &  0    &  0 &   0 &  0    & 0   \\\\ \\hline\n\\end{tabular}\n\\label{table:interior-spec}  \n\\end{table}\n   \n   \nFigure~\\ref{fig:specification_example_3} illustrates an example of \nan $\\sint$-extension $H^*$ of seed graph  $\\GC$ in \nFigure~\\ref{fig:specification_example_1}\nunder the interior-specification  $\\sint$ in \nTable~\\ref{table:interior-spec}.  \n\n\\begin{figure}[h!] \\begin{center}\n\\includegraphics[width=.58\\columnwidth]{fig\/specification_example_3_test.eps}\n\\end{center} \\caption{\nAn illustration of a graph \n$H^*$ that is obtained from  the seed graph  $\\GC$ in \nFigure~\\ref{fig:specification_example_1}\nunder the interior-specification  $\\sint$ in \nTable~\\ref{table:interior-spec},\nwhere the vertices newly introduced by pure paths $P_{a_i}$\nand leaf paths $Q_{v_i}$ are depicted with white squares and circles,\nrespectively.    }\n\\label{fig:specification_example_3} \\end{figure} \n\n\n\n\\subsection*{Chemical-specification\n \n Let $H^*$ be a graph that serves as \n the interior $H^\\inte$ of a target chemical graph $\\C$,\n where the bond-multiplicity of each edge in $H^*$ has be determined.\n\n Finally we introduce a set of rules for constructing \n   a target chemical graph $\\C$ from $H^*$ \n   by choosing  a chemical element $\\ta\\in \\Lambda$ \nand assigning a ${\\rho}$-fringe-tree $\\psi$\n to each interior-vertex $v\\in V^\\inte$. \nWe introduce the following rules for specifying\nthe size of $\\C$, a set of chemical rooted trees  \nthat are allowed to use as  ${\\rho}$-fringe-trees \nand lower and upper bounds on the frequency of\na chemical element, a chemical symbol,\nand an edge-configuration,\nwhere we call the set of prescribed constants   \n a  {\\em chemical specification} $\\sce$:   \n \n\\begin{enumerate}[leftmargin=*]\n\\item[-] \nLower and upper bounds $n_\\LB,  n^*\\in \\mathbb{Z}_+$\non the number of vertices, where $\\nint_\\LB \\leq n_\\LB\\leq n^*$.\n \n\\item[-] \nSubsets  $\\mathcal{F}(v) \\subseteq \\mathcal{F}(D_\\pi), v\\in \\VC$ \nand $\\mathcal{F}_E \\subseteq \\mathcal{F}(D_\\pi)$ \n of chemical rooted trees $\\psi$ with $\\h(\\anpsi)\\leq {\\rho}$, where \n we require that \n every ${\\rho}$-fringe-tree $\\C[v]$ rooted at a vertex $v\\in \\VC$\n (resp., at an internal vertex $v$ not in $\\VC$)   in  $\\C$ \n belongs to $\\mathcal{F}(v)$ (resp.,   $\\mathcal{F}_E$).  \nLet  $\\mathcal{F}^*:=\\mathcal{F}_E\\cup \\bigcup_{v\\in \\VC}\\mathcal{F}(v)$\nand \n$\\Lambda^\\ex$ denote the set of  chemical elements assigned to non-root\nvertices over all chemical rooted trees in $\\mathcal{F}^*$.  \n \n\\item[-] \nA subset  $\\Lambda^\\inte\\subseteq \\Lambda^\\inte(D_\\pi)$, where \n we require that every chemical element $\\alpha(v)$ \n assigned to an interior-vertex  $v$ in $\\C$ belongs to $\\Lambda^\\inte$.\nLet $\\Lambda:= \\Lambda^\\inte\\cup \\Lambda^\\ex$ and\n $\\na_\\ta(\\C)$ (resp., $\\na_\\ta^\\inte(\\C)$ and $\\na_\\ta^\\ex(\\C)$) \n denote the number of vertices   (resp.,   interior-vertices and  exterior-vertices)\n  $v$ such that $\\alpha(v)=\\ta$   in  $\\C$.\n \n\\item[-] \nA set $\\Ldg^\\inte\\subseteq \\Lambda\\times [1,4]$  of chemical  symbols\nand  a set $\\Gamma^\\inte \\subseteq \\Gamma^\\inte(D_\\pi)$  \nof  edge-configurations  $(\\mu,\\mu' ,m)$ with $\\mu \\leq \\mu'$, where \n we require that the edge-configuration $\\ec(e)$ of an interior-edge $e$ in $\\C$ \n belongs to $\\Gamma^\\inte$.\nWe do not distinguish  $(\\mu,\\mu' ,m)$ and $(\\mu' , \\mu,m)$.  \n \n \n\\item[-] \nDefine  $\\Gac^\\inte$ to be the set of   adjacency-configurations such that  \n$\\Gac^\\inte:=\\{(\\ta, \\tb, m) \\mid (\\ta d, \\tb d',m)\\in \\Gamma^\\inte\\}$.   \nLet  $\\ac_\\nu^\\inte(\\C), \\nu\\in \\Gac^\\inte$   \ndenote  the number of  interior-edges $e$ such that $\\ac(e)=\\nu$  in $\\C$.\n  \n\\item[-] \n Subsets $\\Lambda^*(v)\\subseteq \\{\\ta\\in \\Lambda^\\inte\\mid \\val(\\ta)\\geq 2\\}$, \n $v\\in \\VC$,  \n we require that every chemical element $\\alpha(v)$ \n assigned to   a vertex $v\\in  \\VC$\n in the seed graph  belongs to $\\Lambda^*(v)$.  \n\n\\item[-] Lower and upper bound functions \n$\\na_\\LB,\\na_\\UB: \\Lambda\\to  [1,n^*]$  and \n$\\na_\\LB^\\inte,\\na_\\UB^\\inte: \\Lambda^\\inte\\to  [1,n^*]$ \non the number of   interior-vertices  $v$ such that  $\\alpha(v)=\\ta$  in $\\C$. \n\n\\item[-] Lower and upper bound functions  \n$\\ns_\\LB^\\inte,\\ns_\\UB^\\inte: \\Ldg^\\inte\\to  [1,n^*]$ \n  on the number of   interior-vertices $v$ such that $\\cs(v)=\\mu$  in $\\C$.   \n\n\\item[-] Lower and upper bound functions  \n$\\ac_\\LB^\\inte,\\ac_\\UB^\\inte: \\Gac^\\inte \\to  \\mathbb{Z}_+$ \n on the number of  interior-edges $e$ such that $\\ac(e)=\\nu$  in $\\C$. \n\n\\item[-] Lower and upper bound functions  \n$\\ec_\\LB^\\inte,\\ec_\\UB^\\inte: \\Gamma^\\inte \\to  \\mathbb{Z}_+$ \n on the number of  interior-edges $e$ such that $\\ec(e)=\\gamma$  in $\\C$.  \n \n \\item[-] Lower and upper bound functions  \n$\\fc_\\LB,\\fc_\\UB: \\mathcal{F}^*\\to  [0,n^*]$ \n  on the number of   interior-vertices $v$ \n  such that $\\C[v]$ is r-isomorphic to $\\psi\\in \\mathcal{F}^*$  in $\\C$.   \n  \n \\item[-] Lower and upper bound functions  \n$\\ac^\\lf_\\LB,\\ac^\\lf_\\UB: \\Gac^\\lf \\to  [0,n^*]$ \n  on the number of  leaf-edges $uv$ in $\\acC$\n  with adjacency-configuration $\\nu$.  \n\\end{enumerate}\n \nWe call a chemical graph $\\C$ that satisfies a chemical specification $\\sce$\na {\\em $(\\sint,\\sce)$-extension of $\\GC$},\nand denote by $\\mathcal{G}(\\GC, \\sint,\\sce)$ the set of\nall $(\\sint,\\sce)$-extensions of $\\GC$. \n\nTable~\\ref{table:chemical_spec}  shows an example of \na chemical-specification  $\\sce$ to the seed graph  $\\GC$\n in Figure~\\ref{fig:specification_example_1}. \n \n\n\\begin{table}[h!]\\caption{Example~2 of a chemical-specification  $\\sce$.  \n}\n\\begin{tabular}{ |  l |  } \\hline\n $n_\\LB=30$,  $n^* =50$. \\\\\\hline\n  branch-parameter:   ${\\rho}=2$  \\\\\\hline\n\\end{tabular}\n\n\\begin{tabular}{ |  l |  } \\hline\n Each of sets $\\mathcal{F}(v), v\\in \\VC$ and\n $\\mathcal{F}_E$ is set to be \\\\\n the set $\\mathcal{F}$  of chemical rooted trees $\\psi$ with $\\h(\\anpsi)\\leq {\\rho}=2$\nin Figure~\\ref{fig:specification_example_1}(b). \\\\\\hline\n\\end{tabular}\n\n\\begin{tabular}{ |  c | c |   } \\hline\n  $\\Lambda=\\{ \\ttH,\\ttC,\\ttN,\\ttO, \\ttS_{(2)},\\ttS_{(6)}, \\ttP=\\ttP_{(5)}\\}$ & \n  $\\Ldg^\\inte =\\{ \\ttC2 , \\ttC3,  \\ttC4, \\ttN2, \\ttN3, \\ttO2,\n    \\ttS_{(2)}2,  \\ttS_{(6)}3, \\ttP4   \\}$  \n\\\\\\hline\n\\end{tabular}\n\n\\begin{tabular}{ |  c | l |  } \\hline\n  $\\Gac^{\\inte}$ &\n  $ \\nu_1 \\!=\\!(\\ttC   , \\ttC  , 1) ,   \\nu_2 \\!=\\!(\\ttC   , \\ttC  , 2) ,   \n   \\nu_3 \\!=\\!(\\ttC   , \\ttN  , 1) ,  \\nu_4 \\!=\\!(\\ttC  , \\ttO  , 1), \n    \\nu_5 \\!=\\! (\\ttC, \\ttS_{(2)}, 1),\\nu_6 \\!=\\!(\\ttC  , \\ttS_{(6)}, 1), \n    \\nu_7 \\!=\\! (\\ttC  , \\ttP  , 1) $  \\\\ \\hline\n\\end{tabular}\n    \n\n\\begin{tabular}{ |  c | l |  } \\hline\n  $\\Gamma^{\\inte}$ &\n  $ \\gamma_1 \\!=\\! (\\ttC 2 , \\ttC 2, 1) ,\n   \\gamma_2 \\!=\\!(\\ttC 2 , \\ttC 3, 1) ,  \n   \\gamma_3 \\!=\\!(\\ttC 2 , \\ttC 3, 2) ,  \n   \\gamma_4 \\!=\\!(\\ttC 2 , \\ttC 4, 1) , \n   \\gamma_5 \\!=\\!(\\ttC 3 , \\ttC 3, 1) , \n   \\gamma_6 \\!=\\!(\\ttC 3 , \\ttC 3, 2) , $ \\\\\n   &\n  $   \n    \\gamma_7 \\!=\\!(\\ttC 3 , \\ttC 4, 1), \n   \\gamma_8 \\!=\\!(\\ttC 2 , \\ttN 2, 1) ,  \n   \\gamma_9 \\!=\\!(\\ttC 3 , \\ttN 2, 1) ,  \n   \\gamma_{10} \\!=\\!(\\ttC 3 , \\ttO 2, 1), \n    \\gamma_{11} \\!=\\!(\\ttC 2 , \\ttC 2, 2),  \n    \\gamma_{12} \\!=\\!(\\ttC 2 , \\ttO 2, 1) ,$ \\\\\n   &\n  $  \n    \\gamma_{13} \\!=\\!(\\ttC 3 , \\ttN3, 1), \n    \\gamma_{14} \\!=\\!(\\ttC 4, \\ttS_{(2)} 2, 2),  \n    \\gamma_{15} \\!=\\!(\\ttC 2 , \\ttS_{(6)}3, 1), \n   \\gamma_{16} \\!=\\!(\\ttC 3 , \\ttS_{\\tiny (6)}3, 1), \n    \\gamma_{17} \\!=\\!(\\ttC 2, \\ttP4, 2), $ \\\\\n   &\n  $  \n    \\gamma_{18} \\!=\\!(\\ttC 3, \\ttP4, 1)  \n     $ \\\\ \\hline\n\\end{tabular}\n    \n\\begin{tabular}{ |  l|  } \\hline\n$\\Lambda^*(u_1)=\\Lambda^*(u_8)=\\{{\\tt C,  N}\\}$, \n$\\Lambda^*(u_9)=\\{{\\tt C, O}\\}$, \n   $\\Lambda^*(u)=\\{\\ttC\\}$, $u\\in \\VC\\setminus\\{u_1,u_8,u_9\\}$\n   \\\\\\hline\n\\end{tabular}\n\n\\begin{tabular}{ |  c | c c c c  c c c |  } \\hline\n                         & ${\\tt H}$  & ${\\tt C}$ &   ${\\tt N}$ &     ${\\tt O}$ \n                         & $\\ttS_{(2)}$ & $\\ttS_{(6)}$ & $\\ttP$  \\\\\\hline\n $\\na_\\LB(\\ta)$ & 40 &  27 &  1 &   1 & 0 & 0 & 0   \\\\ \\hline \n $\\na_\\UB(\\ta)$ & 65 & 37 & 4 &  8  &   1 &   1 &   1 \\\\\\hline\n\\end{tabular} \n\\begin{tabular}{ |  c | c c c  c c c   |  } \\hline\n   & $\\ttC$ &   $\\ttN$ &     $\\ttO$  & $\\ttS_{(2)}$ & $\\ttS_{(6)}$ & $\\ttP$  \\\\\\hline\n $\\na_\\LB^{\\inte}(\\ta)$ &   9 &  1 &   0  & 0 & 0 & 0      \\\\ \\hline\n $\\na_\\UB^{\\inte}(\\ta) $&  23 & 4 & 5 &   1 &   1 &   1  \\\\\\hline\n\\end{tabular} \n\n\\begin{tabular}{ |  c | c c c c c c  c c c   |  } \\hline\n    & $\\ttC2$ &  $\\ttC3$ &   $\\ttC4$ & $\\ttN2$ &   $\\ttN3$ &   $\\ttO2$\n   & $\\ttS_{(2)}2$ & $\\ttS_{(6)}3$ & $\\ttP4$  \\\\\\hline\n $\\ns_\\LB^{\\inte}(\\mu)$ &  3 &  5 &   0 & 0 &  0 &   0 & 0 &  0 &   0    \\\\ \\hline\n $\\ns_\\UB^{\\inte}(\\mu) $&  8 & 15 & 2 & 2 & 3 &  5  &   1 &   1 &   1   \\\\\\hline\n\\end{tabular} \n\n\\begin{tabular}{ |  c | c c c c c c c |  } \\hline\n         & $\\nu_1 $ &   $\\nu_2 $ & $\\nu_3 $   & $\\nu_4 $\n         &   $\\nu_5 $ & $\\nu_6 $   & $\\nu_7 $ \\\\\\hline\n $\\ac_\\LB^{\\inte}(\\nu)$  &  0  &  0  & 0  & 0   & 0  & 0 & 0     \\\\ \\hline\n $\\ac_\\UB^{\\inte}(\\nu)$ &  30 & 10 & 10 & 10 & 1 & 1 & 1 \\\\\\hline\n\\end{tabular} \n\n\\begin{tabular}{ |  c | c c c c c c c c c c c c c c c c c c |  } \\hline\n    & $\\gamma_1 $ &   $\\gamma_2 $ & $\\gamma_3 $   & $\\gamma_4 $ \n     & $\\gamma_5 $\n    & $\\gamma_6 $ &   $\\gamma_7 $ & $\\gamma_8 $   & $\\gamma_9 $ \n     & $\\gamma_{10} $   & $\\gamma_{11} $       & $\\gamma_{12} $          \n     & $\\gamma_{13} $   & $\\gamma_{14} $       & $\\gamma_{15} $          \n     & $\\gamma_{16} $   & $\\gamma_{17} $       & $\\gamma_{18} $          \n                            \\\\\\hline\n $\\ec_\\LB^{\\inte}(\\gamma)$ &  0 &  0 & 0 &  0  & 0 &  0 &  0 & 0 &  0  & 0 & 0 & 0 \n    &  0 & 0 &  0  & 0 & 0 & 0  \\\\ \\hline\n $\\ec_\\UB^{\\inte}(\\gamma) $& 4 & 15 & 4 &  4  & 10 &  5 & 4 & 4 &  6 & 4 & 4 & 4\n &  2 & 2 &  2  & 2 & 2 & 2  \\\\\\hline\n\\end{tabular} \n\n\n\\begin{tabular}{ |  c | c   c   |  } \\hline \n& $\\psi\\in\\{\\psi_i\\mid i=1,6,11\\}$ \n& $\\psi\\in \\mathcal{F}^*\\setminus \\{\\psi_i\\mid i=1,6,11\\}$ \\\\\\hline\n $\\fc_\\LB(\\psi)$  &  1 &    0   \\\\ \\hline \n $\\fc_\\UB(\\psi)$ &  10 &  3\\\\\\hline\n\\end{tabular} \n \n\n\\begin{tabular}{ |  c | c   c   |  } \\hline \n& $\\nu\\in\\{(\\ttC,\\ttC,1),(\\ttC,\\ttC,2)  \\}$ \n& $\\nu\\in \\Gac^\\lf \\setminus \\{(\\ttC,\\ttC,1),(\\ttC,\\ttC,2)  \\}$   \\\\\\hline\n $\\ac^\\lf_\\LB(\\nu)$  &  0 &    0   \\\\ \\hline \n $\\ac^\\lf_\\UB(\\nu)$ &  10 &  8 \\\\\\hline\n\\end{tabular} \n\n\\label{table:chemical_spec}\n\\end{table}\n\n \nFigure~\\ref{fig:example_chemical_graph} \n illustrates an example $\\Co$ of \na   $(\\sint,\\sce)$-extension of $\\GC$   obtained \nfrom the  $\\sint$-extension $H^*$  \n in Figure~\\ref{fig:specification_example_3} \nunder the chemical-specification $\\sce$ in Table~\\ref{table:chemical_spec}.  \nNote that $\\mathrm{r}(\\Co)= \\mathrm{r}(H^*)= \\mathrm{r}(\\GC)-1=4$\n holds since the edge in $\\Ez$ is discarded in $H^*$. \n  \n  \n\\section{Test Instances for Stages~4 and 5}\\label{sec:test_instances}\n\nWe prepared the following instances (a)-(d) for conducting experiments\nof  Stages~4  and 5 in Phase~2. \n \n In Stages~4 and 5, we  use three properties \n $\\pi\\in \\{${\\sc OptR}, {\\sc SfT}, {\\sc Vis}$\\}$ \n and define a set $\\Lambda(\\pi)$ of chemical elements as follows:  \n $\\Lambda(${\\sc OptR}$)=\\Lambda_5=\\{\\ttH,\\ttC,\\ttO, \\ttN,\\ttS_{(2)},\\ttF \\}$ and \n   $\\Lambda(${\\sc SfT}$)=\\Lambda(${\\sc Vis}$)=\\Lambda_4=\\{\\ttH,\\ttC,\\ttO, \\ttSi_{(4)} \\}$. \n \n\\begin{itemize} \n  \\item[(a)]  $I_{\\mathrm{a}} =(\\GC,\\sint,\\sce)$: The instance\n  introduced in Section~\\ref{sec:specification} to explain the target specification.\nFor each property $\\pi$, we replace\n $\\Lambda=\\{ \\ttH,\\ttC,\\ttN,\\ttO, \\ttS_{(2)},\\ttS_{(6)},\\ttP_{(5)}\\}$\nin Table~\\ref{table:chemical_spec} \n with $\\Lambda(\\pi)\\cap \\{\\ttS_{(2)},\\ttS_{(6)},\\ttP_{(5)}\\}$\n and  remove from the $\\sce$\n all chemical symbols,  edge-configurations and fringe-configurations\n  that cannot be constructed from the replaced element set \n (i.e., those containing a chemical element in \n $\\{\\ttS_{(2)},\\ttS_{(6)}, \\ttP_{(5)}\\}\\setminus \\Lambda(\\pi)$). \n \\end{itemize}\n   \n\n\\begin{itemize} \n  \\item[(b)] $I_\\mathrm{b}^i=(\\GC^i,\\sint^i, \\sce^i)$, $i=1,2,3,4$:\n An instance for inferring chemical graphs with rank at most 2.  \nIn the four instances $I_\\mathrm{b}^i$, $i=1,2,3,4$, \nthe following specifications in $(\\sint,\\sce)$ are common. \n\\begin{enumerate}\n\\item[] \nSet  $\\Lambda:=\\Lambda(\\pi)$\n for a given property $\\pi\\in \\{${\\sc   OptR,  SfT, Vis}$\\}$, \n set $\\Ldg^\\inte$ to be\nthe set of all possible symbols in $\\Lambda\\times[1,4]$  \nthat appear in the data set $D_\\pi$  \nand set $\\Gamma^\\inte$\nto be the set  of  all  edge-configurations that appear in the data set $D_\\pi$. \nSet  $\\Lambda^*(v):= \\Lambda$,  $v\\in \\VC$. \n \n\\item[] \nThe lower bounds  \n $\\ell_\\LB $, $\\bl_\\LB $, $\\ch_\\LB $,  \n $\\bd_{2,\\LB}$,   $\\bd_{3,\\LB}$,  \n $\\na_\\LB$,  $\\na^\\inte_\\LB$,  $\\ns^\\inte_\\LB$,  \n$\\ac^\\inte_\\LB$, $\\ec^\\inte_\\LB$ and $\\ac^\\lf_\\LB$  are all set to be 0.\n\n\\item[] \nSet  upper bounds   \n $\\na_\\UB(\\ta):=n^*, \\na\\in\\{\\ttH,\\ttC\\}$,   \n $\\na_\\UB(\\ta):=5, \\na\\in\\{\\ttO,\\ttN\\}$,\n $\\na_\\UB(\\ta):=2, \\na\\in\\Lambda\\setminus \\{\\ttH,\\ttC,\\ttO,\\ttN\\}$. \nThe other upper bounds  \n $\\ell_\\UB $, $\\bl_\\UB $, $\\ch_\\UB $,  \n $\\bd_{2,\\UB}$,   $\\bd_{3,\\UB}$,  \n $\\na^\\inte_\\UB$,  $\\ns^\\inte_\\UB$,  \n$\\ac^\\inte_\\UB$, $\\ec^\\inte_\\UB$ and $\\ac^\\lf_\\UB$ \nare all set to be an upper bound $n^*$  on $n(G^*)$.\n\n\\item[] \nWe specify $n_\\LB$ as a parameter and\nset\n$n^*:=n_\\LB+10$,\n  $\\nint_\\LB:=\\lfloor (1\/4) n_\\LB \\rfloor$ and\n   $\\nint_\\LB:=\\lfloor (3\/4) n_\\LB \\rfloor$. \n\n\\item[] \n   For each property $\\pi$, let $\\mathcal{F}(D_\\pi)$ denote\n    the set of 2-fringe-trees in the compounds in $D_\\pi$,\n   and select a subset $\\mathcal{F}_\\pi^i\\subseteq  \\mathcal{F}(D_\\pi)$ with\n   $|\\mathcal{F}_\\pi^i|=45-5i$, $i\\in [1,5]$.