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+{"text":"\\section{Introduction}\nHuman intelligence exhibits \\emph{compositional generalization}, the ability to understand and produce unseen combinations of seen components \\cite{chomsky1965aspects,fodor1988connectionism}.\nFor example, if a person knows the meaning of ``run'' and ``run twice'', once she learns the meaning of a new word ``jump'', she can immediately understand the meaning of ``jump twice''.\nThis compositional generalization is essential in human cognition, enabling humans to learn semantics from very limited data and extend to unseen data \\cite{montague1974formal,partee1984compositionality,lake2017building}.\nTherefore, it is widely believed that compositional generalization is a key property towards human-like AI \\cite{mikolov2016roadmap,bengio2019towards,mollica2020composition}.\nHowever, in recent years, accumulating evidences have shown that neural sequence-to-sequence (seq2seq) models exhibit limited compositional generalization ability~\\cite{lake2018generalization,hupkes2020compositionality,keysers2020measuring,furrer2020compositional}.\n\n\n\nOn the other hand, semi-supervised learning \\cite{chapelle2009semi} is a machine learning paradigm that alleviates the limited-data problem through exploiting a large amount of unlabelled data, which has been successfully used in many different areas such as machine translation \\cite{sennrich2016improving,he2016dual,skorokhodov2018semi} and semantic parsing \\cite{yin2018structvae,cao2019semantic}.\nIn sequence-to-sequence (seq2seq) tasks, unlabelled data (i.e., monolingual data) are usually cheap and abundant, containing a large amount of combinations that are unseen in limited labelled data (i.e., parallel data).\nTherefore, we propose a hypothesis that \\emph{semi-supervised learning can enable seq2seq models understand and produce much more combinations beyond labelled data, thus tackling the bottleneck of lacking compositional generalization}.\nIf this hypothesis holds true, the lack of compositional generalization would no longer be a bottleneck of seq2seq models, as we can simply tackle it through exploiting large-scale monolingual data with diverse combinations.\n\nIn this work, we focus on \\emph{Iterative Back-Translation (IBT)} \\cite{hoang2018iterative}, a simple yet effective semi-supervised method that has been successfully applied in machine translation.\nThe key idea behind it is to iteratively augment original parallel training data with pseudo-parallel data generated from monolingual data.\nTo our best knowledge, iterative back-translation has not been studied extensively from the perspective of compositional generalization.\nThis is partially because a concern about \\emph{the quality of pseudo-parallel data}:\ndue to the problem of lacking compositional generalization, for non-parallel data with unseen combinations beyond the parallel training data, pseudo-parallel data generated from them will be error-prone.\nIt is natural to speculate that errors in pseudo-parallel data are going to be reinforced and then even harm the performance.\n\nThis paper broadens the understanding of iterative back-translation from the perspective of compositional generalization, through answering three research questions:\n\\textbf{RQ1. } How does iterative back-translation affect neural seq2seq models' ability to generalize to more combinations beyond parallel data?\n\\textbf{RQ2. } If iterative back-translation is useful from the perspective of compositional generalization, what is the key that contributes to its success?\n\\textbf{RQ3. } Is there a way to further improve the quality of pseudo-parallel data, thereby further improving the performance?\n\n\\textbf{Main Contributions.}\n(1) We empirically show that iterative back-translation substantially improves the performance on compositional generalization benchmarks (CFQ and SCAN) (Section \\ref{section:c1}).\n(2) To understand what contributes to its success, we carefully examine the performance gains and observe that iterative back-translation is effective to correct errors in pseudo-parallel data  (Section \\ref{section:c2}).\n(3) Motivated by this analysis, we propose \\emph{curriculum iterative back-translation} to further improve the quality of pseudo-parallel data, thereby improving the performance of iterative back-translation (Section \\ref{section:c3}).\n\n\\section{Background}\n\n\\subsection{Compositional Generalization Benchmarks}\n\\label{section:tasks}\n\nIn recent years, the compositional generalization ability of DNN models has become a hot research problem in artificial intelligence, especially in Natural Language Understanding (NLU), because compositional generalization ability has been recognized as a basic but essential capability of human intelligence~\\cite{lake2017building}.\nTwo benchmarks, SCAN \\cite{lake2018generalization} and CFQ \\cite{keysers2020measuring}, have been proposed for measuring the compositional generalization ability of different machine learning-based NLU systems.\nOur experiments are also conducted on these two benchmarks.\n\n\\subsubsection{SCAN}\nThe SCAN dataset consists of input natural language commands (e.g., ``\\emph{jump and look left twice}'') paired with output action sequences (e.g., ``\\emph{JUMP LTURN LOOK LTURN LOOK}'').\nDifficult tasks are proposed based on different data split settings, e.g.,\n(1)~\\textbf{ADD\\_JUMP}: the pairs of train and test are split in terms of the primitive \\emph{JUMP}.\nThe train set consists of \\emph{(jump, JUMP)} and all pairs without the primitive \\emph{JUMP}.\nThe rest forms the test set.\n(2)~\\textbf{LENGTH}: Pairs are split by action sequence length into train set ($\\leq 22$ tokens) and test set ($\\geq 24$ tokens). (3)~\\textbf{AROUND\\_RIGHT}: The phrase ``\\emph{around right}\" is held out from the train set, while ``around\" and ``right\" appear separately.\n(4)~\\textbf{OPPOSITE\\_RIGHT}: This task is similar to AROUND\\_RIGHT, while the held-out phrase is ``\\emph{opposite right}''.\n\n\\subsubsection{CFQ}\nThe \\emph{Complex Freebase Questions (CFQ)} benchmark \\cite{keysers2020measuring} contains input natural language questions paired with their output meaning representations (SPARQL queries against the Freebase knowledge graph).\nTo comprehensively measure a learner's compositional generalization ability, CFQ dataset is splitted into train and test sets based on two principles:\n(1) \\emph{Minimizing primitive divergence}: all primitives present in the test set are also present in the training set, and the distribution of primitives in the training set is as similar as possible to their distribution in the test set.\n(2) \\emph{Maximizing compound divergence}: the distribution of compounds (i.e., logical substructures in SPARQL queries) in the training set is as different as possible from the distribution in the test set.\nThe second principle guarantees that the task is compositionally challenging.\nFollowing these two principles, three different \\emph{maximum compound divergence (MCD)} dataset splits are constructed.\nThe mean accuracy of standard neural seq2seq models on the MCD splits is below 20\\%.\nMoreover, this benchmark also constructs 3 MCD data splits for SCAN dataset, and the mean accuracy of standard neural seq2seq models is about 4\\%.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth,clip=true]{pictures\/IBT\/bt.png}\n\\caption{Iterative back-translation: exploiting monolingual data to augment parallel training data.\nThe solid lines mean that the src2trg\/trg2src model generates pseudo-parallel data from monolingual data;\nthe dashed lines mean that parallel and pseudo-parallel data are used to train the models.}\n\\label{fig:ibt}\n\\end{figure}\n\n\\subsection{Iterative Back-Translation~(IBT)}\n\nIterative back-translation~\\cite{hoang2018iterative} has been proved as an effective method utilizing bi-directional monolingual data to improve the performance of machine translation models.\nAlgorithm~\\ref{alg:ibt} describes the training process of iterative back-translation.\nFirstly, the source-to-target (src2trg) model and target-to-source (trg2src) model are initially trained on the original parallel data ($D_p$) for $K$ steps (Line 1).\nAfter this initial stage, each training step samples a batch of monolingual data from one direction and back-translate them to the other direction.\nThen it would perform two mini-batches to train source-to-target model using the given parallel data and pseudo backtranslated parallel data separately, and also another two mini-batches for training target-to-source model.\nFigure~\\ref{fig:ibt} visualizes the iterative training process.\n\n\\begin{algorithm}[h]\n\\small\n\t\\caption{Iterative Back-Translation}\n\t\\textbf{Input:} parallel data $D_p$, monolingual data $D_{src}$, $D_{trg}$ \\\\\n\t\\textbf{Output:} source-to-target model $M_{\\rightarrow}$ and target-to-source model $M_{\\leftarrow}$\n\t\\begin{algorithmic}[1]\n\t\t\\State \\textbf{Initial Stage}: Separately train source-to-target model $M_{\\rightarrow}$ and target-to-source model $M_{\\leftarrow}$ on $D_{p}$ for K steps\n\t\t\\While{$M_{\\rightarrow}$ and $M_{\\leftarrow}$ have not converged  \\textbf{or} step $\\leq$ N}\n\t\t    \\State Sample a batch $B_{t}$ from $D_{trg}$ and use  $M_{\\leftarrow}$ to create $B_p=\\{(\\hat{s}, t) | t\\in B_t\\}$ \n\t\t    \\State Sample a batch $B_{s}$ from $D_{src}$ and use  $M_{\\rightarrow}$ to create $B_p'=\\{(s, \\hat{t}) | s\\in B_s\\}$\n\t\t \n\t\t \n\t\t \n\t\t    \\State Train source-to-target model $M_{\\rightarrow}$ on $D_{p}$\n\t\t    \\State Train source-to-target model $M_{\\rightarrow}$ on $B_p$\n\t\t    \\State Train target-to-source model $M_{\\leftarrow}$ on $D_{p}$\n\t\t    \\State Train target-to-source model $M_{\\leftarrow}$ on $B_{p}'$\n\t    \\EndWhile\n\t\\end{algorithmic}\n\\label{alg:ibt}\n\\end{algorithm}\n\n\\begin{table*}[htbp]\n\\small\n\\caption{Performance (accuracy) on SCAN Tasks. Cells with a white background are results obtained in previous papers; cells with a grey background are results obtained in this paper.}\n\\centering\n\\begin{tabular}{lccccccccc}\n\\toprule[1pt]\n Models  & ADD\\_JUMP & LENGTH  &AROUND\\_RIGHT& OPPOSITE\\_RIGHT & MCD1 & MCD2 & MCD3\\\\\n\\midrule\nLSTM+Att &$0.0\\pm0.0$&$14.1$ & $0.0\\pm0.0$ & $16.5\\pm6.4$ &$6.5\\pm3.0$ & $4.2\\pm1.4$& $1.4\\pm0.2$\\\\\nTransformer &$1.0\\pm0.6$& $0.0$& $53.3\\pm10.9$ & $3.0\\pm6.8$ & $0.4\\pm0.2$ & $1.6\\pm0.3$&$0.88\\pm0.4$\\\\\nUni-Transformer& $0.3\\pm0.3$ & $0.0$ & $47.0 \\pm10.0$ &15.2$\\pm$13.0& $0.5\\pm0.1$ & $1.5\\pm0.2$&$1.1\\pm0.4$\\\\\n\\midrule\nSyn-att&$91.0\\pm27.4$&$15.2\\pm0.7$&$28.9\\pm34.8$&$10.5\\pm8.8$&-&-&-\\\\\nCGPS&$98.8\\pm1.4$& $20.3\\pm1.1$ &$83.2\\pm13.2$ &$89.3\\pm15.5$ &$1.2\\pm1.0$ & $1.7\\pm2.0$& $0.6\\pm0.3$ \\\\\nEquivariant&$99.1\\pm0.0$&$15.9\\pm3.2$&$\\mathbf{92.0\\pm0.2}$&-&-&-&-\\\\\nGECA & $87.0$ &- & $82.0$&- &-&-&-\\\\\nMeta-seq2seq&$99.9$&$99.9$&$16.6$&-&-&-&-\\\\\nT5-11B&$98.3$&$3.3$&$49.2$&$\\mathbf{99.1}$ &$7.9$&$2.4$ &$16.8$ \\\\\n\\midrule[1pt]\n GRU+Att~(Ours) & \\cellcolor{lightgray}$0.6$ &\\cellcolor{lightgray}$14.3$&\\cellcolor{lightgray} $23.8$ & \\cellcolor{lightgray}$0.27$  &\\cellcolor{lightgray} $18.7$ &\\cellcolor{lightgray}$32.0$& \\cellcolor{lightgray}$42.4$ \\\\\n\\quad +mono30&\\cellcolor{lightgray}$\\mathbf{99.6}$&\\cellcolor{lightgray}$\\mathbf{77.7}$&\\cellcolor{lightgray}$37.8$&\\cellcolor{lightgray}$95.8$ &\\cellcolor{lightgray} $\\mathbf{64.3}$&\\cellcolor{lightgray}$\\mathbf{80.8}$&\\cellcolor{lightgray}$\\mathbf{52.2}$ \\\\\n\\midrule\n\\quad +mono100&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$99.9$&\\cellcolor{lightgray}$42.9$&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$87.1$&\\cellcolor{lightgray}$99.0$&\\cellcolor{lightgray}$64.7$\\\\\n\\quad +transductive&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$99.3$&\\cellcolor{lightgray}$34.5$&\\cellcolor{lightgray}$100.0$&\\cellcolor{lightgray}$74.8$ &\\cellcolor{lightgray}$99.7$ &\\cellcolor{lightgray}$77.4$  \\\\\n\\bottomrule[1pt]\n\\end{tabular}\n\\label{tab:scan}\n\\end{table*}\n\n\\section{IBT for Compositional Generalization}\n\\label{section:c1}\n\nTo examine the effectiveness of IBT for compositional generalization, we conduct a series of experiments on two compositional generalization benchmarks (SCAN and CFQ, introduced in Section~\\ref{section:tasks}).\n\n\\subsection{Setup}\n\\label{sec:ibt_setup}\n\\subsubsection{Monolingual Data Settings}\nIn our experiments, we conduct different monolingual data settings as follows:\n\\begin{itemize}\n    \\item \\textbf{+transductive}:\n    \\emph{use all test data as monolingual data.}\n    We use this setting to explore how iterative back-translation performs with the most ideal monolingual data.\n    \\item \\textbf{+mono100}:\n    \\emph{use all dev data as monolingual data.}\n    In this setting, test data do not appear in the monolingual data, while the monolingual data contains much more unseen combinations beyond the parallel training data.\n    \\item \\textbf{+mono30}:\n    \\emph{randomly sample 30\\% source sequences and 30\\% target sequences from dev data.}\n    This setting is more realistic than ``+tranductive'' and ``+mono100'', because there is no implicit correspondence between source-side and target-side monolingual data (i.e., for a sequence in source-side monolingual data, the target-side monolingual data do not necessarily contain its corresponding target sequence, and vice versa).\n    Therefore, in our analysis of experimental results, we regard results of \\textbf{``+mono30\"} as the actual performance of iterative back-translation.\n\\end{itemize}\n\nNote that because there are no dev data in SCAN tasks, we randomly hold out half of the original test data as the dev data (the held-out dev data and the remaining test data are not overlapped).\n\n\\subsubsection{Implementation Details} We implement iterative back-translation based on the code of UNdreaMT\\footnote{https:\/\/github.com\/artetxem\/undreamt}~\\cite{artetxe2018iclr}. For CFQ, we use a 2-layer GRU encoder-decoder model equipped with attention. We set the size of both word embedding and hidden states to 300. We use a dropout layer with the rate of 0.5 and the training process lasts 30000 iterations with batch size 128. For SCAN, we also use a 2-layer GRU encoder-decoder model with attention. Both of the embedding size and hidden size are set to 200. We use a dropout layer with the rate of 0.5 and the training process lasts 35000 iterations with batch size 64. We set $K=5000$ steps in Algorithm~\\ref{alg:ibt} for both CFQ and SCAN benchmarks.\n\\subsubsection{Baselines}\n(i) \\textbf{General seq2seq models}:~LSTM, Transformer and Uni-Transformer~\\cite{keysers2020measuring}. (ii)~\\textbf{SOTA models on SCAN\/CFQ benchmarks}:  Syn-att~\\cite{russin2019compositional}, CGPS~\\cite{li2019compositional},  Equivariant~\\cite{gordon2019permutation}, GECA~\\cite{andreas-2020-good}, Meta-seq2seq~\\cite{lake2019compositional} and T5-11B~\\cite{furrer2020compositional}. \n\n\\begin{table}[tbp]\n\\small\n\\caption{Performance (accuracy) on CFQ tasks.}\n\\centering\n\\begin{tabular}{lcccc}\n\\toprule[1pt]\n Models & MCD1 & MCD2 & MCD3 \\\\\n\\midrule\nLSTM+Attn & $28.9\\pm1.8$ & $5.0\\pm0.8$ & $10.8\\pm0.6$  \\\\\nTransformer & $34.9\\pm1.1$ & $8.2\\pm0.3$ & $10.6\\pm1.1$ \\\\\nUni-Transformer & $37.4\\pm2.2$ & $8.1\\pm1.6$ & $11.3\\pm0.3$\\\\\nCGPS & $13.2\\pm3.9$ & $1.6\\pm0.8$ & $6.6\\pm0.6$\\\\\nT5-11B & $61.4\\pm4.8$&$30.1\\pm2.2$&$31.2\\pm5.7$\\\\\n\\midrule[1pt]\nGRU+Attn~(Ours) &\\cellcolor{lightgray}$32.6\\pm0.22$ &\\cellcolor{lightgray}$6.0\\pm0.25$ & \\cellcolor{lightgray}$9.5\\pm0.25$\\\\\n\\quad +mono30 & \\cellcolor{lightgray}$\\mathbf{64.8\\pm4.4}$&\\cellcolor{lightgray}$\\mathbf{57.8\\pm4.9}$&\\cellcolor{lightgray}$\\mathbf{64.6\\pm4.9}$\\\\\n\\midrule\n\\quad +mono100&\\cellcolor{lightgray}$83.2\\pm3.1$&\\cellcolor{lightgray}$71.5\\pm6.9$&\\cellcolor{lightgray}$81.3\\pm1.6$\\\\\n\\quad +transductive &\\cellcolor{lightgray}$88.4\\pm0.7$ &\\cellcolor{lightgray}$81.6\\pm6.5$ &\\cellcolor{lightgray}$88.2\\pm2.2$\\\\\n\\bottomrule[1pt]\n\\end{tabular}\n\\label{tab:cfq}\n\\end{table}\n\n\\subsection{{Observations}}\n\\begin{figure*}[t]\n\\small\n\\centering\n\\includegraphics[width=0.9\\textwidth]{pictures\/IBT\/accuracy.jpg}\n\\caption{Quality~(accuracy scores) of generated target data during the training procedure.}\n\\label{fig:accuracy_scores}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{pictures\/IBT\/bleu.jpg}\n\\caption{Quality~(BLEU scores) of generated source data during the training procedure.}\n\\label{fig:bleu_scores}\n\\end{figure*}\nWe report the experimental results in Table~\\ref{tab:scan} (SCAN) and~\\ref{tab:cfq} (CFQ).\nNotice that we pay more attention to MCD splits (on both SCAN and CFQ), since it has been proven that they are more comprehensive for evaluating compositional generalization than other splits \\cite{keysers2020measuring,furrer2020compositional}.\nOur observations are as follows:\n\n\\textit{1.Iterative Back-translation substantially improves the performance on compositional generalization benchmarks.}\nFirstly, IBT can always bring large performance gains to our baseline (GRU+Att).\nThen, compared to the previous state-of-the-art models, IBT can consistently achieve remarkable performance on all tasks, while those SOTA models only show impressive performances on ADD\\_JUMP \/ LENGTH \/ AROUND\\_RIGHT \/ OPPOSITE\\_RIGHT tasks of SCAN, but perform poorly on MCD tasks of SCAN\/CFQ.\nMoreover, IBT also achieves higher performance than T5-11B, which is a strong baseline that incorporates rich external knowledge during pre-training stage.\n\n\n\n\n\\textit{2. Better monolingual data, better results.} \nAs described in section~\\ref{sec:ibt_setup}, the quality of monolingual data are gradually improved from  ``+mono30\" to ``+mono100\" then to ``+transductive\". In most cases, it's as expected that ``+transductive\" performs better than ``+mono100\", and ``+mono100\" performs better than ``+mono30\".\n\n\n\n\\section{Secrets behind Iterative Back-Translation}\n\\label{section:c2}\n\nSection \\ref{section:c1} shows that iterative back-translation substantially improves seq2seq models' ability to generalize to more combinations beyond parallel data, but it is still unclear how and why iterative back-translation succeeds (RQ 2).\n\nTo answer this research question, we first empirically analyze pseudo-parallel data quality during the training process of iterative back-translation, and find that errors are increasingly corrected (Section \\ref{section:quality}).\nThen, we conduct ablation experiments to further illustrate this observation.\nWe find that:\n\\emph{even the error-prone pseudo-parallel data are beneficial.}\nWe speculate the reason is that knowledge of unseen combinations are implicitly injected into the model, thereby the src2trg model and the trg2src model can boost each other rather than harm each other (Section \\ref{section:static}).\nMoreover, pseudo-parallel data in iterative back-translation contain perturbations, helping models correct errors more effectively (Section \\ref{section:otf}).\nIn this section, all experiments are conducted on the CFQ benchmark, as it is much more complex and comprehensive than SCAN.\n\n\\begin{figure*}[t]\n\\small\n\\centering\t\n    \\begin{subfigure}{.35\\textwidth}\n    \t\\centering\n    \t\\includegraphics[width=\\textwidth]{pictures\/IBT\/acc_hist.pdf}\n    \t\\caption{Accuracy of Src2trg models}\n    \t\\label{fig:acc}\n    \\end{subfigure}\n    \\begin{subfigure}{.35\\textwidth}\n    \t\\centering\n    \t\\includegraphics[width=\\textwidth]{pictures\/IBT\/bleu_hist.pdf}\n    \t\\caption{BLEU of Trg2src models}\n    \t\\label{fig:bleu}\n    \\end{subfigure}\n\\caption{Ablation experiments for understanding the key factors contribute to the performance gain.\nThe comparison of baseline and BT indicates that even error-prone pseudo-parallel data are beneficial due to the injected implicit knowledge of unseen combinations (Section \\ref{section:static}).\nThe comparison of BT and BT+OTF indicates that perturbations brought by the on-the-fly mechanism can prevent models learning specific incorrect bias, thus improving the performance (Section \\ref{section:otf}).\nIBT performs best because it uses both source-side and target-side monolingual data while others use only target-side monolingual data.}\n\\label{fig:stanford_res}\n\\end{figure*}\n\n\\subsection{Quality of Pseudo-Parallel Data}\n\\label{section:quality}\n\nWe use Figure \\ref{fig:accuracy_scores} and \\ref{fig:bleu_scores} to get a better sense of the quality of pseudo-parallel data during the training process.\nThe x-axis represents the number of training steps: in the first 5000 steps, we train the src2trg model and the trg2src model with only parallel data;\nthe iterative back-translation process starts from step 5001 (dashed vertical lines).\nFor each step, we use the model to generate pseudo-parallel data from all monolingual data, and then evaluate the generated data quality.\nSpecifically:\nthe src2trg model generates a SPARQL query for each natural language utterance in the source-side monolingual data, and we define the quality of these generated SPARQL queries (i.e., target-side sequences) as the accuracy of them;\nthe trg2src model generates a natural language utterance for each SPARQL query in the target-side monolingual data, and we define the quality of these generated natural language utterances (i.e., source-side sequences) as the BLEU score~\\cite{papineni2002bleu} of them.\n\nWe observe that \\emph{iterative back-translation can increasingly correct errors in pseudo-parallel data.}\nEven though the pseudo-parallel data are error-prone at the end of initial stage~(dashed horizontal lines), the accuracy\/BLEU score has increased dramatically since the iterative back-translation process starts.\n\n\n\n\n\\subsection{Impact of Error-Prone Pseudo-Parallel Data}\n\\label{section:static}\n\nTo explore the reasons why iterative back-translation can increasingly correct error pseudo-parallel data, we investigate whether models can benefit from the initially generated pseudo-parallel data which are error-prone.\nIf yes, errors are going to be reduced during the iterative training process;\notherwise, errors are going to be reinforced.\n\nToward this end, we conduct two ablation experiments with ``+mono30\" monolingual data setting:\n\n\\begin{itemize}\n\\item \\textbf{BT-Src2trg}: the standard back-translation method for training the src2trg model (i.e., BT in Figure \\ref{fig:acc}).\nIt consists of three phases:\nfirst, train a src2trg model and a trg2src model using parallel data;\nthen, generate pseudo-parallel data from target-side monolingual data using the trg2src model;\nfinally, tune the src2trg model using the union of parallel data and pseudo-parallel data.\n\\item \\textbf{BT-Trg2src}: the standard back-translation method for training the trg2src model (i.e., BT in Figure \\ref{fig:bleu}).\nIt also consists of three phrases dual to those in BT-Src2trg.\n\\end{itemize}\n\n\n\n\n\n\n\nAccording to the performance of BT and baseline (i.e., src2trg\/trg2src models trained only on parallel data) in Figure \\ref{fig:stanford_res}, we observe that \\emph{even error-prone pseudo-parallel data can be beneficial.}\nSpecifically, though the initial pseudo-parallel data are error-prone (on average, 21.9\\% BLEU for Trg2src and 16.0\\% accuracy for Src2trg), they can still improve each other (5.4\\% BLEU and 4.0\\% accuracy gains for Trg2src and Src2trg models, respectively).\n\nA reasonable illustration for the observation is that: even though pseudo-parallel data are error-prone, it can still implicitly inject the knowledge of unseen combinations into the src2trg\/trg2src models, thereby boosting these models rather than harming them.\n\n\\subsection{Impact of Perturbations}\n\\label{section:otf}\n\n\nPrevious studies show that \\emph{perturbations (or noises)} in pseudo data play an important role in semi-supervised learning \\cite{zhang2016exploiting,edunov2018understanding,he2019revisiting,xie2020self}.\n\nIn iterative back-translation, the on-the-fly mechanism (i.e., models will be tuned each mini-batch on-the-fly with a batch of pseudo-parallel data) brings perturbations to pseudo-parallel data.\nDifferent batches of pseudo-parallel data are produced by different models at different steps, rather than the same model.\nThis makes errors more diverse, thus preventing the src2trg\/trg2src model learning specific incorrect bias from specific dual model.\nTherefore, it is reasonable to speculate that some performance gains are brought by such perturbations.\n\nWe conduct two ablation experiments (with ``+mono30\" monolingual data setting) to study the impact of such perturbations:\n\n\\begin{itemize}\n\\item \\textbf{BT-Src2trg-with-OTF}:\nthis method can be seen as iterative back-translation without source-side monolingual data (i.e., BT+OTF in Figure \\ref{fig:acc}).\nSpecifically, during the iterative training process, the trg2src model will only be tuned with parallel data.\nAs Figure \\ref{fig:bleu_scores} shows, the quality of pseudo-parallel data produced by the trg2src model would keep error-prone if it is only tuned on parallel data (the baseline curves in Figure \\ref{fig:bleu_scores}).\nWe want to see whether the src2trg model could be improved if the trg2src model keeps providing error-prone pseudo-parallel data.\n\\item \\textbf{BT-Trg2src-with-OTF}:\nthis method can be seen as iterative back-translation without target-side monolingual data (i.e., BT+OTF in Figure \\ref{fig:bleu}).\nWe want to see whether the trg2src model could be improved if the src2trg model keeps providing error-prone pseudo-parallel data.\n\\end{itemize}\n\n\n\n\n\nAccording to the performance of BT and BT+OTF in Figure \\ref{fig:stanford_res}, we find that perturbations brought by the on-the-fly mechanism do help models benefit from error-prone pseudo-parallel data more effectively.\nFor example, on average, the accuracy of BT-Src2trg-with-OTF is 33.3\\%, while the accuracy of BT-Src2trg is only 20.1\\%.\nAccording to our statistics, each sequence in target-side monolingual data has 49.3 different back-translations during the iterative training process.\nThe perturbations bring 13.3\\% performance gain, even if the quality of pseudo-parallel data has almost no improvement (see the baseline curves in Figure \\ref{fig:bleu_scores}).\n\nThis experimental results support our hypothesis that \\emph{perturbations brought by the on-the-fly mechanism contributes a lot to the success of iterative back-translation in compositional generalization benchmarks}.\n\n\\section{Curriculum Iterative Back-Translation}\n\\label{section:c3}\n\nAs discussed in Section \\ref{section:c2}, the major reason for why iterative back-translation succeeds is that it can increasingly correct errors in pseudo-parallel data.\nBased on this observation, it is reasonable to speculate that iterative back-translation can be further improved through injecting inductive bias that can help errors be reduced more efficiently.\nTowards this end, we consider curriculum learning~\\cite{bengio2009curriculum} as a potential solution: start out iterative back-translation with easy monolingual data (of which the generated pseudo-parallel data are less error-prone), then gradually increase the difficulty of monolingual data during the training process.\nBased on this intuition, we propose \\emph{Curriculum Iterative Back-Translation (CIBT)}.\n\n\\begin{table*}[t]\n\\small\n    \\centering\n    \\caption{Performance~(accuracy) of curriculum iterative back-translation.}\n\t\\begin{tabular}{lcccccc}\n\t    \\hline\n    \t\\multirow{2}{*}{} & \\multirow{2}{*}{IBT} & \\multicolumn{5}{c}{CIBT with hyperparameter $c$ (steps in each stage)} \\\\ \\cline{3-7}\n    \t &  & 2000 & 2500 & 3000 & 3500 & 4000 \\\\ \\cline{1-1} \\cline{2-2} \\cline{3-7}\n         MCD1 & $  64.8 \\pm 4.4 $ & $  66.1 \\pm 5.0 $ & $  66.0 \\pm 4.8 $ & $  \\textbf{66.6} \\pm 5.4 $ & $  65.9 \\pm 3.7 $ & $  65.4 \\pm 3.8 $ \\\\ \n        MCD2 & $  57.8 \\pm 4.9 $ & $  68.6 \\pm 2.6 $ & $  \\textbf{69.1} \\pm 3.1 $ & $  68.0 \\pm 1.9 $ & $  66.8 \\pm 2.4 $ & $  65.4 \\pm 3.1 $ \\\\\n        MCD3 & $  64.6 \\pm 4.9 $ & $  70.2 \\pm 4.9 $ & $  68.4 \\pm 7.0 $ & $  \\textbf{70.4} \\pm 4.8 $ & $  69.2 \\pm 4.1 $ & $  67.0 \\pm 6.3 $ \\\\ \n        Mean & $  62.4 \\pm 6.1 $ & $  \\textbf{68.3} \\pm 4.1 $ & $  67.8 \\pm 4.7 $ & $  \\textbf{68.3} \\pm 4.1 $ & $  67.3 \\pm 3.4 $ & $  65.9 \\pm 4.1 $ \\\\ \\hline\n    \\end{tabular}\n\t\\label{cl_results_summary}\n\\end{table*}\n\n\\subsection{Method}\n\n\\subsubsection{Problem Formulation}\n\nWe denote source-side and target-side monolingual data in iterative back-translation as $D_{src}$ and $D_{trg}$, respectively.\nSuppose that we have a simple heuristic algorithm $\\mathcal{C}$, which can divide $D_{src}$ into $n$ parts (denoted as $D_{src}^{(1)}$, $D_{src}^{(2)}$, ..., $D_{src}^{(n)}$), and $D_{trg}$ can also be divided into $n$ parts (denoted as $D_{trg}^{(1)}$, $D_{trg}^{(2)}$, ..., $D_{trg}^{(n)}$).\nThey are sorted by difficulty in ascending order:\nthe initial src2trg model performs roughly better on $D_{src}^{(i)}$ than on $D_{src}^{(j)}$ if $i<j$;\nand the initial trg2src model performs roughly better on $D_{trg}^{(i)}$ than on $D_{trg}^{(j)}$ if $i<j$.\nWe define that $(D_{src}^{(1)}, D_{trg}^{(1)}), (D_{src}^{(2)}, D_{trg}^{(2)}), ..., (D_{src}^{(n)}, D_{trg}^{(n)})$ are $n$ curriculums from simple to difficult.\nOur goal is to improve the performance of iterative back-translation with this additional curriculum setting.\n\n\\subsubsection{Method Details}\n\nAlgorithm \\ref{cl alg} details the workflow of curriculum iterative back-translation.\n\n\\begin{algorithm}[!h]\n\\small\n\t\\caption{Curriculum Iterative Back-Translation}\n\t\\label{cl alg}\n\t\\textbf{Input:} $M_{\\rightarrow}$ and $M_{\\leftarrow}$, the src2trg and trg2src models that have been initially trained on parallel data; $D_{src}$ and $D_{trg}$, source-side and target-side monolingual data; $\\mathcal{C}$, curriculum segmentation algorithm; $c$, the number of steps each stage should be trained; $n$, number of curriculums.\n\t\\begin{algorithmic}[1]\n\t\t\\State Use $\\mathcal{C}$ to split $D_{src}$ into $D^{(1)}_{src}, D^{(2)}_{src}, ..., D^{(n)}_{src}$\n\t\t\\State Use $\\mathcal{C}$ to split $D_{trg}$ into $D^{(1)}_{trg}, D^{(2)}_{trg}, ..., D^{(n)}_{trg}$\n\t\t\\State $D'_{src}=\\emptyset, D'_{trg}=\\emptyset$\n\t\t\\For{$t= 1, 2, ..., n$} \\algorithmiccomment{$n$ stages}\n\t\t\\State $D'_{src} = D'_{src} \\cup D^{(t)}_{src}$\n\t\t\\State $D'_{trg} = D'_{trg} \\cup D^{(t)}_{trg}$\n\t\t\\State Iterative back-translation ($c$ steps) to update $M_{\\rightarrow}$ and $M_{\\leftarrow}$, with $D'_{src}$ and $D'_{trg}$ as monolingual data\n\t\t\\EndFor\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Experimental Setup}\nWe evaluate the effectiveness of curriculum iterative back-translation on the CFQ ``+mono30\" setting.\n\n\\subsubsection{Curriculum Segmentation.}\nDefining the curriculum segmentation algorithm $\\mathcal{C}$ is critical in our method.\nIn previous research work on curriculum learning, many metrics for curriculum segmentation have been proposed, such as length, frequency, and perplexity.\n\nIn curriculum iterative back-translation, $\\mathcal{C}$ is used to split both $D_{src}$ and $D_{trg}$.\nWe denote that $\\mathcal{C}_i(s)$ is an indicator that takes the value 1 if and only if $\\mathcal{C}$ put the sample $s$ into the $i$-th curriculum.\nIt is a straightforward assumption that: for each $i = 1, 2, ..., n$, the performance of curriculum iterative back-translation would be better if $\\mathcal{C}_i(x)=\\mathcal{C}_i(y)$ for any source-side sequence $x$ paired with its ground truth target-side sequence $y$.\nIntuitively, we want to ensure that $x$ and $y$ are in the same curriculum.\n\nBased on this principle, we define $\\mathcal{C}$ based on the number of entities occurring in each source\/target sample.