\n   For each instance $I_\\mathrm{b}^i$, \n   set $\\mathcal{F}_E :=\\mathcal{F}(v):=  \\mathcal{F}_\\pi^i$, $v\\in \\VC$ and \n$\\fc_\\LB(\\psi):=0, \\fc_\\UB(\\psi):=10, \\psi\\in  \\mathcal{F}_\\pi^i$. \n\\end{enumerate}\n \n  Instance $I_\\mathrm{b}^1$ is given   by the rank-1 seed graph $\\GC^1$ \n  in Figure~\\ref{fig:specification_example_polymer}(i)\n  and   Instances $I_\\mathrm{b}^i$, $i=2,3,4$ are\n   given by  the rank-2 seed graph $\\GC^i$, $i=2,3,4$ in \n   Figure~\\ref{fig:specification_example_polymer}(ii)-(iv). \n   \n   \n\\begin{itemize} \n \\item[(i)]  For instance $I_\\mathrm{b}^1$, select as a seed graph \n  the monocyclic graph   $\\GC^1=(\\VC,\\EC=\\Et\\cup \\Ew)$\n  in Figure~\\ref{fig:specification_example_polymer}(i),\n  where $\\VC=\\{u_1,u_2\\}$, $\\Et=\\{a_1\\}$ and  $ \\Ew=\\{a_2\\}$. \nWe  include a linear constraint \n$\\ell(a_1)\\leq \\ell(a_2)$ \nand $5\\leq \\ell(a_1)+\\ell(a_2) \\leq 15$  as part of the side constraint. \n  \n \\item[(ii)]\n For instance $I_\\mathrm{b}^2$, select as a seed graph \n  the  graph   $\\GC^2=(\\VC,\\EC=\\Et\\cup \\Ew\\cup \\Eew)$ \n  in Figure~\\ref{fig:specification_example_polymer}(ii),\n  where\n$\\VC=\\{u_1,u_2,u_3,u_4\\}$, \n$\\Et=\\{a_1,a_2\\}$, \n$\\Ew=\\{a_3\\}$  and \n$\\Eew=\\{a_4,a_5\\}$. \nWe include a linear constraint $\\ell(a_1)\\leq \\ell(a_2)$ \nand $\\ell(a_1)+\\ell(a_2)+\\ell(a_3)\\leq 15$. \n    \n \\item[(iii)]\n For instance $I_\\mathrm{b}^3$, select as a seed graph \n  the  graph   $\\GC^3=(\\VC,\\EC=\\Et\\cup \\Ew\\cup \\Eew)$ \n  in Figure~\\ref{fig:specification_example_polymer}(iii),   where\n$\\VC=\\{u_1,u_2,u_3,u_4\\}$, \n$\\Et=\\{a_1\\}$, \n$\\Ew=\\{a_2, a_3\\}$  and \n$\\Eew=\\{a_4,a_5\\}$. \nWe include   linear constraints \n$\\ell(a_1)\\leq \\ell(a_2)+\\ell(a_3)$, $\\ell(a_2)\\leq \\ell(a_3)$\nand $\\ell(a_1)+\\ell(a_2)+\\ell(a_3)\\leq 15$.  \n\n \\item[(iv)] \n For instance $I_\\mathrm{b}^4$, select as a seed graph \n  the  graph   $\\GC^4=(\\VC,\\EC=\\Et\\cup \\Ew\\cup \\Eew)$ \n  in Figure~\\ref{fig:specification_example_polymer}(iv),   where\n$\\VC=\\{u_1,u_2,u_3,u_4\\}$, \n$\\Ew=\\{a_1, a_2, a_3\\}$  and \n$\\Eew=\\{a_4,a_5\\}$. \nWe   include   linear constraints \n$\\ell(a_2)\\leq \\ell(a_1)+1$,\n$\\ell(a_2)\\leq \\ell(a_3)+1$,  $\\ell(a_1)\\leq \\ell(a_3)$  \nand $\\ell(a_1)+\\ell(a_2)+\\ell(a_3)\\leq 15$. \n \\end{itemize}\n \\end{itemize}\n\n \n \n We define instances in (c) and (d) \n in order to find chemical graphs that have an intermediate structure of\n given two chemical cyclic graphs \n $G_A=(H_A=(V_A,E_A),\\alpha_A,\\beta_A)$ \nand $G_B=(H_B=(V_B,E_B),\\alpha_B,\\beta_B)$.\nLet\n $\\Lambda_A^\\inte$ and  $\\Lambda_{\\mathrm{dg},A}^\\inte$ \n denote the sets  of chemical elements\n and chemical symbols  of\n the interior-vertices in $G_A$, \n $\\Gamma_A^\\inte$   denote the sets of edge-configurations of\n  the interior-edges in $G_A$,   \n  and \n  $\\mathcal{F}_A$ denote the set of 2-fringe-trees in $G_A$.  \nAnalogously define sets\n $\\Lambda_B^\\inte$,    $\\Lambda_{\\mathrm{dg},B}^\\inte$,   \n $\\Gamma_B^\\inte$ and   $\\mathcal{F}_B$ \n in $G_B$.\n \n \n\\begin{itemize}  \n\\item[(c)]  $I_{\\mathrm{c}}=(\\GC,\\sint,\\sce)$: \nAn instance aimed to infer a chemical graph $G^\\dagger$ such that\nthe core of $G^\\dagger$ is equal to the core of $G_A$ and \nthe frequency of each edge-configuration in the non-core of $G^\\dagger$\nis equal to that of  $G_B$. \nWe use chemical compounds CID~24822711 and CID~59170444 in \n Figure~\\ref{fig:instance_I_c_I_d}(a) and (b)\n for $G_A$ and $G_B$, respectively.  \\\\\nSet   a seed graph $\\GC=(\\VC,\\EC=\\Eew)$ to be the core of $G_A$. \\\\\nSet  $\\Lambda:=\\{{\\tt H,C,N,O}\\}$, \nand  set $\\Ldg^\\inte$ to be\nthe set of all possible chemical symbols in $\\Lambda\\times[1,4]$.\\\\\nSet \n$\\Gamma^\\inte:=\\Gamma_A^\\inte\\cup \\Gamma_B^\\inte$ and \n  $\\Lambda^*(v):=\\{\\alpha_A(v)\\}$, $v\\in \\VC$.  \\\\\nSet \n$\\nint_\\LB:=\\min\\{\\nint(G_A), \\nint(G_B)\\}$, \n$\\nint_\\UB:=\\max\\{\\nint(G_A), \\nint(G_B)\\}$, \\\\\n$n_\\LB:=\\min\\{n(G_A), n(G_B)\\}-10=40$ \nand   $n^*:=\\max\\{n(G_A), n(G_B)\\}+5$. \\\\\nSet  lower bounds  \n $\\ell_\\LB $, $\\bl_\\LB $, $\\ch_\\LB $,  \n $\\bd_{2,\\LB}$,   $\\bd_{3,\\LB}$,  \n $\\na_\\LB$,  $\\na^\\inte_\\LB$,  $\\ns^\\inte_\\LB$, \n$\\ac^\\inte_\\LB$ and  $\\ac^\\lf_\\LB$  to be 0.\\\\\nSet  upper bounds   \n $\\na_\\UB(\\ta):=n^*, \\na\\in\\{\\ttH,\\ttC\\}$,   \n $\\na_\\UB(\\ta):=5, \\na\\in\\{\\ttO,\\ttN\\}$,\n $\\na_\\UB(\\ta):=2, \\na\\in\\Lambda\\setminus \\{\\ttH,\\ttC,\\ttO,\\ttN\\}$ \nand set the other upper bounds\n $\\ell_\\UB $, $\\bl_\\UB $, $\\ch_\\UB $,  \n $\\bd_{2,\\UB}$,   $\\bd_{3,\\UB}$,  \n$\\na^\\inte_\\UB$,  $\\ns^\\inte_\\UB$, \n$\\ac^\\inte_\\UB$   and  $\\ac^\\lf_\\UB$ to be  $n^*$. \\\\\nSet $\\ec_\\LB^\\inte(\\gamma)$ \nto be the number of core-edges  in $G_A$ with $\\gamma\\in \\Gamma^\\inte$ and  \n $\\ec_\\UB^\\inte(\\gamma)$  \nto be the number interior-edges in $G_A$ and  $G_B$ \nwith edge-configuration $\\gamma$. \\\\\nLet $\\mathcal{F}_B^{(p)}, p\\in [1,2]$ denote the set of chemical rooted \ntrees r-isomorphic $p$-fringe-trees in $G_B$; \\\\\nSet $\\mathcal{F}_E :=\\mathcal{F}(v):= \n \\mathcal{F}_B^{(1)}\\cup \\mathcal{F}_B^{(2)}$, $v\\in \\VC$ and\n$\\fc_\\LB(\\psi):=0, \\fc_\\UB(\\psi):=10, \\psi\\in \\mathcal{F}_B^{(1)}\\cup \\mathcal{F}_B^{(2)}$. \n \n  \\item[(d)] $I_{\\mathrm{d}}=(\\GC^1,\\sint, \\sce)$:     \nAn instance aimed to infer a chemical monocyclic graph $G^\\dagger$ such that\nthe frequency vector of  edge-configurations in  $G^\\dagger$\nis a vector obtained by merging those of $G_A$ and $G_B$.