\nFor example, consider the source sequence \\textit{``Were M1 and M3 distributed by M0's employer and distributed by M2?\"}.\nIt contains 4 entities (M0, M1, M2, and M3), and its corresponding target sequence (a SPARQL query) also contains these 4 entities.\nTherefore, we put them into the same curriculum.\nSpecifically, we define 6 curriculums, which contain monolingual data with $(\\leq 1) \/ 2 \/ 3 \/ 4 \/ 5 \/ (\\geq 6)$ entities.\n\n\nFigure \\ref{baseline_acc_in_each_curriculum} shows that: in this entity-based curriculum setting, curriculum 1-6 are indeed arranged in order from simple to difficult.\nSpecifically, we evaluate the performance of the baseline model (i.e., the seq2seq model trained on parallel data) on $D_{src}^{(1)}, D_{src}^{(2)}, ..., D_{src}^{(6)}$, respectively.\nWe observe that the baseline model performs better on curriculum 1-2 than on curriculum 3-6, which indicates that this entity-based curriculum setting is reasonable.\n\n\\subsubsection{Hyperparameters}\nWe evaluate the performance of curriculum iterative back-translation with varying $c$ (the number of steps each stage should be trained): 2000\/2500\/3000\/3500\/4000.\nTo be fair, after $n\\times c$ steps, we continue to train on complete monolingual data (i.e., $D_{src}$ and $D_{trg}$) for $(25,000-n\\times c)$ steps, thus keeping the number of overall training steps the same.\n\n\\subsection{Results and Analysis}\n\\subsubsection{Curriculum learning benefits iterative back-translation.}\n\n\\begin{figure}[t]\n\\centering\n\\small\n\\includegraphics[width=0.9\\columnwidth,clip=true]{pictures\/CL\/baseline_accuracy_in_each_curriculum.pdf}\n\\caption{Performance of baseline model in each curriculum.}\n\\label{baseline_acc_in_each_curriculum}\n\\end{figure}\n\n\\begin{figure}[t]\n\\small\n\\centering\n\\includegraphics[width=0.7\\columnwidth,clip=true]{pictures\/CL\/performance_on_first_k_curriculums.pdf}\n\\caption{Performance on different subsets. This figure indicates that curriculum learning is more beneficial to difficult data (larger $k$) than simple data (smaller $k$).}\n\\label{influence_of_each_curriculum}\n\\end{figure}\n\nTable \\ref{cl_results_summary} shows the main results. CIBT consistently outperforms IBT in all three data splits (MCD1\/MCD2\/MCD3) and all hyperparameter settings ($c$ =$2000 \/ 2500 \/ 3000 \/ 3500 \/ 4000$). For MCD1\/MCD2\/MCD3 tasks of CFQ, CIBT outperforms IBT by 1.8\\%\/11.3\\%\/5.8\\% accuracy scores.\nAn interesting observation is that incorporating curriculum learning into iterative back-translation brings more stable performance:\nIBT performs much worse on MCD2 (57.8\\%) than on MCD1\/MCD3 (64.8\\%\/64.6\\%), while CIBT's improvement on MCD2 (11.3\\%) is much better than on MCD1\/MCD3 (1.8\\%\/5.8\\%), making the performances on MCD1\/MCD2\/MCD3 comparable (66.6\\%\/69.1\\%\/70.4\\%).\n\n\\subsubsection{Curriculum learning is more beneficial to difficult data than simple data.}\nFigure \\ref{influence_of_each_curriculum} further illustrates how curriculum learning improves the performance.\nWe use $k$ to denote test cases that would be put into the first $k$ curriculums, then obtain 6 subsets of test data: $T_1$, $T_2$, ..., $T_6$.\nWe have $T_1\\subset T_2\\subset ... \\subset T_6$.\nAs we can observe in Figure \\ref{influence_of_each_curriculum}, on the simplest subset ($T_1$), the performances of IBT and CIBT are very close.\nIf we consider $T_2$, a subset which is a bit more difficult than $T_1$, CIBT performs slightly better than IBT.\nSimilarly, we observe that: more difficult a subset of test data is, more gains CIBT brings.\nTherefore, we can conclude that curriculum learning is more beneficial to difficult data than simple data.\n\n\n\n\\section{Related Work}\n\n\\subsection{Compositional Generalization}\n\\label{sec:related_cg}\nRecently, exploring compositional generalization of DNN models has attracted a large attention on different topics, e.g., visual-question answering \\cite{hudson2018compositional}, algebraic compositionality \\cite{veldhoen2016diagnostic,saxton2018analysing}, and logic inference \\cite{bowman2015tree,mul2019siamese}.\nIn this work, we focus on compositional generalization in language, which is benchmarked by SCAN \\cite{lake2018generalization} and CFQ \\cite{keysers2020measuring} tasks.\n\nWhile mainstream DNN models are proved to exhibit limited compositional generalization ability in language \\cite{lake2018generalization,hupkes2020compositionality,keysers2020measuring,furrer2020compositional}, there have been many efforts to address this limitation, including\ndesign specialized architectures~\\cite{russin2019compositional,li2019compositional,gordon2019permutation,guo2020hierarchical,liu2020compositional}, data augmentation~\\cite{jia-liang-2016-data,andreas-2020-good}, meta-learning~\\cite{lake2019compositional,nye2020learning} and pre-training models~\\cite{furrer2020compositional}.\nComparing to these previous work, this paper incorporates semi-supervised learning into compositional generalization, which is task-agnostic and model-agnostic, thus improving the performance on both SCAN and CFQ benchmarks.\n\n\\subsection{Iterative Back-translation}\nThe idea of iterative back-translation dates back at least to \\emph{back-translation}, which simply augments parallel training data using pseudo-parallel data generated from only target-side monolingual data.\nBack-translation has been proven effective in statistical machine translation \\cite{goutte2009learning,bojar2011improving} and neural machine translation \\cite{sennrich2016improving,edunov2018understanding,imamura2018enhancement,xia2019generalized,wang2019improving}.\nIterative back-translation \\cite{hoang2018iterative,cotterell2018explaining,lample2018unsupervised,conneau2019cross} is an extension of back-translation, in which forward and backward direction models generate pseudo-parallel data for each other and improve each other, bringing stronger empirical performance.\n\nTo our best knowledge, this paper is the first to study iterative back-translation from the perspective of compositional generalization.\nWe demonstrate that iterative back-translation can help seq2seq models make better generalizations to much more combinations, thus broadening the understanding of iterative back-translation.\n\n\\section{Conclusion}\nIn this paper, we revisit iterative back-translation from the perspective of compositional generalization.\nWe find that iterative back-translation is an effective method that exploits a large amount of monolingual data to enable seq2seq models better generalize to more combinations beyond limited parallel data.\nWe also find that iterative back-translation has the ability to increasingly correct errors in pseudo-parallel data, thus contributing to its success.\nTo encourage this behaviour, we propose curriculum iterative back-translation, which explicitly improves the quality of pseudo-parallel data, thus further improving the performance. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{Sec:Intro}\n\\par Wireless communication has become an increasingly important part of human existence, a reality re-emphasized by the recent pandemic \\cite{saeed2020wireless}. The ubiquitous presence of wireless signals brings about significant advantages in terms of connectivity for communication and environmental awareness. However, this comes at the potential cost of users' data privacy and security \\cite{furqan2021wireless}. In the current work, we focus on this problem in the context of cooperative communication in the presence of untrusted relays.\n\n\\par \\Ac{MIMO} is considered to be an essential enabler for \\ac{5G} (and even Wi-Fi) and beyond wireless networks \\cite{IEEE_specturm}. However, services such as \\ac{mMTC} and \\ac{IoT} cannot always ensure the availability of hardware capabilities that can support \\ac{MIMO} operation. This led to the emergence of \\textit{cooperative communication} paradigm around the turn of the century, where the aim is to improve the \\ac{QoS} of the users through cooperation. Cooperative communication, while generally applicable to different network realizations, is more suited to \\textit{ad hoc} and \\acp{WSN} compared to the cellular systems \\cite{nosratinia2004cooperative}. Unlike the latter, the former two systems lack a sophisticated authentication mechanism. This may lead to violation of a user's data confidentiality by the very nodes that are expected to improve the communication via spatial diversity. Accordingly, there exists a plethora of approaches for \\ac{PLS} in cooperative communication, which can be broadly categorized into \\textit{cooperative relaying}, \\textit{cooperative jamming} and their hybrid combinations. Both these approaches usually comprise of two phases, where the first one is broadcast of information from the source. In the second step, the appropriate relay forwards the message to the destination. In the case of relaying, the selection of the helper (relay) is done to degrade the eavesdropper's interception capability. In the case of jamming, the source itself, the destination, or any other helper node transmits noise-like signals such that the eavesdropper cannot hear the message \\cite{pahuja2019cooperative}.\n\n\\par Despite the clear advantage in terms of degraded eavesdropping capability \\cite{jameel2018comprehensive}, the cooperative jamming techniques have various limitations. For instance, in the case of friendly jamming the presence of a trustable node is necessary \\cite{zhang2012physical}. Furthermore, the jamming signal from this helper can also degrade the destination's performance. Destination-based jamming techniques resolve this issue by exploiting their prior knowledge of the jamming signal. However, they require full-duplexing capabilities to achieve spatial diversity \\cite{zhao2018secrecy}. In both cases, jamming signals are to be transmitted throughout the first phase (where the source broadcasts the signal) of cooperative communication, which is unsuitable for power-limited devices. To this end, we propose the design of a jamming signal which is only transmitted for part of the broadcast phase, yet its effect is spread throughout the signal duration by exploiting the properties of \\ac{OFDM} signals and \\ac{FFT} operation. Since the jamming signal is only transmitted during the \\ac{CP} part at the destination, it also eliminates the need for full-duplexing capability. \n\n\\par The rest of this article is organized as follows. Section \\ref{Sec:SysModel} describes the general system model for \\ac{AAF} cooperative relaying. The proposed approach for \\ac{CP} jamming is discussed in Section \\ref{sec:Proposed}. Simulation results and related discussions are covered in \\ref{sec:Results} while Section \\ref{sec:Conclusion} concludes the work and provides some future directions. \n\n\\section{System Model}\n\\label{Sec:SysModel}\n\\par Without loss of generality, we assume a dual-hop half-duplex relay-aided system with \\ac{AAF} protocol \\cite{liu2009cooperative}. As shown in Fig. \\ref{fig:SysModel}, it consists of a source that wants to communicate with a destination in the presence of an untrusted relay, where each node consists of a single antenna. Here, the goal is to use the advantages provided by the relay without letting it decode any information. The system is based on \\ac{OFDM} modulation with $N$ subcarriers. The channels corresponding to source-destination ($H_{sd}$), source-relay ($H_{sr}$), and relay-destination ($H_{rd}$) links are assumed to be slowly varying Rayleigh fading with $L$ exponentially decaying taps. A \\ac{CP} of length $L$ is introduced to convert the multipath frequency channel into flat fading subchannels. \n\n\\par At the transmitter, the frequency domain \\ac{OFDM} symbols $\\mathbf{{S}}=\\begin{bmatrix} S_{0} & S_{1} &...& S_{N-1}\\end{bmatrix}^{\\textrm T} \\in \\mathbb{C}^{[N \\times 1]}$ are passed through the \\ac{IFFT} process as $s[n]=\\sum_{k=0}^{N-1} S_{k} \\exp^{\\frac{j 2 \\pi nk}{N}}$ and a \\ac{CP} of length $L$ is added. The transmitted signal with \\ac{CP} can be written as \n$\\mathbf{{s}}=[s[N-L+1], \\dots, s[N-1], s[0], s[1], \\dots, s[N-L], s[N-L+1], \\dots,  s[N-1] ]$. Finally, the source node broadcasts the information to relay and destination. The received signal at the destination in the frequency domain on $k$-th subcarrier can be represented by\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Y_sd}\nY_{sd}(k) &= \\sqrt{P_1} H_{sd}(k) S(k) + V_{sd}(k) ,\\\\k &= 1,\\dots, K;~n = 1, \\dots, N,\n\\end{aligned}\n\\end{equation}\nwhere $H_{sd}(k)$ is the \\ac{CFR} of the $k$-th subcarrier between source and destination, the transmitted power by the source is represented by $P_{1}$, $S(k)$ is the symbol transmitted by source node on $k$-th subcarrier, and $V_{sd}(k)$ represents \\ac{AWGN} on the $k$-th subcarrier with variance $N_{0}\/2$. Similarly, the received signal at relay is given by\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Y_sr}\nY_{sr}(k) &= \\sqrt{P_1} H_{sr}(k) S(k) + V_{sr}(k),\n\\end{aligned}\n\\end{equation}\nwhere $H_{sr}(k)$ is the \\ac{CFR} of the $k_{th}$ subcarrier for source-relay and $V_{sr}(k)$ represents the \\ac{AWGN}. The relay node will normalize the received signal before retransmission by the factor  \n\\begin{equation}\n\\beta (k)=\\sqrt \\frac{1}{P_1 H_{sr}(k)+ N_{0}}.\n\\end{equation}\n\nFinally, the received signal at destination from the relay can be given as \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Y_rd}\nY_{rd}(k) &= \\sqrt{P_2} H_{rd}(k) \\Big(\\beta (k) Y_{sr}(k))\\Big) + V_{rd}(k), \n\\end{aligned}\n\\end{equation}\nwhere $H_{rd}(k)$ is the frequency domain channel attenuation on the $k$-th subcarrier between relay and destination, the transmitted power is represented by $P_{2}$, and  $V_{rd}(k)$ shows  \\ac{AWGN} on the $k$-th subcarrier. The receiver will finally combine $Y_{rd}(k)$ signal with $Y_{sd}(k)$ given in (\\ref{eq:Y_sd}) by applying \\ac{MRC} and decode the information \\cite{seo2006maximum}.\n\n\\section{Proposed Approach}\n\\label{sec:Proposed}\n\\begin{figure}[t!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.9\\columnwidth]{Figures\/SysModelv3.pdf}\\label{fig:SysModel}} \n    \\subfigure[]{\\includegraphics[width=0.9\\columnwidth]{Figures\/Proposedv3.pdf}\\label{fig:Proposed}} \n    \\caption{Illustration of the proposed approach in a relaying scenario. (a) A typical dual-hop relay aided system comprising of a single source, relay and destination each. (b) The user signal is transmitted from source ($S$) at time $t_0$, reaching the relay ($R$) and destination ($D$) at times $t_1$ and $t_2$, respectively, while the jamming signal is transmitted by $D$ at $t_2$ and reaches $R$ at $t_3$.}\n    \\label{fig:eqqAPP}\n\\end{figure}\n\n\\par This section presents the details of the proposed algorithm. The basic idea here is to devise a method that enables us to utilize the advantages of the untrusted relay while keeping the information secure from being eavesdropped on by the same node. Moreover, we want to achieve this goal without using an external helper or full-duplex receiver. It should be noted that in the case of half-duplex destination, diversity gain is nullified since the destination transmits jamming signal throughout the first phase rendering it incapable of receiving the broadcast copy of the signal. The proposed approach, on the other hand, provides the required spatial diversity obtained from the signals received in both phases while ensuring that the relay is unable to intercept the data signal. This is achieved by the transmission of jamming signal over the \\ac{CP} duration of the received \\ac{OFDM} signal in the first phase.\n\nThe proposed algorithm has two phases.\n\\begin{itemize}\n    \\item \\textbf{Phase-1}: Source transmits a communication signal while destination sends a jamming signal during a portion of signal duration.\n    \\item \\textbf{Phase-2}: The relay forwards a signal that it receives during phase-1 to the destination for final combining and decoding at the destination. \n\\end{itemize}\n\\noindent The following text provides details of the aforementioned phases of the proposed approach.\n\\subsection{Phase-1}\n\n\\par In phase-1, the source node broadcasts an \\ac{OFDM} signal to destination and untrusted relay over multi-path Rayleigh fading channel. The received signals at destination and relay are given in (\\ref{eq:Y_sd}) and (\\ref{eq:Y_sr}), respectively. The relay forwards the received signals using \\ac{AAF} protocol to the destination in phase-2 to enhance the reliability of communication. However, there is a possibility that the relay may try to intercept and decode the data signal for malicious purposes. To ensure reliable communication while ensuring security from the untrusted relay, we propose that the destination transmits the jamming signal over the \\ac{CP} duration of the received signal right before receiving the \\ac{OFDM} signal. Even though the jamming signal overlaps with only a portion of the time-domain transmitted signal, its effect spreads over all the data-containing subcarriers once the \\ac{FFT} process is applied at the relay. This received signal at the relay, containing the jamming signal arriving from the destination is given by\n\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:y_sr_hat}\n\\hat{Y}_{sr}(k) &= \\sqrt{P_1} H_{sr}(k) S(k) + V_{sr}(k) \\\\&+\\sqrt{P_j} H_{rd}(k) J(k)+ V_{rd}(k),\n\\end{aligned}\n\\end{equation}\nwhere $J(k)$ is the jamming signal and the power of the jamming signal transmitted by the destination is represented by $P_{j}$. It should be noted that (\\ref{eq:y_sr_hat}) contains the additional jamming term compared to (\\ref{eq:Y_sr}), hence the term $\\hat{Y}_{sr}(k)$ is used instead of ${Y}_{sr}(k)$.\n\n\\par Figure \\ref {fig:Proposed} provides a temporal illustration of phase-1 and how the signals arrive at both destination and relay. The transmission from the source initiates at time $t_0$. The signal arrives at relay and destination nodes at times times $t_1$ and $t_2$, respectively. The destination then transmits the jamming signal, equal in length to the \\ac{CP} itself and represented by a checkered block in the figure, immediately upon receiving its signal. The jamming signal reaches the relay at $t_3$. The starting point of the jammed portion of the relay's signal in time can be calculated as\n\\begin{equation}\nt_d = t_3 - t_1 = \\frac{d_{sd}+d_{rd}-d_{sr}}{c},\n\\end{equation}\nwhere $d_{sd}$ indicates the distance between source and destination, $d_{sr}$ is the distance between source and relay, and $d_{rd}$ is the distance between relay and destination. As mentioned earlier, as long as part of the jamming signal overlaps with the data portion at the relay, its decoding capability is degraded. The overlap itself is ensured by the propagation delay between the transceivers. \n\n\\subsection{Phase-2}\n\\par In phase-2, the relay transmits the signal to the destination after employing \\ac{AAF} protocol. The received signal at the destination from the relay without jamming is given by (\\ref{eq:Y_rd}). Replacing ${Y}_{sr}(k)$ with $\\hat{Y}_{sr}(k)$ in (\\ref{eq:Y_rd}) and expanding it gives us\n\\begin{equation}\n\\begin{aligned}\n\\hat{Y}_{rd}(k) &= \\sqrt{P_2} H_{rd}(k)\\big(\\beta(k) \\sqrt{P_1} H_{sr}(k) S(k)\\\\& +  V_{sr}(k)+\\sqrt{P_j} H_{rd}(k) J(k)+ V_{rd}(k)\\big).\\\\ \n\\end{aligned}\n\\end{equation}\n\n\\par The receiver will first remove the jamming signal from $\\hat{Y}_{rd}(k)$. Since the jamming signal is known at the destination and \\ac{CP} is discarded in the receiver anyway, the jamming signal does not affect the signal of the legitimate receiver. However, some reflected components of the jamming signal may leak into the actual data and cause a self-interference. In this study, we assume that the receiver is able to cancel that interference by estimating the channel around itself. Afterwards, it employs \\ac{MRC} \\cite{seo2006maximum} to combine the resultant signal with the $Y_{sd}(k)$ that it received during phase-1.  \n\n\\par Here it is important to highlight that unlike conventional \\ac{PLS} techniques \\cite{hamamreh2018classifications}, the proposed method does not require channel state information at the transmitter \\cite{ankarali2016channel}. This renders the proposed approach suitable for scenarios where the transmitter\/source node has limited capabilities in terms of computation, channel estimation, etc. \n\n\n\\begin{table}[t!]\n\\centering \\renewcommand{\\arraystretch}{1.25}\n\\caption{Simulation parameters and assumptions} \\label{tab:sim_parameters}\n\\resizebox{0.9\\columnwidth}{!}{%\n\\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Parameter}      & \\textbf{Value}                \\\\ \\hline\nSimulation environment             & Urban macro                   \\\\ \\hline\nCarrier frequency $(f_c)$     & 2 GHz                         \\\\ \\hline\nShadow fading standard deviation $(\\sigma)$        & 4 dB               \\\\ \\hline\nSource\/relay transmit power ($P_{Tx}$)  & 23 dBm                        \\\\ \\hline\nNoise power density $(N_0)$     & -174 dBm\/Hz                    \\\\ \\hline\nSource-destination distance     & 1000 m \n\\\\ \\hline\nSource-relay distance     & (100, 200, ... , 900) m \n\\\\ \\hline\nCIR length ($L$)      & 32 \n\\\\ \\hline\nModulation      & QPSK\n\\\\ \\hline\nFFT Size ($N$)      & 256\n\\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\section{Simulation Results and Discussion}\n\\label{sec:Results}\n\\par As mentioned earlier, this work considers the presence of a single antenna source, destination, and relay nodes in an urban macro environment. As the ratio between received data and jamming signals depends heavily on the position of the relay, the following pathloss model is used \\cite{3GPP_38_901}:\n\\begin{equation}\n\\label{pathloss}\n    PL = 22\\log_{10}(d)+20\\log_{10}(f_c)+28+\\sigma,\n\\end{equation}\nwhere $d$ is the distance between the transceivers in meters, $f_c$ is the carrier frequency in GHz, and $\\sigma$ represents the shadow fading modeled as a zero-mean log-normal distribution. The source and destination nodes are assumed to be $1$km apart, with the relay occupying possible positions at multiples of $100$m from them. The transmission power of both source and relay is selected as $23$dBm, which is typical of a cellular \\ac{UE}. Each link experiences a slow Rayleigh fading channel, with an exponentially decaying power delay profile containing a \\ac{CIR} of $L = 32$ taps, which is also the length of the \\ac{CP} used (unless mentioned otherwise). Perfect synchronization and channel estimation is assumed for all links. Noise \\ac{PSD}, $N_0$, is taken to be $-174$dBm\/Hz. The simulation results are averaged over $5000$ \\ac{OFDM} blocks, where each block contains $256$ \\ac{QPSK} symbols. A summary of these simulation parameters is provided in Table \\ref{tab:sim_parameters}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{Figures\/Relay_Dest.pdf}\n    \\caption{BER performance comparison of relay and destination. Relay's performance is seen to be degraded severely with the proposed CP jamming scheme.}\n    \\label{fig:rel_dest_comparison}\n\\end{figure}\n\n\\par Figure \\ref{fig:rel_dest_comparison} shows the performance of relay and destination nodes in terms of \\ac{BER}. This is used to represent the \\textit{security gap}, which is quantified by the gap in error rates (bit, symbol, packet, etc.) of the legitimate and illegitimate receivers \\cite{hamamreh2018classifications}. The horizontal axis represents the position of the relay, where zero is the location of the source itself, while $1$ represents the location of the destination. For this simulation, both data and jamming signals are assumed to have the same power levels ($23$ dBm). The two curves for relay represent the two cases of normal relay operation (without jamming) and with the proposed jamming approach, respectively, while the curves for destination represent its performance in phase-1 (where it only receives the broadcast signal from the source) and when it applies \\ac{MRC} to the two copies received via source and relay. It can be seen that the relay's performance is significantly better than the destination when neither jamming nor combining is performed. This is simply due to the physical closeness of the relay to the source, and the consequent decrease in pathloss. It is shown in \\cite{chen2009multi} that distance-based cooperative protocol selects the relay closest to the destination as the optimal one. Our results also re-iterate this, as the performance of destination after \\ac{MRC} is significantly improved when the relay moves closer to it. The proposed algorithm also increases the \\ac{BER} of the untrusted relay at this point, leading to increased security gap.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=1\\columnwidth]{Figures\/CP_Pjam.pdf}\n    \\caption{Effect of jamming power and \\ac{CP} duration.}\n    \\label{fig:CP_Pjam}\n\\end{figure}\n\n\\par The interception capability of the eavesdropping relay depends on the jamming signal received. Accordingly, we look at the effect of different power levels of the jamming signals ($P_J$) as well as its different durations, as shown in Fig. \\ref{fig:CP_Pjam}. For the former, we have considered three cases, i.e., $P_J = [0.25, 0.5, 1]*P_{Tx}$. It can be seen that variation in $P_J$ has a clearly distinguishable effect on the relay's \\ac{BER}. For instance, when the relay is at the middle of the source and destination nodes, its \\ac{BER} goes from $0.019$ to $0.05$ as $P_J$ increases from $0.25*P_{Tx}$ to $P_{Tx}$. To see the effect of the length of the jamming signal, we looked at three cases where both the length of \\ac{CP} and jamming were varied, i.e., \\ac{CP} ratios of $1\/16, 1\/8, 1\/4$ were used (which were kept equal to \\ac{CIR} itself). It is observed that the length of the jamming signal itself has no visible effect on the \\ac{BER} performance. This observation also encourages us to use the minimum possible \\ac{CP} length in the system without compromising on the security performance of the proposed algorithm.\n\n\\section{Conclusion and Future Directions}\n\\label{sec:Conclusion}\nWireless communication has become exceedingly important in our daily lives. However, one of the major concerns in future networks is the privacy of user data. To this end, this work tries to address the eavesdropping problem caused by untrusted relays. The proposed approach involves the transmission of a jamming signal from the destination in the interval while it receives the \\ac{CP} part of the signal. Unlike the conventional solutions which require jamming signals to be transmitted throughout the broadcast phase, the proposed method only requires the signal to be transmitted for a fraction of it.\n\n\\par It should be noted that even though the current work focuses on the cooperative communication scenario, specifically untrusted relays, the concept presented in this work is much more general and can be applied to the eavesdropping problem in any \\ac{OFDM} system. Moreover, in such systems the jamming signal may also corrupt the (blind) synchronization attempted by any illegitimate receiver. Further work can also be carried out to optimize the relay placement and jamming power allocation to maximize the security gap in cooperative systems. Moreover, the proposed technique can be used for secure key exchange between the legitimate nodes in the presence of an untrusted relay. Furthermore, the proposed algorithm can also be applied in cases when there is no direct path between source and destination.\n\n\\section*{Acknowledgment}\nThis work was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant No. 120C142. The work of Haji M. Furqan was supported by HISAR Lab at TUBITAK BILGEM, Gebze, Turkey.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Lorenz 1984 Model}\nThe study of atmospheric dynamics is challenging due to its complex and chaotic nature. Lorenz \\cite{Lorenz:1984atmosphere} proposed in 1984 a simplified model for representing Hadley circulation of air in the atmosphere:\n\\begin{align}\n\\frac{dX}{dt}&=-Y^2-Z^2-aX+aF\\\\\n\\frac{dY}{dt}&=XY-bXZ-Y+G\\\\\n\\frac{dZ}{dt}&=bXY+XZ-Z\n\\end{align}\n\nThe equations are, in his words, \"the simplest model capable of representing an unmodified or modified Hadley circulation, determining its stability, and, if it is unstable, representing a stationary or migratory disturbance\" \\cite{Lorenz:1984atmosphere}. Lorenz uses the variable $X$ to represent the intensity of the symmetric globe-encircling westerly wind current, and also the poleward temperature gradient. The variables $Y$ and $Z$ represent the cosine and sine phase of a chain of superposed large-scale eddies. The parameters $a$, $b$, $F$ and $G$ may be chosen within certain bounds.\n\nThe model demonstrates chaotic behaviour for certain sets of parameters. In want of analytical solutions, initial-value problems are traditionally solved purely numerically by the use of finite steps in the temporal domain. The time steps of explicit time advance methods for general, space-dependent problems are restricted to small values through constraints such as the Courant-Friedrichs-Lewy (CFL) condition, and implicit schemes require time-consuming matrix operations at each time step. Semi-implicit methods allow large time steps and more efficient matrix inversions than those of implicit methods, but may feature limited accuracy. In this work we suggest an alternative, time-spectral approach for solution of equations (1)-(3). \n\n\\subsection{Generalized Weighted Residual Method}\nThe Generalized Weighted Residual Method (GWRM) differs from traditional spectral methods for initial-value problems \\cite{Gottlieb:1} in that also the time domain is treated spectrally \\cite{Scheffel:GWRM1}. As a result the GWRM eliminates grid causality conditions such as the CFL condition, being associated with time stepping algorithms. Although the problems to be solved are assumed causal, the method is acausal in the sense that the time dependence is calculated by a global minimization procedure (the weighted residual formalism) acting on the time integrated problem. Recall that in standard WRM \\cite{Fletcher:1}, initial value problems are transformed into a set of of coupled ordinary, linear or non-linear, differential equations for the time-dependent expansion coefficients. These are solved using finite differencing techniques. \n\nThe GWRM enables semi-analytical solutions;  finite approximate solutions are obtained as analytical Chebyshev expansions. Not only temporal and spatial, but also a physical parameter domain may be treated spectrally, being of interest for carrying out scaling dependence in a single computation. Chebyshev polynomials are used for the spectral representation. These have several desirable qualities. They converge rapidly to the approximated function, they are real and can straightforwardly be converted to ordinary polynomials and vice versa, their minimax property guarantees that they are the most economical polynomial representation, they can be used for non-periodic boundary conditions (being problematic for Fourier representations) and they are particularly apt for representing boundary layers where their extrema are locally dense \\cite{Scheffel:GWRM1}.\n\nIn standard text books on spectral methods for differential equations, time-spectral methods are usually touched upon only briefly and dismissed on the grounds that they are expensive \\cite{Canuto:1,Boyd:1}. Historically, a number of authors have investigated various suggestions for and aspects of spectral methods in time. \nIn 1979 a pseudo-spectral method, based on iterative calculation and an approximate factorization of the given equations, was suggested in \\cite{Morchoisne:1}. Also, some early ideas were not developed further by Peyret and Taylor in \\cite{Peyret:1}.