\nWe use chemical monocyclic compounds CID~10076784 and CID~44340250\nin   Figure~\\ref{fig:instance_I_c_I_d}(c) and (d) \n for $G_A$ and $G_B$, respectively.  \nSet a seed graph to be   the monocyclic seed graph  \n $\\GC^1=(\\VC,\\EC=\\Et\\cup \\Ew)$ with \n  $\\VC=\\{u_1,u_2\\}$, $\\Et=\\{a_1\\}$ and  $ \\Ew=\\{a_2\\}$ \n  in Figure~\\ref{fig:specification_example_polymer}(i). \\\\\nSet  $\\Lambda:=\\{{\\tt H,C,N,O}\\}$,  \n $\\Ldg^\\inte:=\\Lambda_{\\mathrm{dg},A}^\\inte \n                 \\cup \\Lambda_{\\mathrm{dg},B}^\\inte$ and \n$\\Gamma^\\inte:=\\Gamma_A^\\inte\\cup \\Gamma_B^\\inte$. \\\\\nSet \n$\\nint_\\LB:=\\min\\{\\nint(G_A), \\nint(G_B)\\}$, \n$\\nint_\\UB:=\\max\\{\\nint(G_A), \\nint(G_B)\\}$, \\\\\n  $n_\\LB:=\\min\\{n(G_A),n(G_B)\\}=40$ and  \n  $n^*:=\\max\\{n(G_A),n(G_B)\\}$. \\\\\nSet  lower bounds  \n $\\ell_\\LB $, $\\bl_\\LB $, $\\ch_\\LB $,  \n $\\bd_{2,\\LB}$,   $\\bd_{3,\\LB}$,  \n $\\na_\\LB$,  $\\na^\\inte_\\LB$,  $\\ns^\\inte_\\LB$, \n$\\ac^\\inte_\\LB$  and  $\\ac^\\lf_\\LB$ to be 0.\\\\\nSet  upper bounds   \n $\\na_\\UB(\\ta):=n^*, \\na\\in\\{\\ttH,\\ttC\\}$,   \n $\\na_\\UB(\\ta):=5, \\na\\in\\{\\ttO,\\ttN\\}$,\n $\\na_\\UB(\\ta):=2, \\na\\in\\Lambda\\setminus \\{\\ttH,\\ttC,\\ttO,\\ttN\\}$ \nand set the other  upper bounds  \n $\\ell_\\UB $, $\\bl_\\UB $, $\\ch_\\UB $,  \n $\\bd_{2,\\UB}$,   $\\bd_{3,\\UB}$,  \n  $\\na^\\inte_\\UB$,  $\\ns^\\inte_\\UB$,\n$\\ac^\\inte_\\UB$ and  $\\ac^\\lf_\\UB$  to be   $n^*$. \\\\\nFor each edge-configuration\n $\\gamma \\in \\Gamma^\\inte$,  \nlet  $\\x^*_A(\\gamma^\\inte)$  (resp., $\\x^*_B(\\gamma^\\inte)$)   denote\n the number of interior-edges with $\\gamma$ in $G_A$ (resp., $G_B$), \n $\\gamma \\in \\Gamma^\\inte$ and   \nset \\\\\n$\\x^*_{\\min}(\\gamma):=\\min\\{\\x^*_A(\\gamma), \\x^*_B(\\gamma)\\}$, \n $\\x^*_{\\max}(\\gamma):=\\max\\{\\x^*_A(\\gamma), \\x^*_B(\\gamma)\\}$, \\\\\n$\\ec_\\LB^\\inte(\\gamma):=\n\\lfloor (3\/4)\\x^*_{\\min}(\\gamma)+(1\/4)\\x^*_{\\max}(\\gamma) \\rfloor$\nand  \\\\\n$\\ec_\\UB^\\inte(\\gamma):=\n\\lceil (1\/4)\\x^*_{\\min}(\\gamma)+(3\/4)\\x^*_{\\max}(\\gamma) \\rceil$. \\\\\nSet $\\mathcal{F}_E :=\\mathcal{F}(v):=  \\mathcal{F}_A\\cup \\mathcal{F}_B$, \n$v\\in \\VC$ and \n$\\fc_\\LB(\\psi):=0, \\fc_\\UB(\\psi):=10, \\psi\\in \\mathcal{F}_A\\cup \\mathcal{F}_B$. \\\\\nWe  include a linear constraint \n$\\ell(a_1)\\leq \\ell(a_2)$ \nand $5\\leq \\ell(a_1)+\\ell(a_2) \\leq 15$  as part of the side constraint. \n \\end{itemize}\n \n \n\\section{Two-layered Model}\\label{sec:2LM\nThis section reviews the two-layered model introduced  by \n Zhu~et~al.~\\cite{ZAHZNA21}. \n\n Let  $\\C=(H,\\alpha,\\beta)$ be a chemical graph\n and  ${\\rho}\\geq 1$ be an integer, which we call a {\\em branch-parameter}.\n \n  A {\\em two-layered model} of $\\C$ is a partition of\n the hydrogen-suppressed chemical graph $\\anC$ into\n an ``interior'' and an ``exterior'' in the following way. \n We call a vertex $v\\in V(\\anC)$\n   (resp., an edge $e\\in E(\\anC))$ of   $\\C$\n   an {\\em exterior-vertex} (resp.,    {\\em exterior-edge}) if\n    $\\h(v)< {\\rho}$ (resp., $e$ is incident to an  exterior-vertex)\nand denote the sets of exterior-vertices and exterior-edges \nby $V^\\ex(\\C)$ and $E^\\ex(\\C)$, respectively\nand denote  $V^\\inte(\\C)=V(\\anC)\\setminus  V^\\ex(\\C)$ and \n$E^\\inte(\\C)=E(\\anC)\\setminus E^\\ex(\\C)$, respectively.\nWe call a vertex in $V^\\inte(\\C)$ (resp.,   an edge in $E^\\inte(\\C)$) \n   an {\\em interior-vertex} (resp.,    {\\em interior-edge}). \n The set  $E^\\ex(\\C)$ of  exterior-edges forms \na collection of connected graphs each of which is\nregarded as a rooted tree $T$ rooted at \nthe vertex $v\\in V(T)$ with the maximum $\\h(v)$. \nLet $\\mathcal{T}^\\ex(\\anC)$ denote \nthe set of these chemical rooted trees in $\\anC$. \nThe {\\em interior} $\\C^\\inte$ of $\\C$ is defined to be the subgraph\n $(V^\\inte(\\C),E^\\inte(\\C))$ of $\\anC$. \n\nFigure~\\ref{fig:example_chemical_graph}\n illustrates an example of a hydrogen-suppressed chemical graph $\\anC$.