\n\nIn 1986 and 1989, Tal-Ezer \\cite{Tal-Ezer:1,Tal-Ezer:2},  proposed time-spectral methods for linear, periodic hyperbolic and parabolic equations, respectively, using a polynomial approximation of the evolution operator in a Chebyshev least square sense. Periodicity was assumed for the spatial domain through use of the Fourier spectral approximation.\nThe method extends the traditional ${\\Delta}t=O(1\/N{^2})$ stability criterion for explicit algorithms, where the space resolution parameter $N=O(1\/{\\Delta}x)$, to higher efficiency resulting in the stability condition ${\\Delta}t=O(1\/N)$). This approach to extend the time step in explicit methods was further studied in \\cite{Kosloff:1}. The method is not widely used; a reason for this may be its complexity and its restriction to certain classes of problems. Later, Luo extended the method to more general boundary conditions and multiple spatial dimensions \\cite{Luo:1}.\n\nIerley et al. \\cite{Ierley:1} solved a a class of nonlinear parabolic partial differential equations with periodic boundary conditions using a Fourier representation in space and a Chebyshev representation in time. Similarly as for the GWRM, the Burger equation and other problems were solved with high resolution. Tang and Ma \\cite{Tang:1} also used a spatial Fourier representation for solution of parabolic equations, but introduced Legendre Petrov-Galerkin methods for the temporal domain.\n\nIn 1994, Bar-Yoseph et al \\cite{Zrahia:1,Bar-Yoseph:1} used space-time spectral element methods for solving one-dimensional nonlinear advection-diffusion problems and second order hyperbolic equations. Chebyshev polynomials were later employed in space-time least-squares spectral element methods \\cite{Maerschalck:1}.\n\nA theoretical analysis of Chebyshev solution expansion in time and one-dimensional space, for equal spectral orders, was given in \\cite{Dutt:1}. The minimized residuals employed were however different from those of the GWRM.\n\nMore recently Dehghan and Taleei \\cite{Dehghan:1} found solutions to the non-linear Schr\\\"{o}dinger equation, using a time-space pseudo-spectral method where the basis functions in time and space were constructed as a set of Lagrange interpolants.\n\nTime-spectral methods feature high order accuracy in time. For implicit finite difference methods, deferred correction may provide high order temporal accuracy \\cite{Gustafsson:1,Gustafsson:2}. A relatively recent approach to increase the temporal efficiency of finite difference methods is time-parallelization via the parareal algorithm \\cite{Lions:1}. This method, however, features rather low parallel efficiency and improvements have been suggested, for example the use of spectral deferred corrections \\cite{Emmett:1}.\n\nAn interesting Jacobian-free Newton-Krylov method for implicit time-spectral solution of the compressible Navier-Stokes equations has recently been put forth by Attar \\cite{Attar:1}. \n\nA time-spectral method for periodic unsteady computations, using a Fourier representation in time, was suggested in \\cite{Gopinath:1} and further developed in \\cite{Sicot:1} and \\cite{Luder:1}. A generalization to quasi-periodic problems was developed in \\cite{Mavriplis:1}. \n\nIn summary, although time-spectral methods have been explored in various forms by several authors during the last few decades, and were found to be highly accurate, the GWRM as described in \\cite{Scheffel:GWRM1} has not been pursued. The present work contributes to the evaluation of this method.\n\n\nThe structure of the paper is as follows. In section 2 we briefly review the general GWRM formalism for solving a set of pde's but subsequently restrict us to a discussion of an optimized solution of the ode's (1)-(3). A major goal of this study is to evaluate possible advantages of the GWRM in relation to finite difference methods (FDM). Thus a comparative numerical study of convergence, accuracy, and efficiency for short time interval solutions are presented in section 3. We have selected the explicit fourth order Runge-Kutta method (RK4) to represent efficient FDM. Comparisons will also be made with the implicit second order Lobatto IIIA (trapezoidal rule) and fourth order Lobatto IIIC methods, the latter being particularly apt for stiff problems \\cite{Brenan:Lobatto}. The chaotic character of the Lorenz equations is predominant in longer time calculations. In section 4 we determine long time accuracy and efficiency as well as the Liapunov exponent for the scenario studied here, employing both the RK4 and the GWRM. In section 5 the predictability of the GWRM is studied. We find that the efficiency of the GWRM compares well with and may exceed that of the RK4 method. A discussion follows in section 6 and conclusions are given in section 7.\n\n\\section{GWRM formalism}\n\nThe GWRM \\cite{Scheffel:GWRM1} solves a system of initial-value partial differential equations\n\\begin{equation}\n\\frac{\\partial\\mathbf{u}}{\\partial t}=D\\mathbf{u}+f\n\\end{equation}\nHere $D$ denotes a linear or nonlinear matrix differential operator that may depend on physical variables ($t$, $\\mathbf{x}$, and $\\mathbf{u}$) as well as on physical parameters (denoted $\\mathbf{p}$), and $f$=${f}(t,\\mathbf{x};\\mathbf{p})$ is a known source or forcing term. An approximate solution ansatz is assumed as a truncated multivariate series of first kind Chebyshev polynomials $T$, which becomes\n\\begin{equation}\n\\mathbf{u}(t,x;p)=\\sum_{q=0}^{Q}\\sum_{r=0}^{R}\\sum_{s=0}^{S}\\mathbf{a}_{qrs}T_q(\\tau)T_r(\\xi)T_s(P)\n\\end{equation}\n\nfor the case of one spatial dimension $x$ and one parameter $p$. The coefficients $\\mathbf{a}_{qrs}$ of this analytical expression are obtained from the Galerkin weighted residual method. For a single differential equation in $u$ it involves integration\n\\begin{equation}\n\\int_{t_0}^{t_1}\\int_{{x_0}}^{{x_1}}\\int_{{p_0}}^{{p_1}}RT_q(\\tau)T_r(\\xi)T_s(P)w_tw_xw_pdtd{x}d{p}=0\n\\end{equation}\nover the entire computational domain and solution for all indices $q,r,s$. The following weight factors are used \\cite{Mason:1}:\n\\begin{equation}\nw_t=(1-\\tau^2)^{\\frac{1}{2}},\nw_x=(1-\\xi^2)^{\\frac{1}{2}},\nw_p=(1-P^2)^{\\frac{1}{2}}\n\\end{equation}\n$R$ is the residual of the approximation \\cite{Fletcher:1}, defined as\n\\begin{equation}\nR=u(t,{x};{p})-\\left(u(t_0,{x};{p})+\\int_{t_0}^{t}(Du+f)dt'\\right)\n\\end{equation}\nand $\\tau$, $\\xi$ and $P$ are given by ($z$ can be any of $t$, $x$ or $p$ and $z_0$ and $z_1$ represent interval boundaries.)\n\\begin{equation}\n\\tau=\\frac{t-A_t}{B_t},\n\\xi=\\frac{x-A_x}{B_x},\nP=\\frac{p-A_p}{B_p},\n\\end{equation}\n\\begin{equation}\nA_z=\\frac{z_1+z_0}{2},\nB_z=\\frac{z_1-z_0}{2}\n\\end{equation}\n\nCarrying out the integrations in equation (6), the following system of algebraic equations results for the Chebyshev coefficients: \n\\begin{equation}\n\ta_{qrs}=2\\delta_{q0}b_{rs}+A_{qrs}+F_{qrs}\n\\end{equation}\nfor which $b_{rs}$ are the Chebyshev coefficients of the initial condition, and $A_{qrs}$ are obtained from the expansion\n\\begin{equation}\n\\int_{t_0}^{t}Dudt'=\\sum_{q=0}^{Q}\\sum_{r=0}^{R}\\sum_{s=0}^{S}A_{qrs}T_q(\\tau)T_r(\\xi)T_s(P)\n\\end{equation}\nshowing that equation (11) is a linear\/nonlinear equation in the parameters $a_{qrs}$ depending on the form of equation (4). The coefficients $F_{qrs}$ are obtained by a similar procedure to that in (12). Boundary conditions enter by replacing high-end $r$ indexed coefficients of (11) with corresponding spectral boundary equations. For the system of ode's (1)-(3), the forcing term and boundary conditions are not applicable and equations (11)-(12) are replaced by\n\\begin{equation}\n\t\\mathbf{a}_{q}=2\\delta_{q0}\\mathbf{b}+\\mathbf{A}_{q}\n\\end{equation}\n\\begin{equation}\n\\mathbf{b}=(X(0),Y(0),Z(0)),  ~~~ \\int_{t_0}^{t}D\\mathbf{u}dt'=\\sum_{q=0}^{Q}\\mathbf{A}_{q}T_q(\\tau)\n\\end{equation}\nwhere the array $\\mathbf{a}_q$ contains the Chebyshev coefficients of all $X$, $Y$ and $Z$ variables.\n\nTo iteratively solve the system of non-linear algebraic equations for the Chebyshev coefficients in the GWRM, a semi-implicit root-solver (SIR) has been developed \\cite{Scheffel:SIR}. This solver has better global convergence characteristics than Newton's method and avoids landing on local minima, as may occur when employing Newton's method with line-search. In GWRM applications, the handling of matrix equations in SIR for solution of equations (11) or (13) is the bottleneck both in terms of memory use and efficiency. In unoptimized cases, memory scales with the total number of Chebyshev coefficients to the second power, here as $(3(Q+1))^2$, and the number of operations scales to the third power, here $(3(Q+1))^3$. There exist, however, optimizing metods not touched upon here to reduce these scalings substantially. \n\nFor the present ode's to be solved, the maximum Chebyshev order $Q$ will be 10 or less, so that the main factor governing efficiency is rather the rate of convergence, determined by the number of SIR iterations which, in turn, depend on the length of the time interval used. A single GWRM solution for temporal domains exceeding about two Lorenz time units can generally not be computed due to the complexity of the present equations and the spurious roots that are generated by the nonlinear algebraic terms obtained by transforming equations (1)-(3) to equation (13). The time domain is thus divided into a number of partially overlapping subintervals of typical length 0.5-2 Lorenz time units. End conditions at each time interval are used as initial conditions for the subsequent interval. The solutions are finally joined to constitute a piece-wise analytical solution for the entire time domain. It should be noted that the lengths of the time intervals exceed the length of finite difference time steps with orders of magnitude. In this work, GWRM time intervals are typically a factor 100 longer than the RK4 time steps for comparable accuracy. \n\nFurthermore, the GWRM code employed here uses an adaptive algorithm to control the length of each time interval in order to guarantee convergence. If the accuracy requirements are not fulfilled, a shorter time interval will be chosen and the coefficients will be recalculated. The amount of overlap between subsequent time domains can be arbitrarily chosen in order to provide optimized two-point contact and enhanced convergence. \n\nGenerally, the convergence of iterative solvers like SIR are strongly dependent on the choice of starting conditions. An important property of the GWRM is that convergence is guaranteed as the time interval is decreased. This is because the solution then approaches the initial state, the coefficients of which are used as initial iterate for SIR. Moreover we will see that for simulations that are performed as perturbations of a \"base run\", use of the latter solution as initial state significantly reduces the number of required SIR iterations. The GWRM efficiency then exceeds that of RK4 and implicit methods by a wide margin. \n\n\n\\section{Convergence, accuracy, and efficiency}\n\nWe are interested in solving equations (1)-(3) accurately for long times up to some 100 time units. Consequently it is of interest to optimize the GWRM solution in individual time intervals. We may thus ask: is it preferable to use higher order approximations in long time intervals rather than to use lower order approximations in shorter intervals? To find out,  we solve equations (1)-(3) for $a = 0.25$, $b = 4.0$, $F = 8.0$ and $G = 1.0$, which parameters correspond to chaotic behaviour. The initial conditions used are well inside the attractor domain; $(X(0),Y(0),Z(0))=(0.96,-1.1,0.5)$. A corresponding GWRM solution for the time interval $[0,30]$ is shown in FIG. 1.\n\n\\begin{figure}[h!]\n\t\\includegraphics[width=5.0in]{Lorenz_1984_XYZ.eps}\n\t\\caption{GWRM solution of Lorenz equations (1)-(3) for initial values $(X(0),Y(0),Z(0))=(0.96,-1.1,0.5)$ and parameters $a = 0.25$, $b = 4.0$, $F = 8.0$ and $G = 1.0$. From top to bottom; $X(t)+8, Y(t)+4, Z(t)$. The time domain was divided into $N=71$ subintervals during the time-adaptive computation.}\n\\end{figure}\n\nIn order to control GWRM accuracy we primarily need to consider the parameters used in SIR, the accuracy of which is essentially determined by three parameters. The first is the round-off error which, for the Maple computations of this paper, is globally set to $10^{-14}$. Second, a parameter $tol$ governs the mean absolute difference between successively iterated Chebyshev coefficients. Third, a parameter $\\epsilon$ controls the numerical ratio between the highest and lowest two mode number Chebyshev coefficients (absolute values); this ratio is small for well resolved solutions. \n\nIn TABLE~\\ref{Table_maxinterval} approximate maximum time interval lengths, corresponding to convergent GWRM solutions, are shown for different Chebyshev orders $Q$. CPU times for each case are also given. We have here set $tol=10^{-5}$. SIR is run in standard mode, without sub-iterations; see \\cite{Scheffel:SIR}. The convergence of Newton's method would be insufficient for some of these cases. When solving for longer time intervals than those of TABLE~\\ref{Table_maxinterval}, the GWRM may still converge but to spurious solutions that are not causally related to the initial conditions assumed. For the same reason, values of $Q$ beyond 8 do not result in convergent solutions in longer time intervals. We can now answer the question posed above; it is usually more efficient to employ lower values of $Q$ in time subintervals of some optimized length than to use a higher value in a single interval with the same total length. \n\n\\begin{table}[h!]\n\t\\caption{Maximum time interval lengths allowing GWRM convergence, and corresponding CPU times in seconds.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t$Q$&Time interval&CPU time\\\\\\hline\n\t\t$4$&$1.0$&0.031\\\\\\hline\n\t\t$5$&$1.3$&0.046\\\\\\hline\n\t\t$6$&$1.6$&0.047\\\\\\hline\n\t\t$7$&$1.9$&0.047\\\\\\hline\n\t\t$8$&$2.0$&0.062\\\\\\hline\n\t\t$9$&$1.7$&0.062\\\\\\hline\n\t\t$10$&$1.8$&0.078\\\\\\hline\n\t\\end{tabular}\\label{Table_maxinterval}\n\\end{center}\n\\end{table}\n\nWe need also to consider accuracy. In order to determine the values required of the parameters $Q$, $tol$ and $\\epsilon$ for a certain accuracy, comparisons with highly accurate solutions of (1)-(3) should be carried out. The solutions are obtained from Maple's built-in ode solver dsolve, for which we have set the relative error to $10^{-10}$. A series of computations have been made for the time interval $[0,2]$. The automatic time-adaptive algorithm, here using a time interval overlap of length $10^{-8}$ time units, checks for convergence. If convergence is not obtained, the time interval is halved. Every $M$th interval, where $M$ can be set (it is here 10), the code attempts to extend the time interval by a given factor, here 1.5. Obtained accuracy, in terms of maximum error of either of the $X$, $Y$, or $Z$ variables, are displayed in TABLE~\\ref{Table_max_error_at_2}. Again we have set $tol=10^{-5}$. The number of time subintervals used is denoted by $N$.\n\n\\begin{table}[h!]\n\t\\caption{Maximum GWRM absolute error of $X,Y,Z$ at $t=2$ for $tol=10^{-5}$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\t$\\epsilon$&$Q$&$N$&CPU time&Error\\\\\\hline\n\t\t$10^{-1}$&4&$7$&0.078&$7.9\\times10^{-2}$\\\\\\hline\n\t\t$10^{-1}$&6&$3$&0.062&$1.6\\times10^{-2}$\\\\\\hline\n\t\t$10^{-1}$&8&$3$&0.078&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-1}$&10&$3$&0.109&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-2}$&4&$12$&0.109&$1.9\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&6&$3$&0.078&$1.6\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&8&$3$&0.094&$1.1\\times10^{-3}$\\\\\\hline\n\t\t$10^{-2}$&10&$3$&0.109&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-3}$&4&$28$&0.218&$1.7\\times10^{-3}$\\\\\\hline\n\t\t$10^{-3}$&6&$7$&0.125&$3.1\\times10^{-4}$\\\\\\hline\n\t\t$10^{-3}$&8&$3$&0.078&$1.1\\times10^{-3}$\\\\\\hline\n\t\t$10^{-3}$&10&$3$&0.110&$5.6\\times10^{-5}$\\\\\\hline\n\t\t$10^{-5}$&4&$176$&0.998&$1.2\\times10^{-5}$\\\\\\hline\n\t\t$10^{-5}$&6&$18$&0.250&$1.5\\times10^{-5}$\\\\\\hline\n\t\t$10^{-5}$&8&$7$&0.156&$1.2\\times10^{-6}$\\\\\\hline\n\t\t$10^{-5}$&10&$7$&0.250&$6.8\\times10^{-7}$\\\\\\hline\n\t\t$10^{-5}$&12&$3$&0.140&$2.9\\times10^{-6}$\\\\\\hline\n\t\\end{tabular}\\label{Table_max_error_at_2}\n\\end{center}\n\\end{table}\n\nIt is seen that, as far as accuracy is concerned, low $Q$ values are undesirable. In order to satisfy the $\\epsilon$ criterion, the corresponding solutions need a large number of time subintervals and are thus costly. The solution error at the end of the computation (\"Error\" in TABLE~\\ref{Table_max_error_at_2}) need be of order $10^{-3}$ for the computations of sections IV-V of this study. Thus $Q$-values in the range of 8-10 are found to be optimal with respect to both accuracy and efficiency. The root solver SIR is here best run in Newton mode to reduce the number of iterations required. The initial time interval is set to $[0,2\/N_t]$ with $N_t=3$.\n\nNext, we compare with the RK4 method. The single parameter that controls accuracy is the step length parameter $h$. In TABLE~\\ref{Table_RK4_at_20} the maximum absolute error of $X,Y,Z$ at $t=20$ is computed for various step lengths. Corresponding results for the GWRM are displayed for different values of $\\epsilon$ and $tol$ (with $N_t=50$) in TABLE~\\ref{Table_GWRM_fixed_tol_at_20} and TABLE~\\ref{Table_GWRM_fixed_epsilon_at_20} respectively. It is seen that for low accuracy RK4 is somewhat more efficient than GWRM. For higher accuracy (better than $10^{-3}$) the GWRM is more efficient than RK4. This is particularly pronounced when the time overlap (\"handshaking\") procedure in GWRM is omitted. It should be noted that similar results as in TABLE~\\ref{Table_GWRM_fixed_tol_at_20} and TABLE~\\ref{Table_GWRM_fixed_epsilon_at_20} are obtained when the initial conditions $(X(0),Y(0),Z(0))$ are varied. The results are thus representative. \n\nRK4 is an explicit method. Are implicit algorithms like the second-order Trapezoid or the fourth-order Lobatto IIIC methods possibly more efficient? In TABLES \\ref{Table_Trapezoid_at_20} and \\ref{Table_LobattoIIIC_at_20} we show results for these methods for the same step lengths that were used for RK4 in TABLE III. It is seen that both methods are more costly than RK4 and GWRM for the same accuracy. The Lobatto IIIC method is more than an order of magnitude slower.\n\n\\begin{table}[h!]\n\t\\caption{RK4 method maximum absolute error of $X,Y,Z$ at $t=20$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$h$&Steps&CPU time&Error\\\\\\hline\n\t\t$0.1$&200&$0.21$&$1.3\\times10^{0}$\\\\\\hline\n\t\t$0.05$&400&$0.41$&$1.5\\times10^{-1}$\\\\\\hline\n\t\t$0.02$&1000&$1.01$&$4.8\\times10^{-3}$\\\\\\hline\n\t\t$0.01$&2000&$1.96$&$3.2\\times10^{-4}$\\\\\\hline\n\t\t$0.005$&4000&$3.91$&$5.7\\times10^{-5}$\\\\\\hline\t\n\t\\end{tabular}\\label{Table_RK4_at_20}\n\\end{center}\n\\end{table}\n\\begin{table}[h!]\n\t\\caption{GWRM maximum absolute error of $X,Y,Z$ at $t=20$ for $tol=10^{-5}$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\t$\\epsilon$&$Q$&$N$&CPU time&Error\\\\\\hline\n\t\t$10^{-1}$&6&$38$&0.48&$9.1\\times10^{-1}$\\\\\\hline\n\t\t$10^{-1}$&8&$37$&0.80&$1.8\\times10^{-1}$\\\\\\hline\n\t\t$10^{-1}$&10&$37$&1.12&$5.5\\times10^{-3}$\\\\\\hline\n\t\t$10^{-2}$&6&$51$&0.64&$1.3\\times10^{-1}$\\\\\\hline\n\t\t$10^{-2}$&8&$38$&0.73&$2.8\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&10&$37$&1.14&$5.5\\times10^{-3}$\\\\\\hline\t\t\n\t\t$10^{-3}$&6&$72$&0.84&$7.2\\times10^{-2}$\\\\\\hline\n\t\t$10^{-3}$&8&$50$&0.91&$2.3\\times10^{-2}$\\\\\\hline\n\t\t$10^{-3}$&10&$38$&1.05&$2.4\\times10^{-3}$\\\\\\hline\t\t\n\t\t$10^{-5}$&6&$185$&1.84&$1.2\\times10^{-3}$\\\\\\hline\n\t\t$10^{-5}$&8&$83$&1.42&$3.8\\times10^{-4}$\\\\\\hline\n\t\t$10^{-5}$&10&$55$&1.44&$5.6\\times10^{-5}$\\\\\\hline\n\t\\end{tabular}\\label{Table_GWRM_fixed_tol_at_20}\n\\end{center}\n\\end{table}\n\\begin{table}[H]\n\t\\caption{GWRM maximum absolute error of $X,Y,Z$ at $t=20$ for $\\epsilon=10^{-5}$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|}\\hline\n\t\t$tol$&$Q$&$N$&CPU time&Error\\\\\\hline\n\t\t$10^{-1}$&6&$173$&1.20&$1.1\\times10^{0}$\\\\\\hline\n\t\t$10^{-1}$&8&$95$&1.12&$1.5\\times10^{0}$\\\\\\hline\n\t\t$10^{-1}$&10&$58$&1.08&$1.9\\times10^{0}$\\\\\\hline\n\t\t$10^{-2}$&6&$200$&1.65&$4.8\\times10^{-2}$\\\\\\hline\n\t\t$10^{-2}$&8&$84$&1.22&$2.0\\times10^{-1}$\\\\\\hline\n\t\t$10^{-2}$&10&$55$&1.17&$2.5\\times10^{-2}$\\\\\\hline\n\t\t$10^{-3}$&6&$185$&1.73&$1.8\\times10^{-3}$\\\\\\hline\n\t\t$10^{-3}$&8&$83$&1.28&$4.6\\times10^{-4}$\\\\\\hline\n\t\t$10^{-3}$&10&$55$&1.30&$4.2\\times10^{-4}$\\\\\\hline\n\t\t$10^{-5}$&6&$185$&1.84&$1.2\\times10^{-3}$\\\\\\hline\n\t\t$10^{-5}$&8&$83$&1.42&$3.8\\times10^{-4}$\\\\\\hline\n\t\t$10^{-5}$&10&$55$&1.44&$5.6\\times10^{-5}$\\\\\\hline\n\t\\end{tabular}\\label{Table_GWRM_fixed_epsilon_at_20}\n\\end{center}\n\\end{table}\n\\begin{table}[H]\n\t\\caption{Trapezoid method maximum absolute error of $X,Y,Z$ at $t=20$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$h$&Steps&CPU time&Error\\\\\\hline\n\t\t$0.1$&200&$0.84$&$1.1\\times10^{0}$\\\\\\hline\n\t\t$0.05$&400&$1.44$&$6.3\\times10^{-1}$\\\\\\hline\n\t\t$0.02$&1000&$3.33$&$8.0\\times10^{-1}$\\\\\\hline\n\t\t$0.01$&2000&$6.89$&$2.1\\times10^{-1}$\\\\\\hline\n\t\t$0.005$&4000&$12.6$&$8.0\\times10^{-2}$\\\\\\hline\n\t\t\\end{tabular}\\label{Table_Trapezoid_at_20}\n\\end{center}\n\\end{table}\n\n\\begin{table}[H]\n\t\\caption{Lobatto IIIC method maximum absolute error of $X,Y,Z$ at $t=20$.}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$h$&Steps&CPU time&Error\\\\\\hline\n\t\t$0.1$&200&$4.99$&$5.8\\times10^{0}$\\\\\\hline\n\t\t$0.05$&400&$9.32$&$4.8\\times10^{-2}$\\\\\\hline\n\t\t$0.02$&1000&$20.9$&$1.3\\times10^{-3}$\\\\\\hline\n\t\t$0.01$&2000&$40.4$&$4.6\\times10^{-5}$\\\\\\hline\n\t\t$0.005$&4000&$78.0$&$3.5\\times10^{-5}$\\\\\\hline\t\n\t\\end{tabular}\\label{Table_LobattoIIIC_at_20}\n\\end{center}\n\\end{table}\n\n\\section{Long time behaviour}\nTo examine the accuracy of numerical solutions in absence of analytical solutions, a common practice is to study the convergence of different solutions, controlled by some parameter, towards a high accuracy solution. This approach will now be used in a comparative study of the accuracy of RK4, Lobatto IIIC and the GWRM for the Lorenz 1984 problem in the time interval $[0,30]$. Chaotic systems like the Lorenz 1984 equations are characterized by the fact that two initially adjacent states (with separation $E_{initial}$) will, even in absence of numerical errors, deviate at a rate \n\\begin{equation}\nE(t)=E_{initial}e^{\\lambda t}\n\\end{equation}\nwhere $\\lambda$ is the Lyapunov exponent \\cite{Lyapunov:1892thesis}. \n\nOur numerical results show that $\\ln E$ typically grows rapidly when $t$ is small and then linearly when $t$ is large. This suggests an approximate dependence of the form $E(t)=F(t)+Ce^{At}$ where $F(t)$ is some non-exponential function, playing a dominant role when $t$ is small. For large $t$, the exponential factor takes over and the logarithmic graph is linear. \n\nIn light of the above, we have found the following numerical procedure to be useful. For each of the numerical methods, a set of 100 solutions have been produced employing a set of selected accuracies. Each solution is compared with a high accuracy solution that uses exactly the same initial conditions, and the deviation as a function of time has been computed as described below. To optimize statistics, the attractor space is scanned by letting the end state of each high accuracy solution constitute the initial state for the new pair of solutions to be computed. The graphs $\\ln(E)$ vs $t$ are subsequently generated by taking the arithmetic mean of the calculations.\n\nThe expression\n\\begin{equation}\n\\ln(E)=E_0+At\n\\end{equation}\nis fitted to the graphs over the entire temporal domain. We find that the slope $A$ is quite independent of the choice of method or accuracy, which suggests that it is determined by the Lorenz system (1)-(3) itself indicating that $A=\\lambda$. The accuracy of the method can thus be solely estimated from $E_0$. Since the $\\ln E(t)$ graph is similar for each variable $X,Y,Z$, as can be seen from FIG. 2, we will mainly focus our discussion on the $X$ dimension.  \n\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{GWRM_wohs_rel_error_Ex.eps}\n\t\\includegraphics[width=2.3in]{GWRM_wohs_rel_error_Ey.eps}\n\t\\includegraphics[width=2.3in]{GWRM_wohs_rel_error_Ez.eps}\n\t\\caption{GWRM: $\\ln E_{X,\\epsilon,tol}$, $\\ln E_{Y,\\epsilon,tol}$ and $\\ln E_{Z,\\epsilon,tol}$ vs $t$ for $tol=10^{-5}$ and different $\\epsilon$.}\n\\end{figure}\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{GWRM_wohs_tol_Ex.eps}\n\n\n\t\\caption{GWRM: $\\ln E_{X,\\epsilon,tol}$ vs $t$ for $\\epsilon=10^{-5}$ and different $tol$.}\n\t\\end{figure}\nTwo factors control the accuracy of the GWRM method: the SIR parameters $tol$ and $\\epsilon$.\nThe deviation for each pair of GWRM computations is calculated by\n\\begin{equation}\nE_{k,\\epsilon,tol}(t)=|{u_{k}(t)}_{\\epsilon,tol}-{u_{k}(t)}_{10^{-5},10^{-5}}|\n\\end{equation}\nwhere $u_{k}$ denote the components of the vector $(X,Y,Z)$. Plots of $ln(E_{k,\\epsilon,tol}(t))$ for various $\\epsilon$ are given in FIG. 2 and a plot showing the deviations due to different $tol$ is shown in FIG. 3. Least square fitted parameters of (16) are listed in TABLE~\\ref{Table_errorfit_GWRM}.\n\\begin{table}[h!]\n\t\\caption{GWRM: Fitted parameters of Equation(17).}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|}\\hline\n\t\t$\\varepsilon$&tol&$A$&$E_0$\\\\\\hline\n\t\t$1\\times 10^{-5}$&$1\\times 10^{-4}$&0.27&-17.9\\\\\\hline\n\t\t$1\\times 10^{-5}$&$1\\times 10^{-3}$&0.27&-12.9\\\\\\hline\n\t\t$1\\times 10^{-5}$&$1\\times 10^{-2}$&0.27&-7.4\\\\\\hline\n\t\t$1\\times 10^{-4}$&$1\\times 10^{-5}$&0.27&-14.5\\\\\\hline\n\t\t$1\\times 10^{-3}$&$1\\times 10^{-5}$&0.27&-11.4\\\\\\hline\n\t\t$1\\times 10^{-2}$&$1\\times 10^{-5}$&0.27&-9.1\\\\\\hline\t\n\t\\end{tabular}\\label{Table_errorfit_GWRM}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{RK4_Ex.eps}\n\n\n\t\\caption{RK4: $\\ln E_{X,h}$ vs $t$ with a variation of $h$.}\n\\end{figure}\n\\begin{figure}[h!]\n\\center\n\t\\includegraphics[width=2.3in]{Lobatto3C_Ex.eps}\n\n\n\t\\caption{Lobatto IIIC: $\\ln E_{X,h}$ vs $t$ with a variation of $h$.}\n\\end{figure}\nThe explicit classical RK4 and implicit Lobatto IIIC finite difference methods are chosen for comparisons.\nFor FDM, the controlling factor of accuracy is the time step $h$. The deviation is thus calculated as\n\\begin{equation}\nE_{k,h}(t)=|{u_{k}(t)}_{h}-{u_{k}(t)}_{5\\times 10^{-3}}|\n\\end{equation}\nDeviations $E(t)$ for different time step lengths are plotted in FIG:s 4 and 5 respectively and the fitted parameters are listed in TABLE~\\ref{Table_errorfit_RK4} and TABLE~\\ref{Table_errorfit_LobattoIIIC}.\n\n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\caption{RK4: fitted parameters of Equation(17).}\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t$h$&$A$&$E_0$\\\\\\hline\n\t\t\t$1\\times 10^{-1}$&0.27&-4.1\\\\\\hline\n\t\t\t$5\\times 10^{-2}$&0.27&-6.9\\\\\\hline\n\t\t\t$2\\times 10^{-2}$&0.27&-10.8\\\\\\hline\n\t\t\t$1\\times 10^{-2}$&0.27&-13.6\\\\\\hline\n\t\\end{tabular}\\label{Table_errorfit_RK4}\n\t\\end{center}\n\\end{table}\n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\caption{Lobatto IIIC: fitted Value of Equation(17).}\n\t\t\\begin{tabular}{|c|c|c|}\\hline\n\t\t\t$h$&$A$&$E_0$\\\\\\hline\n\t\t\t$1\\times 10^{-1}$&0.27&-6.8\\\\\\hline\n\t\t\t$5\\times 10^{-2}$&0.27&-9.1\\\\\\hline\n\t\t\t$2\\times 10^{-2}$&0.27&-12.8\\\\\\hline\n\t\t\t$1\\times 10^{-2}$&0.27&-15.4\\\\\\hline\n\t\\end{tabular}\\label{Table_errorfit_LobattoIIIC}\n\t\\end{center}\n\\end{table}\nThe accuracy in terms of $E(0)$ of the fourth order FDM, the RK4 and Lobatto IIIC methods, is found to depend strongly on the step size; approximately $E(0)\\propto h^{3.8}$. The accuracy scalings of the GWRM are $E_0\\propto \\text{$\\epsilon$}^{1.2}$ for $tol=10^{-5}$ and $E_0\\propto \\text{$tol$}^{2.3}$ for $\\epsilon=10^{-5}$; see TABLE XI. Comparing TABLES \\ref{Table_errorfit_GWRM} and \\ref{Table_errorfit_RK4} with run times in TABLES \\ref{Table_RK4_at_20}, \\ref{Table_GWRM_fixed_tol_at_20} and \\ref{Table_GWRM_fixed_epsilon_at_20} it is again seen that the GWRM is more efficient at higher accuracies than the finite difference methods. \n\n\\begin{table}[h!]\n\\center\n\\caption{Accuracy comparison.}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tMethod&$\\epsilon$&$tol$&$h$&$E_0$\\\\\\hline\n\t\tRK4&&&$1\\times 10^{-1}$&-4.1\\\\\\hline\n\t\t&&&$5\\times 10^{-2}$&-6.9\\\\\\hline\n\t\t&&&$2\\times 10^{-2}$&-10.8\\\\\\hline\n\t\t&&&$1\\times 10^{-2}$&-13.6\\\\\\hline\n\t\tLobatto IIIC&&&$1\\times 10^{-1}$&-6.8\\\\\\hline\n\t\t&&&$5\\times 10^{-2}$&-9.1\\\\\\hline\n\t\t&&&$2\\times 10^{-2}$&-12.8\\\\\\hline\n\t\t&&&$1\\times 10^{-2}$&-15.4\\\\\\hline\n\t\tGWRM&$1\\times 10^{-2}$&$1\\times 10^{-5}$&&-9.1\\\\\\hline\n\t\t&$1\\times 10^{-3}$&$1\\times 10^{-5}$&&-11.4\\\\\\hline\n\t\t&$1\\times 10^{-4}$&$1\\times 10^{-5}$&&-14.5\\\\\\hline\n\t\t&$1\\times 10^{-5}$&$1\\times 10^{-2}$&&-7.4\\\\\\hline\n\t\t&$1\\times 10^{-5}$&$1\\times 10^{-3}$&&-12.9\\\\\\hline\n\t\t&$1\\times 10^{-5}$&$1\\times 10^{-4}$&&-17.9\\\\\\hline\n\t\\end{tabular}\\label{Table_efficiency}\n\\end{table}\n\nIn weather forecasting, running a number of simulations with slightly perturbed initial conditions is a common practice to enhance predictability. The GWRM is particularly well suited for these tasks, since the Chebyshev coefficients for the \"base\" run solution may be used as good initial iterates in SIR when computing the perturbed scenarios. This substantially reduces the number of iterations and the CPU time. In the next section predictability is studied in a comparison between the GWRM and the RK4 methods.  \n\n\\section{Predictability}\n\n\\noindent The Lorenz equations, and many weather related phenomena, are inherently chaotic. In other words, it is improbable that a long-term solution is representative if the initial condition varies slightly from the \"exact\" value. The error will eventually saturate at some level which implies that a random initial condition can be chosen without affecting the error growth, thus predictability is lost. The characteristic time $T$ when a perturbed solution diverges significantly from the true solution is thus of importance. It is in the time interval $[0,T]$ that predictions are meaningful.\n\nIt is of interest to compare the GWRM and RK4 method with regards to predictability. Our analysis goes as follows: a solution $\\mathbf{u_{1}}$ is obtained for $t=50$, where $\\mathbf{u}$ again denotes the vector $(X(t),Y(t),Z(t))$, and $\"1\"$ specifies a base run of the Lorenz equations with initial conditions  $\\mathbf{V}=(X(0),Y(0),Z(0))$. Next we perturb the initial conditions slightly; $\\mathbf{V}'=\\mathbf{V}+\\mathbf{\\delta}$, where the three components of the vector $\\mathbf{\\delta}$ are randomly distributed so that $0<\\delta<0.01$. The simulation is subsequently run again to obtain a perturbed solution $\\mathbf{u}'_{n}$. The deviations $e_{k,n}=(u'_{k,n}-u_{k,1})^2$ between the two simulations are then calculated for $k=1,2,3$. The same procedure is used for all perturbed scenarios, and then summed over the scenarios in order to obtain an error growth $E_{k}(t)=\\frac{1}{N}{\\sum}(u'_{k,n}-u_{k,1})^2, ~n=2..N$, where $N$ is the number of scenarios. Both RK4 and GWRM undergo the same analysis with comparable accuracies, which is chosen as the average accuracy of the three variables X, Y and Z, whereby the CPU times used are collected in TABLE XII. \n\nThe GWRM is found to be more than four times as efficient as RK4. As was shown in section III, this is partly due to the high efficiency of the GWRM for high accuracy, but mainly due to that for these perturbative runs, the initial iterates used by SIR are closer to the solution thus reducing the number of iterations needed.