\nFor a branch-parameter ${\\rho}=2$, \nthe interior of  the chemical graph $\\anC$ in Figure~\\ref{fig:example_chemical_graph} \nis obtained by removing the set of vertices with degree 1 ${\\rho}=2$ times; i.e., \nfirst remove  \nthe set  $V_1=\\{w_1,w_2,\\ldots,w_{14}\\}$ of vertices of degree 1 in $\\anC$ \nand then remove  the set\n $V_2=\\{w_{15},w_{16},\\ldots,w_{19}\\}$ of vertices of degree 1 in $\\anC-V_1$,\n where the removed vertices become the exterior-vertices of $\\anC$. \n\n\n\n  \nFor each interior-vertex $u\\in V^\\inte(\\C)$,\nlet $T_u\\in \\mathcal{T}^\\ex(\\anC)$ denote the chemical tree rooted at $u$\n(where possibly $T_u$ consists of vertex $u$)\nand \ndefine the {\\em $\\rho$-fringe-tree} $\\C[u]$ \nto be  \nthe chemical rooted tree obtained from $T_u$ by putting back\n the hydrogens originally attached with $T_u$ in $\\C$. \nLet $\\mathcal{T}(\\C)$ denote the set of $\\rho$-fringe-trees \n$\\C[u], u \\in V^\\inte(\\C)$. \n %\nFigure~\\ref{fig:example_fringe-tree}  illustrates\nthe set  $\\mathcal{T}(\\C)=\\{\\C[u_i]\\mid i\\in [1,28]\\}$ of the 2-fringe-trees \n  of the example $\\C$ with $\\anC$\nin Figure~\\ref{fig:example_chemical_graph}. \n\n\\begin{figure}[h!] \\begin{center}\n\\includegraphics[width=.84\\columnwidth]{fig\/example_fringe-tree.eps}\n\\end{center} \\caption{\nThe set $\\mathcal{T}(\\C)$ of  2-fringe-trees  $\\C[u_i], i\\in [1,28]$ of the example $\\C$ with $\\anC$ \nin Figure~\\ref{fig:example_chemical_graph}, \nwhere the root of each tree is depicted with a gray circle and\n the hydrogens attached to non-root vertices are omitted in the figure.  \n }\n\\label{fig:example_fringe-tree} \\end{figure} \n\n \n\\smallskip\n\\noindent {\\bf Feature Function~} \n The feature of an  interior-edge $e=uv\\in E^\\inte(\\C)$ \n such that $\\alpha(u)=\\ta$, $\\deg_{\\anC}(u)=d$, \n $\\alpha(v)=\\tb$, $\\deg_{\\anC}(v)=d'$  and $\\beta(e)=m$  is represented by \n a tuple $(\\ta d, \\tb d', m)$, which is called the {\\em edge-configuration} \n  of the edge $e$, where \n  we call the tuple $(\\ta, \\tb, m)$ \n the {\\em adjacency-configuration} of the edge $e$. \n \nFor an integer $K$, a feature vector $f(\\C)$ of a chemical graph $\\C$\nis defined by a {\\em feature function} $f$ that consists of $K$ descriptors. \nWe call  $\\RK$ {\\em  the feature space}.\n\n Tanaka et~al.~\\cite{TZAHZNA21} \n  defined  a feature vector $f(\\C)\\in \\RK$  \nto be a combination of the frequency \nof edge-configurations of   the interior-edges  and\nthe frequency of chemical rooted trees among the set \nof  chemical rooted trees $\\C[u]$ over all interior-vertices $u$. \n \n\\smallskip\n\\noindent {\\bf Topological Specification~}   \nA topological specification is described\nas a set of the following rules proposed by \nShi et al.~\\cite{SZAHZNA21}\nand modified by Tanaka et~al.~\\cite{TZAHZNA21}:\n\\begin{enumerate}[nosep]\n\\item[(i)]\na {\\em seed graph} $\\GC$ as an  abstract form of  a target chemical graph $\\C$;\n\\item[(ii)]\n a set $\\mathcal{F}$ of chemical rooted trees  as candidates\n for a tree  $\\C[u]$ rooted at each interior-vertex $u$ in $\\C$; \nand \n\\item[(iii)]\nlower and upper bounds on the number of components \n in a target chemical graph such as  chemical elements, \ndouble\/triple bonds and the interior-vertices in $\\C$. \n\\end{enumerate} \n\n\\begin{figure}[h!] \\begin{center}\n\\includegraphics[width=.98\\columnwidth]{fig\/specification_example_1_test.eps}\n\\end{center} \\caption{\n(a) An illustration of a seed graph $\\GC$ with $\\mathrm{r}(\\GC)=5$, \nwhere the vertices in $\\VC$ are depicted with gray circles,\nthe edges in $\\Et$ are depicted with dotted lines,\nthe edges in $\\Ew$ are depicted with dashed lines,\nthe edges in $\\Ez$ are depicted with gray bold lines and  \nthe edges in $\\Eew$ are depicted with black solid lines;\n(b) A set $\\mathcal{F}=\\{\\psi_1,\\psi_2,\\ldots,\\psi_{30}\\}\\subseteq\n\\mathcal{F}(D_\\pi)$ of 30 chemical rooted trees\n$\\psi_i, i\\in [1,30]$, where the root of each tree is depicted with a gray circle, \nwhere  the hydrogens attached to non-root vertices are omitted in the figure.    }\n\\label{fig:specification_example_1} \\end{figure}  \n\nFigure~\\ref{fig:specification_example_1}(a) and (b)\n illustrate  examples of  a  seed graph  $\\GC$ and \n a set $\\mathcal{F}$ of chemical rooted trees, respectively. \n Given a seed graph $\\GC$, \n the interior of   a target chemical graph $\\C$ is constructed\n from $\\GC$ by replacing some edges $a=uv$ \n with paths $P_a$ between the end-vertices\n $u$ and $v$ and by attaching new paths $Q_v$ to some vertices $v$.  \nFor example, a chemical graph $\\C$ with $\\anC$ \nin Figure~\\ref{fig:example_chemical_graph} is constructed\nfrom the seed  graph  $\\GC$ in Figure~\\ref{fig:specification_example_1}(a)\nas follows.\n\\begin{enumerate}[nosep,  leftmargin=*]\n\\item[-]\nFirst replace  five edges\n $a_1=u_1 u_{2},  a_2=u_1 u_{3},  a_3=u_4 u_{7}, a_4=u_{10}u_{11}$\nand $a_5=u_{11}u_{12}$ in  $\\GC$ \n with new paths  \n$P_{a_1}=(u_1,u_{13},u_{2})$, \n$P_{a_2}=(u_{1},u_{14},u_{3})$,\n$P_{a_3}=(u_{4},u_{15},u_{16},u_{7})$, \n $P_{a_4}=(u_{10},u_{17},u_{18},u_{19},u_{11})$ and\n $P_{a_5}=(u_{11},u_{20},u_{21},u_{22},u_{12})$, respectively\n to obtain a subgraph $G_1$ of $\\anC$. \n\\item[-]\nNext attach to this graph  $G_1$ three new paths \n$Q_{u_5}=(u_5,u_{24})$, \n$Q_{u_{18}}=(u_{18},u_{25},u_{26},u_{27})$ and \n$Q_{u_{22}}=(u_{22},u_{28})$\nto obtain  \nthe interior of  $\\anC$ in Figure~\\ref{fig:example_chemical_graph}.