\n\n\\begin{table}[H]\n\\center\n\\caption{Error growth comparison between GWRM and RK4.}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tNumerical Method & Av. acc. & Steps & $\\epsilon$ & \\# Runs & CPU time [min] \\\\\\hline\n\t\tGWRM & 5.6e-6 & & $10^{-5}$ & 250 & 17.9 \\\\\\hline\n\t\tRK4 & 5.9e-6 & 15000 & & 250 & 83.5 \\\\\\hline\n\t\\end{tabular}\\label{Table_errorgrowth}\n\\end{table}\n\nThe most evident advantage that GWRM has over RK4 is that the time sub-intervals are typically two orders of magnitude larger. The time-adaptive scheme also allows for longer time intervals in smooth regions and shorter intervals in high gradient regions. This reduces the computation time for the method. The error growth in these studies show that the Lorenz equations are highly sensitive to perturbed initial conditions, and this will have a dominating effect where weather predictions are concerned, as is well established. This shows the need for a numerical model that is efficient at higher accuracies. As can be seen from FIG. \\ref{fig:1}, RK4 and GWRM give very similar error growths for the same accuracies. It is also seen that the characteristic time $T$ before which predictions are valid is about $T=10$ time units, whereafter the error growths tend to saturate.\n\n\\begin{figure}[H]\n\\center\n\t\\includegraphics[width=2.3in]{ETx_t50_P3.eps}\n\t\\includegraphics[width=2.3in]{ETy_t50_P3.eps}\n\t\\includegraphics[width=2.3in]{ETz_t50_P3.eps}\n\t\\caption{Error growth analysis using 250 perturbed scenarios, with comparable accuracy for the GWRM and RK4. GWRM: \\textit{brown line}, $m=8$, $tol=10^{-5}$ and $\\epsilon=10^{-5}$. RK4: \\textit{black dash line}, steps=15000.}\n\t\\label{fig:1}\n\\end{figure}\n\n\n\\section{Discussion}\nThe GWRM is a time-spectral method and differs substantially from traditional time-stepping methods. In this study a first assessment is made of its potential for modelling advanced NWP problems by addressing the Lorenz 1984 chaotic equations. The results are only tentative with respect to NWP, but motivate further studies where space dependence is taken into account. Of particular interest here is that recent development of the GWRM for problems in magnetohydrodynamics has lead to algorithms where high efficiency in handling the spatial dimensions have been obtained, using subdomains. The physical equations of each spatial subdomain may here be solved locally whereas the internal boundary condition equations are solved globally. This enables parallelization of the temporal domain.\n\nAn interesting feature of the GWRM is the possibility to average over the fast time scale for problems with several time scales \\cite{Scheffel:GWRM1}. A consequence is enhanced efficiency, since only lower order Chebyshev expansions are needed. This approach would be of considerable interest when performing multiple perturbation analysis of a base scenario because fewer scenarios would have to be computed, enhancing efficiency. For the present strongly nonlinear problem, however, high accuracy was found necessary for convergence to the correct solution. The multiple roots of the nonlinear algebraic system of equations (13) are closely distributed. Thus SIR easily departs from the correct solution, corresponding to the initial conditions given, for low temporal resolution. An effort to circumvent this tendency was made by linearising equations (1)-(3) before solution using SIR and gradually adding the nonlinear terms during the iterations. This approach is again not successful because of the tendency of root solvers to lock at the positions of the roots corresponding to the linear equations when the nonlinear terms are included. \n\nAnother question of interest is whether very long time intervals ($>>1$ time unit) could be employed, using the GWRM. The lesson learned from various problems solved so far is that the manageable interval length is indeed problem dependent. For non-smooth problems like the present, the time interval length is limited even if the Chebyshev order is increased because, again, of the spurious solutions found.\n\nAlthough a causal problem is solved, the GWRM need not necessarily be used as a causal method since both initial and end conditions may be applied. This is an interesting possibility for time-spectral methods which, however, requires further theoretical understanding. Various combination of initial and end conditions have indeed been tried, all within the attractor domains of the solutions, but no improvement of convergence was found.  \n\nUsually overlapping time intervals, employing two point contact, improves convergence. It is notable, and worthy of further study, that the best results (maximum efficiency) were here obtained using one point contact.\n\nSummarizing, the fact that the GWRM competes well with standard finite difference methods in solving the demanding Lorenz 1984 equations and that very efficient GWRM techniques have been developed for pde's where spatial dimensions are included, suggests that usage of time-spectral methods for NWP is well worth further exploration.  \n\n\n\\section{Conclusion}\nIn this work, a preliminary evaluation of a recently developed time-spectral method GWRM is carried out with respect to potential use for numerical weather prediction. The Lorenz (1984) chaotic equations are solved. The efficiency and the behaviour of the error growth with time is compared to traditional explicit and implicit finite difference methods. In particular, the optimal length of the solution time intervals have been determined, the accuracy of the method has been studied in detail and predictability has been investigated. \n\nIt is found that GWRM efficiency is in parity of, or better than, the finite difference methods. Efficiency is further enhanced for cases where several perturbed scenarios need be computed. This is mainly due to that GWRM time intervals are two orders of magnitude larger than those of finite difference methods. Furthermore, the GWRM solutions are analytical Chebyshev series expansions. These findings, and the existence of efficient algorithms for time parallelisation of spatially dependent pde's, are encouraging for future studies of time-spectral methods for NWP. \n\n\\section{Acknowledgement}\n\\noindent The authors would like to thank Dr {\\AA}ke Johansson at SMHI (Swedish Meteorological and Hydrological Institute) for inspiration and for pointing at the relevance of solving the Lorenz equations for a primary evaluation of the GWRM as a method for NWP.\n\n\\label{}\n\n\n\n\n\\section*{References}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Categories of causal r\\textsc{pes}\\ and \\textsc{rcn}}\\label{sec:category}\nOccurrence nets and \\textsc{pes} es are equipped with morphisms and turned  into categories that are\nrelated by suitable functors. \nIn this section, we extends such constructions to \\textsc{rcn}\\ and causal-r\\textsc{pes}. \nWe start by recalling the notions of morphisms for occurrence nets and prime event structures.\n\\begin{definition}\\label{de:occ-net-morph}\n Let $C_0 = \\langle B_0, E_0, F_0, \\mathsf{c}_0\\rangle$ and $C_1 = \\langle B_1, E_1, F_1, \\mathsf{c}_1\\rangle$\n be two occurrence nets. Then, an occurrence net \\emph{morphism} from $C_0$ to $C_1$ is a pair $(\\beta,\\eta)$ where\n $\\beta$ is a relation between $B_0$ and $B_1$, $\\eta : E_0\\to E_1$ is a partial mapping such that\n \\begin{itemize}\n   \\item $\\beta(\\mathsf{c}_0) = \\mathsf{c}_1$ and for each $b_1\\in \\mathsf{c}_1$ there exists a unique\n         $b_0\\in \\mathsf{c}_0$ such that $b_0 \\beta b_1$, and\n   \\item for all $e_0\\in E_0$, when $\\eta(e_0)$ is defined and equal to $e_1$ then \n         $\\beta(\\pre{e_0}) = \\pre{\\eta(e_0)} = \\pre{e_1}$\n         and $\\beta(\\post{e_0}) = \\post{\\eta(e_0)} = \\post{e_1}$.\n \\end{itemize}\n\\end{definition}\nAs we consider just occurrence nets, we have also that if $b_0 \\beta b_1$ and $(e_1,b_1)\\in F_1$, then there\nexists a unique event $e_0\\in E_0$ such that $\\eta(e_0) = e_1$ and $(e_0,b_0)\\in F_0$. \nConsequently, if a condition $b_1$ in $C_1$ is related to a condition $b_0$ in $C_0$ \nby $\\beta$, then the events producing these conditions are related \nby $\\eta$; moreover, there is a unique event $e_0$ in $C_0$ mapped to the event $e_1$. \nMorphisms on occurrence nets compose and the identity mapping is defined as\nthe identity relation on conditions and the \nidentity mapping on events. \nWe have the category having as objects the occurrence nets and as morphisms the \noccurrence nets morphisms, which will be denoted with $\\mathbf{Occ}$.\n\n\\begin{definition}\\label{de:pes-morph}\n Let $P_0 = (E_0, <_0, \\#_0)$ and $P_1 = (E_1, <_1, \\#_1)$ be two \\textsc{pes} es. Then,  a\n \\emph{\\textsc{pes}\\ morphism} from $P_0$ to $P_1$ is a partial mapping $f : E_0 \\to E_1$ such that\n \\begin{enumerate}\n   \\item for all $e_0\\in E_0$ if $f(e_0)$ is defined then $\\hist{f(e_0)} \\subseteq f(\\hist{e_0})$, \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and\n         $f(e_0) \\#_1 f(e_0')$ then $e_0 \\#_0 e_0'$, and \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and equal, then\n         $e_0 \\neq e_0'$ implies $e_0 \\#_0 e_0'$.    \n \\end{enumerate}\n\\end{definition}\n\nAgain \\textsc{pes}\\ morphisms compose, and we have the category $\\mathbf{PES}$ \n whose objects are \\textsc{pes} es\nand whose arrows are the \\textsc{pes}\\ morphisms.\n\nThe  categories $\\mathbf{Occ}$ and $\\mathbf{PES}$ are related as follows:  each occurrence net is associated with a prime event structure \nas stated\nin Proposition~\\ref{pr:on-to-pes}; each occurrence net morphism\n $(\\beta,\\eta)$ is associated with  $\\eta$, which turns out to be a \\textsc{pes}\\ morphism. We denote such functor as\n$\\mathcal{P}$  (as in Proposition~\\ref{pr:on-to-pes}).\nConversely, the mapping $\\mathcal{E}$ introduced in Proposition~\\ref{pr:pes-to-on} can be turned into a functor, \nas shown in \\cite{Win:ES}.  \nA \\textsc{pes}\\ morphism $f$ is turned into an occurrence net morphism as follows:\nthe relation $\\beta$ on conditions is defined such that (i) $(a,A) \\beta (a',A')$ when $a = \\bot = a'$, \n$A' = f(A) \\neq \\emptyset$, and $|A| = 1$, and (ii) $(e,A) \\beta (f(e),A')$ when $f(e)$ is defined and\n$A' = f(A) \\neq \\emptyset$;  the partial mapping on events is just $f$. It is then easy to\nsee that such definition indeed conforms an occurrence net morphism. \nThe nice result is that these functors form a coreflection, where $\\mathcal{C}$ is the right\nadjoint and $\\mathcal{E}$ the left adjoint.\n\nWe now recall the notion of r\\textsc{pes}\\ morphisms introduced in \\cite{GPY:CatRES}.\n\n\\begin{definition}\\label{de:rpes-morph}\n  Let $P_0 = (E_0, U_0, <_0, \\#_0, \\prec_0, \\triangleright_0)$ and $P_1 = (E_1, U_1, <_1, \\#_1, \\prec_0, \\triangleright_0)$ be two r\\textsc{pes}. Then, an\n \\emph{r\\textsc{pes}\\ morphism} from $P_0$ to $P_1$ is a partial mapping $f : E_0 \\to E_1$ such that\n \\begin{enumerate}\n   \\item for all $e_0\\in E_0$ if $f(e_0)$ is defined then \n         $\\setcomp{e_1\\in E_1}{e_1 <_1 f(e_0)} \\subseteq \\setcomp{f(e)}{e <_0 e_0}$, \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and\n         $f(e_0) \\#_1 f(e_0')$ then $e_0 \\#_0 e_0'$, \n   \\item for all $e_0, e_0'\\in E_0$ such that  $f(e_0)$ and $f(e_0')$ are both defined and equal, then\n         $e_0 \\neq e_0'$ implies $e_0 \\#_0 e_0'$, \n   \\item for all $e_0\\in E_0$ and $e\\inU_0$ such that\n         $f(e_0)$ and $f(e)$ are both defined  and $f(e_0) \\triangleright_1 f(e)$ then\n         $e_0 \\triangleright_0 e$, and\n   \\item for all $e_0\\inU_0$ if $f(e_0)$ is defined then \n         $\\setcomp{e_1\\in E_1}{e_1 \\prec_1 \\underline{f(e_0)}} \\subseteq \\setcomp{f(e)}{e <_0 \\underline{e_0}}$.          \n \\end{enumerate}\n\\end{definition}\nThe notion of morphism above generalise the one on \\textsc{pes}\\ by requiring that prevention is preserved as\nwell as the reverse causality relation. \nIn \\cite{GPY:CatRES} it is also shown that \nr\\textsc{pes}\\ and r\\textsc{pes}\\ morphisms form a category, which is called $\\mathbf{RPES}$. \nWe restrict our attention to the subcategory $\\mathbf{cRPES}$, which \n has causal-r\\textsc{pes} as objects and r\\textsc{pes}\\ morphisms as arrows.\n\n\\begin{proposition}\\label{pr:full-subcat}\n $\\mathbf{cRPES}$ is a full subcategory of $\\mathbf{RPES}$.\n\\end{proposition}\n\nWe now turn our attention to reversible causal nets and introduce the notion of morphisms for reversible causal nets.\n\n\\begin{definition}\\label{de:rcn-morph}\n Let $R_0 = \\langle B_0, E_0, U_0, F_0, \\mathsf{c}_0\\rangle$ and \n $R_1 = \\langle B_1, E_1, U_1, F_1, \\mathsf{c}_1\\rangle$ be two \\textsc{rcn}{s}.\n Then an \\textsc{rcn}\\ \\emph{morphism} from $R_0$ to $R_1$ is the pair $(\\beta,\\eta)$ such that\n \\begin{itemize}\n   \\item $(\\beta,\\eta)$ restricted to the occurrence nets  \n         $\\langle B_0, E_0\\setminus U_0, F'_0, \\mathsf{c}_0\\rangle$ to\n         $\\langle B_1, E_1\\setminus U_1, F'_1, \\mathsf{c}_1\\rangle$ is an occurrence net morphism, \n         and\n   \\item $\\eta(U_0) \\subseteq U_1$ and if $e\\in U_0$ and $\\eta(e)$ is defined then\n         also $f(e')$ is defined, where $e' = h(e)$.      \n \\end{itemize}\n\\end{definition}\nIt is straightforward to check that \\textsc{rcn}\\ morphisms compose and that the identity relation and the identity mapping on events\nconforms a \\textsc{rcn}\\  morphism. Hence,  \\textsc{rcn}\\ together with \\textsc{rcn}\\ morphism form a category, which we call\n$\\mathbf{rOcc}$. \n\nThen, the definition of $\\mathcal{C}$ in Proposition~\\ref{pr:rce-to-rpes} can be \\changed{easily}{} extended to a functor.\n\\begin{proposition}\\label{pr:C-is-functor} \n $\\mathcal{C} : \\mathbf{rOcc} \\to \\mathbf{cRPES}$ acting on objects as in Proposition~\\ref{pr:rce-to-rpes}\n and on morphisms $(\\beta,\\eta) : R_0 \\to R_1$ as $\\eta$ restricted to the events that are not\n reversing ones, is a functor.\n\\end{proposition}\n\nAlso the construction in Definition~\\ref{de:rpestorcn} can be extended to a functor. \n\n\\begin{proposition}\\label{pr:E-is-functor} \n $\\mathcal{E}_{r} : \\mathbf{cRPES} \\to \\mathbf{rOcc}$ acting on objects as in Definition~\\ref{de:rpestorcn}\n and on morphisms as stipulated in for occurrence net, requiring that reversing events are\n preserved, is a functor.\n\\end{proposition}\n\nAlong the lines of \\cite{Win:ES} we can establish a relation between the categories\n$\\mathbf{cRPES}$ and  $\\mathbf{rOcc}$.\n\n\\begin{theorem}\\label{th:coreflection} \n $\\mathcal{E}_{r}$ and $\\mathcal{C}_{r}$ form a coreflection, where $\\mathcal{C}_{r}$ is the right\nadjoint and $\\mathcal{E}_{r}$ the left adjoint.\n\\end{theorem}\n\n\n\n\\subsection{From r\\textsc{pes}\\ to \\textsc{rcn}}\nCorrespondingly to what is usually done when relating nets to event structures, we show that \nif we focus on causal r\\textsc{pes}{es} then we can relate them to reversible occurrence nets.\nThe construction is indeed quite standard (see \\cite{Win:ES,BCP:LenNets} among many others),\nbut we do need a further observation on causal r\\textsc{pes}.\n\n\\begin{proposition}\\label{pr:pes-associated-to-crpes}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a causal r\\textsc{pes}\\ and\n let $<^{+}$ be the transitive closure of $<$. Then, $\\#$ is inherited along  $<^{+}$,\n \\emph{i.e.} $e\\ \\#\\ e' <^{+} e''\\ \\Rightarrow\\ e\\ \\#\\ e''$.\n\\end{proposition}\nA consequence of this proposition is that the conflict relation is fully characterized by the\ncausality relation, and the same intuition for introducing reversible causal net can be used in\nassociating a net to a causal r\\textsc{pes}\\ like the one used to associate an occurrence net to a \\textsc{pes}. \n\n\\begin{definition}\\label{de:rpestorcn}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a causal r\\textsc{pes}, and \n  $\\bot\\not\\in E$ be a new symbol. Define \n $\\mathcal{E}_{r}(\\mathsf{P})$ as the Petri net $\\langle B, \\hat{E}, F, \\mathsf{c}\\rangle$ where\n \\begin{itemize}\n    \\item $B = \\setcomp{(a,A)}{a\\in E\\cup\\setenum{\\bot} \\land A \\subseteq E \\land \\CF{A} \\land \n    a \\neq \\bot\\ \\Rightarrow\\ \\forall e\\in A.