\n\\item[-]\nFinally  attach to the interior   28 trees selected from the set $\\mathcal{F}$ \nand assign chemical elements and bond-multiplicities in the interior\nto  obtain a chemical graph $\\C$ with $\\anC$ in Figure~\\ref{fig:example_chemical_graph}. \nIn Figure~\\ref{fig:example_fringe-tree},  \n  $\\psi_1\\in \\mathcal{F}$ is selected for $\\Co[u_i]$, $i\\in\\{6,7,11\\}$.\n    Similarly \n  $\\psi_2$  for  $\\Co[u_9]$,\n  $\\psi_4$   for $\\Co[u_1]$, \n  $\\psi_6$   for $\\Co[u_i]$,   \n  $i\\in\\{3,4,5,10,19,22,25,26\\}$,\n  $\\psi_8$   for $\\Co[u_8]$, \n  $\\psi_{11}$    for $\\Co[u_i]$, $i\\in\\{2,13,16,17,20\\}$,\n  $\\psi_{15}$   for $\\Co[u_{12}]$,\n   $\\psi_{19}$    for $\\Co[u_{15}]$,\n   $\\psi_{23}$    for $\\Co[u_{21}]$,\n   $\\psi_{24}$    for $\\Co[u_{24}]$,\n   $\\psi_{25}$    for $\\Co[u_{27}]$, \n   $\\psi_{26}$   for $\\Co[u_{23}]$, \n   $\\psi_{27}$  for $\\Co[u_{14}]$     \n   and \n    $\\psi_{30}$  for $\\Co[u_{28}]$. \n\\end{enumerate} \n\n\nOur definition of a topological specification is analogous with the one  by \n Tanaka et~al.~\\cite{TZAHZNA21} \n   except for a necessary modification due to the introduction \n   of multiple valences of chemical elements, cations and anions \n(see Appendix~\\ref{sec:specification} for a full description of topological specification).  \n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\texttt{Ripper} \\cite{DBLP:conf\/icml\/Cohen95} was the first rule learning system that effectively countered the overfitting problem and is still a state-of-the-art method in inductive rule learning \\cite{DBLP:series\/cogtech\/FurnkranzGL12}. It scales nearly linearly with the number of examples in a dataset and therefore outperforms other rule learning algorithms such as \\texttt{C4.5rules} in terms of runtime. While linear complexity is uncritical for small datasets, in bigger datasets it might be insufficient. \n\nNeural networks cope with this problem and often even score a better accuracy, whereas at the same time they have the disadvantage of being difficult to interpret. There are already some approaches that combine the efficiency of neural networks with the interpretability of rules. Algorithms like ENDER \\cite{DBLP:journals\/datamine\/DembczynskiKS10}, Neural Logic Programming \\cite{DBLP:journals\/corr\/YangYC17}, or BOOMER \\cite{mr:ECML-PKDD-2020} learn rule sets or lists by minimizing a loss by using gradient descent and differentiable models like in neural networks. Another approach \\cite{DBLP:conf\/icml\/RuckertK03} applies stochastic local search to find good solutions in a reasonable period of time. With very fast decision rules (\\texttt{VFDR}) \\cite{DBLP:journals\/datamine\/KosinaG15} there already exists an algorithm especially adapted to data streams, which can also handle changes in the data over time. \n\nIn this short communication, we focus on an evaluation of the mini-batch approach as one of the efficiency-improving techniques of neural networks. A network tailored to the learning of rule sets could use mini-batches to efficiently learn rule sets itself. Moreover, repeated fine-tuning on small datasets could replace the expensive optimization part of the \\texttt{Ripper} algorithm. \nIt could as well be expanded to even deeper rule structures with invention of new features in hidden layers.\nFinally, this approach also brings the benefit of being an anytime algorithm.\n\n\\section{Network for Mini-Batch Rule Learning}\nWe created a network whose output should be structured like a rule set in \\texttt{DNF}. Therefore, the network consists of three layers, the input layer, one hidden layer (= \\texttt{AND} layer) and the output layer (= \\texttt{OR} layer). The \\texttt{AND} layer combines the literals to rules and the \\texttt{OR} layer rules to a rule set. \nThe network is designed for binary classification problems and produces a single prediction output that is either \\texttt{True} or \\texttt{False} whereby \\texttt{True} means that the less frequent class is chosen. \n\nIn the input layer, the network currently only takes nominal attributes $\\mathcal{A}$ into account and converts them by one-hot-encoding to $|A_i|$ binary features (= literals) where $|A_i|$ is the number of possible values for attribute $A_i \\in \\mathcal{A}$. We also do this for Boolean attributes or other attributes with $|A_i|=2$ because it provides an option to encode unknown values by setting all corresponding features to \\texttt{False}.\nFor the \\texttt{AND} and \\texttt{OR} layers, Boolean matrix multiplication is used to propagate the data through the network. Since matrix multiplication computes the sum of component-wise products, it corresponds to a disjunction. To get the conjunction, according to De Morgan's law, the values can be inverted before and after the addition. \n\nBefore the network is trained, the \\texttt{AND} layer is initialized by adding literals randomly to the rules, i.e., by setting the corresponding layer variable to \\texttt{True}. Each attribute is added with the probability $3\/|\\mathcal{A}|$ so that on average three literals of different attributes are added. The \\texttt{OR} layer is initialized by \\texttt{True}s, which means all rules belong to the rule set. \n\nDuring the training the data is processed in mini-batches, where in each iteration, greedy adjustments can be made to the current rule set. For each mini-batch a maximum of two flips is established to prevent overfitting. A flip is understood to be a change in the \\texttt{AND} layer from \\texttt{True} to \\texttt{False} (= removal of a literal) or from \\texttt{False} to \\texttt{True} (= addition of a literal). In this case, other literals of the same attribute are removed automatically to avoid the conjunction of contradictory literals. The flip with the biggest improvement of the accuracy on this mini-batch is selected (if any improvement is possible). \n\nThe last selection of the training is the re-selection of rules based on not only a single mini-batch, but on all training data. The \\texttt{OR} layer is filled with \\texttt{False}s and then the computed rules are greedily added as before the literals in the \\texttt{AND} layer. Therefore, the rule set only contains rules that improve the accuracy on the total training data.\n\n\\section{Experiments}\nFor our first experiments, we selected the \\textsf{adult} dataset from the UCI Repository with almost 50,000 instances of which 20\\% were retained as a test set. The number of rules was set to 20 and the mini-batch size to 400 which resulted in nearly 100 mini-batches. The development of the accuracy across the mini-batches is shown in Fig.~\\ref{fig1}. The accuracy on the current mini-batch is shown in blue, the accuracy on the training set in orange and the accuracy on the test set in green. After a sharp increase in the first mini-batch, the accuracies on the training and test set continue to rise until after 25 mini-batches their maximum values are almost reached. The training set accuracy runs slightly above the test set whereas the evaluation values on the mini-batches are furthest up since the rule set was adapted on this mini-batch immediately before.\n\nFurthermore, the rule network and \\texttt{Ripper} were applied and compared on ten datasets from Kaggle challenges and the UCI repository (Table~\\ref{tab1}). For \\texttt{Ripper}, the \\texttt{Java} implementation \\texttt{JRip} is used whereas the rule network is implemented in Python. The number of rules was again set to 20, the batch size to the square root of the number of instances. The accuracies were obtained via 1$\\times$10-fold cross validation.\n\n\\vspace{-0.9em}\n\\begin{table}\n\\begin{minipage}{0.58\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{adult.png}\n  \\captionof{figure}{Accuracy over number of mini-batches.}\n  \\label{fig1}\n\\end{minipage}\n\\begin{minipage}{0.37\\linewidth}\n  \\centering\n  \\caption{Accuracies Network vs \\texttt{Ripper}.}\n  \\label{tab1}\n  \\medskip\n  \\begin{tabular}{lrr} \n  \\hline\n  Dataset       & Network & RIPPER  \\\\ \n  \\hline\n  adult         & 0.7645  & 0.8234  \\\\\n  airline       & 0.7783  & 0.7820  \\\\\n  alpha-bank    & 0.8735  & 0.8734  \\\\\n  bank          & 0.8830  & 0.8924  \\\\\n  breast-cancer & 0.6946  & 0.6958  \\\\\n  credit-g      & 0.7020  & 0.7110  \\\\\n  hepatitis     & 0.7925  & 0.8129  \\\\\n  kr-vs-kp      & 0.8477  & 0.9919  \\\\\n  mushroom      & 0.9745  & 1.0000  \\\\\n  vote          & 0.9290  & 0.9540  \\\\\n  \\hline\n  \\end{tabular}\n\\end{minipage}\n\\end{table}\n\\vspace{-0.3em}\n\n\\texttt{Ripper} performs better than the rule network on nine out of ten datasets and almost the same on the remaining one. For half of the datasets, the rule network performs at least 2\\% worse and for two the difference even exceeds 5\\%. In both the adult dataset and the kr-vs-kp dataset (king rook vs king pawn in chess endgame) the rule network seems to suffer from an unsuitable initialization of the rules. \n\nThe training time of the rule network is ten times longer than for \\texttt{Ripper}. Note, however, that a good level of accuracy is typically reached much earlier (e.g., at about 25\\% for the \\textsf{adult} dataset in Fig.~\\ref{fig1}). Finding a good stopping criterion for the iterative refinement is a question we are currently working on.\n\n\\section{Conclusions}\nWe have proposed a technique to use mini-batches for learning rule sets. Even though the approach is still outperformed by \\texttt{Ripper}, we think that the achieved performance level justifies a deeper investigation.\nIn future work, we thus plan to improve the mini-batch approach by the usage of gradient descent or another more targeted search of flips and the establishment of an early-stopping criterion.\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}