\\ a \\ll e}$,\\smallskip\n   \\item $\\hat{E} = E\\times\\setenum{\\mathtt{f}}\\  \\cup\\  U\\times\\setenum{\\mathtt{r}}$,\n   \\item $F = \\setcomp{(b,(e,\\mathtt{f}))}{b = (a,A)\\ \\land\\ e\\in A} \\quad \\cup\\quad\n              \\setcomp{((e,\\mathtt{f}),b)}{b = (e,A)}\\quad\\cup\\quad$ \\phantom{asdfa}\n              $\\setcomp{(b,(e,\\mathtt{r}))}{b = (e,A)}\\quad\\cup\\quad\n              \\setcomp{((e,\\mathtt{r}),b)}{b = (a,A)\\ \\land\\ e\\in A}$,\n                          and\n   \\item $\\mathsf{c} = \\setcomp{(\\bot,A)}{A \\subseteq E\\ \\land\\ \\CF{A}}$,                                               \n \\end{itemize}\n\\end{definition} \nIn essence the construction above takes the \\textsc{pes}\\ associated to an r\\textsc{pes}\\ and\nconstructs the associated occurrence net, which is then \\emph{enriched} with \nthe reversing events (transitions). The result is a reversible occurrence net.\n\n\\begin{proposition}\\label{prp:rpes_to_cnet}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a causal r\\textsc{pes}. Then  \n $\\mathcal{E}_{r}(\\mathsf{P}) = \\langle B, \\hat{E}, U\\times\\setenum{\\mathtt{r}}, F, \\mathsf{c}\\rangle$ as defined\n in Definition \\ref{de:rpestorcn} is a reversible occurrence net with respect to $U\\times\\setenum{\\mathtt{r}}$.\n\\end{proposition} \n\\begin{theorem}\\label{th:cccrpestorcn}\n Let $\\mathsf{P}$ be a causal \n r\\textsc{pes}. Then  \n $X'$ is a configuration of $\\mathcal{E}_{r}(\\mathsf{P})$ iff $X$ \n is a configuration of $\\mathsf{P}$, where $X' = \\setcomp{(e,\\mathtt{f})}{e\\in X}$.\n\\end {theorem}\nClearly, if we start from a reversible causal net, we get a r\\textsc{pes}\\ from which a reversible causal\nnet can be obtained having the same states (up to renaming of events).\n\\begin{corollary}\n Let $R$ be a \\textsc{rcn}. Then $\\states{\\mathcal{E}_{r}(\\mathcal{C}_{r}(R))} = \\states{R}$.\n\\end{corollary}\n\n\\begin{example}\n Consider the r\\textsc{pes}\\ with four events $\\setenum{e_1,e_2,e_3,e_4}$ such that $e_1 < e_3$ and $e_2 < e_4$,\n $e_1$ is in conflict with $e_2$ and this conflict is inherited along $<$. Furthermore, let \n $e_1$ and $e_3$ be reversible, and $e_3 \\triangleright \\underline{e}_1$. The construction \nin Definition~\\ref{de:rpestorcn} gives the net below.\n \\begin{center}\n  \\input{rpesrcnfig.tex}\n  \\end{center}\n\\end{example}\n\n\n\n\\section{Conclusions and future works}\\label{sec:conc}\nThe constructions we have proposed to associate a reversible causal net to a causal \nreversible prime event structure, and vice versa, are certainly driven by the classical ones \n(see \\cite{Win:ES}) for relating occurrence nets and prime event structures. \nThe consequence of this approach is that the causality relation, either the one given in a r\\textsc{pes}\\ or\nthe one induced by the flow relation in the occurrence net obtained ignoring the reversing\nevents, is the one driving the construction. One of the other two relations of an r\\textsc{pes}\\ is substantially\nignored (and we obtain from a \\textsc{rcn}\\ a causal r\\textsc{pes}\\ where the reverse causality relation just \nsays that an event can be reversed only after it has occurred) whereas the second (prevention)\nis tightly related to the causality relation: $b$ is caused by $a$ precisely when $b$ prevents \nof undoing of $a$.\nThe notion of reversible causal net we have proposed suggests this construction, so the problem \nof finding which kind of net would correspond to, for example, a cause-respecting or even\nan arbitrary r\\textsc{pes}\\ remains open and\ncertainly deserves to be investigated.\n\nIt is however interesting to observe that the construction in Definition \\ref{de:rpestorcn}\ngives a reversible causal net even when the r\\textsc{pes}\\ one started with is not a causal r\\textsc{pes}. \n Consider the r\\textsc{pes}\\ with two events $\\setenum{e_1, e_2}$ such that\n $e_1 < e_2$ and where the conflict and the prevention relations are empty.\n The only reversible event is $e_1$ and $e_1 \\prec \\underline{e}_1$.\n The set $\\{e_2\\}$ is a reachable configuration: we can remove $e_1$ from a reachable configuration \n$\\setenum{e_1, e_2}$ by performing the event $\\underline{e}_1$. \nThis is an example of out-of-causal order computation. \nGiven this r\\textsc{pes}, our construction produces the following \\textsc{rcn}, which does not have $\\{e_2\\}$ among its configurations.\n %\n \\begin{center}\n   \\input{counterex.tex}\n \\end{center}\n\n\\noindent\nThe constructions we have proposed are somehow the more adherent to what is usually done, based on the \ninterpretation that \\emph{causality} implies that the event causing some other event somehow\nproduces something that it is used by the latter. \nThis is not the only interpretation of what causality could mean. \nIn fact, causality is often confused with the observation that two causally related events appear\nordered in the same way in each possible execution, and when we talk about ordered execution, \nit should be stressed that this can be achieved in several ways, for instance using \\emph{inhibitor}\narcs. \nConsider the net $C$\n\\begin{center}\n\\input{inhiscausal.tex}\n\\end{center}\nthe event $e_2$ can be executed only after the event $e_1$ has been executed. However, $e_1$ \ndoes not produce a token (resource) that must be used by $e_2$. If we simply make\nthe event $e_1$ reversible but do nothing to prevent reversing of $e_1$ before $e_2$ is reversed,\nthen we would obtain the net $C'$. We could do better in $C''$ where we model the prevention using the\nso-called \\emph{read arcs} \\cite{MR:CN}. Hence, using the inhibitor or read arcs seem feasible way\nforwards to capture more precise the new relations of  r\\textsc{pes} es, including prevention. \nA similar approach has been already pursued in\n\\cite{CP:soap17} to model the so called modifiers that are able to change the causality pattern of\nan event.\nThis suggests that, for arbitrary r\\textsc{pes} es, we need to find relations different from the flow \nrelation to capture faithfully (forward and reverse) causal and prevention dependencies.\nThis will be the subject of future research.\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nEvent structures and nets are closely related. Since the seminal papers\nby Nielsen, Plotkin and Winskel \\cite{NPW:PNES} and Winskel \\cite{Win:ES}, the relationship among\nnets and event structures has been considered as a pivotal characteristic of\nconcurrent systems, as testified by numerous papers in the literature addressing\n event structures and nets. \nThe ingredients of an event structure are, beside a set of events, a number of relations that are\nused to express which events can be part of a configuration (the snapshot of a concurrent \nsystem), modelling a consistency predicate, and how events can be added to reach another \nconfiguration, modelling the dependencies among the (sets of) events.\nOn the net side, the ingredients boil down to constraints on how transitions may be executed,\nand usually have a structural flavour.\n\nSince the introduction of event structures there has been a flourish of\ninvestigations into the possible relations among events, giving rise to a number\nof different definitions of event structures.\nWe recall some of them, without the claim of completeness. \nFirst to mention are the classical \\emph{prime} event structures \n\\cite{Win:ES} where\nthe dependency between events, called \\emph{causality}, is given by a partial order and the consistency\nis determined by a \\emph{conflict} relation.\n\\emph{Flow} event structures \\cite{Bou:FESFN} drop the requirement that the dependency should\nbe a partial order, and \\emph{bundle} event structures \\cite{Langerak:1992:BES} are able to represent \nOR-causality by allowing each event to be caused by a member of a bundle of events.\n\\emph{Asymmetric} event structures \\cite{BCM:CNAED} introduce the notion of weak causality that can \nmodel asymmetric conflicts. \\emph{Inhibitor} event structures \\cite{BBCP:rivista} are able\nto faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. \nIn \\cite{BCP:LenNets} event structures where the causality relation may be circular are\ninvestigated, and in \\cite{AKPN:lmcs18} the notion of dynamic causality is considered.\nFinally, we mention the quite general approach presented in \\cite{GP:CSESPN}, where there is\na unique relation, akin to a \\emph{deduction relation}. \nTo each of the aforementioned event structures \na particular class of nets corresponds. To prime event structures we have \n\\emph{occurrence nets}, to flow event structures we have flow nets, to bundle event structures\nwe have \\emph{unravel nets} \\cite{CaPi:PN14}, to asymmetric and inhibitor event structures we\nhave \\emph{contextual nets} \\cite{BCM:CNAED,BBCP:rivista}, to event structures with circular\ncausality we have \\emph{lending nets} \\cite{BCP:LenNets}, to those with dynamic causality we have\n\\emph{inhibitor unravel nets} \\cite{CP:soap17} and finally to the ones presented in \n\\cite{GP:CSESPN}  \\emph{1-occurrence nets} are associated.\n\nRecently a new type of event structure tailored to model \\emph{reversible} computation has been\nintroduced and studied \\cite{PU:jlamp15,UPY:RES-NGC18}. In particular, in \\cite{PU:jlamp15},\n\\emph{reversible prime event structures} have been introduced. In this kind of event structure \ntwo relations are added: \nthe \\emph{reverse causality} relation and the \\emph{prevention} relation. The first \none is a standard dependency relation: in order to reverse an event some other events must be\npresent. The second relation, on the contrary, identifies those events whose presence \\emph{prevents}\nthe event being reversed. \nThis kind of event structure is able to model different\nflavours of reversible computation\nsuch as causal-consistent reversibility~\\cite{rccs,ccsk,rhotcs} and out-of-causal-order \nreversibility~\\cite{PhiUliYuen12,UK16}. \nCausally consistent reversibility relates reversibility with causality: an event can be undone provided that all of its effects have been undone. This allows the system to get back to a past state, which was possible to reach by just the normal (forward) computation.\nThis notion of reversibility is natural in reliable distributed systems since when an error occurs the system tries to go back to a past consistent state. Examples of how causal consistent reversibility is used to model reliable systems are  \ntransactions~\\cite{DanosK05,LaneseLMSS13} and rollback protocols~\\cite{VassorS18}.\nAlso, causally consistent reversibility can be used for program analysis and  debugging \\cite{GiachinoLM14,LanesePV19}. \n On the other hand, out-of-causal-order reversibility  does not preserve causes, and it is suitable to model biochemical reaction where, for example, a bond can be undone leading to a different state which was not present before.\n\nReversibility in Petri nets has been studied in \\cite{PhilippouP18,MMU:coordination19} with two different approaches. \nIn \\cite{PhilippouP18} reversibility for acyclic Petri net is solved by\nrelying on a new kind of tokens, called bonds, that\nkeep track of the execution history. Bonds are rich enough for allowing other approaches to\nreversibility such as out-of-causal order and causal consistent reversibility. In\n \\cite{MMU:coordination19}\na notion of \\emph{unfolding} of a P\/T (place\/transition) net, where all the transitions can be reversed, has been\nproposed. In particular,  by resorting to standard notions of the Petri net  theory \\cite{MMU:coordination19} provides a causally-consistent reversible semantics \nfor P\/T nets. This exploits the well-known unfolding of P\/T nets into occurrence nets \n\\cite{Win:ES}, and is done by adding for each transition its reversible counterpart. \n \n\nIn this paper we start\nour research quest towards studying what kind of nets can be\nassociated to reversible prime event structures. To this aim we introduce the notion of\na \\emph{reversible causal net} which is an occurrence net enriched with \ntransitions which operationally\n\\emph{reverse} the effects of executing some others\n\n\nWe associate to each reversing transition (event in the occurrence net dialect) a unique transition, \nwhich is in charge of producing\n the effects that the reversing transition\nhas to undo.\nTo execute a reversing event in a reversible causal net \nthe events caused by the event to be reversed (the one associated to the reversing one)\nhave to be reversed as well, and if this is not possible then the reversing event cannot be\nexecuted. This corresponds, in the reversible event structure, to the fact that such \nevents \\emph{prevent} the reversing event from happening. A reversible causal \nnet where the reversing events \nhave been removed is just an occurrence net.\nThis discussion suggests the easiest way of relating\nreversible causal nets and reversible prime event structure: the causal relation is\nthe one induced by the occurrence net, and the prevention relation is the one induced\nby the events that are in the future of the one to be reversed. The reversible causality relation\nis the basic one: in order to reverse an event the event itself must be present. \nWhat is obtained from a reversible causal net is a \\emph{causal} reversible prime event structure,\nwhich is a subclass of reversible prime event structures.  \n\nWhen we start from a causal reversible prime event structure, it is possible to obtain\na reversible causal net. The ingredients that are used are just the causal relation and\nthe set of reversible events. \nThus, if we start from a causal\nreversible prime event structure, the obtained reversible causal net  \nhas the same set of configurations.\nHence, the precise correspondence is between causal reversible prime event structures and \nreversible causal nets. \nThis relation is made clear also by turning causal reversible prime event structures and\nreversible causal nets into categories. Then the constructions associating reversible causal nets to \ncausal reversible prime event structures can be turned into functors and these functors form a coreflection.\nThis implies that the notion of reversible causal net is the appropriate one when dealing with causal\nprime event structures.\n\n\\paragraph{Structure of the paper.}\nSection \\ref{sec:preliminaries} reviews some preliminary notions for nets and\nevent structures, including reversible prime event structures. Section \\ref{sec:cn-and-pes} recalls\nthe well-known relationship between prime event structures and occurrence nets. The core of the\npaper is Section \\ref{sec:rcn-and-rpes} where we first introduce reversible causal nets and then\nwe show how to obtain a reversible causal net from an occurrence net. We then \nshow how to associate a causal reversible prime event structure to a reversible causal net, \nand vice versa. \nWe sum up our findings in Section \\ref{sec:category} where we introduce a notion of morphism for \nreversible causal nets and show that this gives a category which is related to the subcategory of causal\nreversible prime event structures.\nSection \\ref{sec:conc} concludes the paper\n\n\n\\section{Omitted Proofs}\\label{sec:proof}\n\\input{proofs}\n\\end{document}\n\n\\section{Occurrence nets and prime event structures}\\label{sec:cn-and-pes}\nWe now review the notion of occurrence nets~\\cite{NPW:PNES,Win:ES}. \nGiven a net $N = \\langle S, T, F, \\mathsf{m}\\rangle$,\nwe \nwrite $<_N$ for transitive closure of $F$, and $\\leq_N$ for the reflexive closure of $<_N$.\n We say $N$ is \\emph{acyclic} if  $\\leq_N$ is a partial order.\nFor occurrence nets, we adopt the usual convention and refer to places and transitions respectively \n as {\\em conditions} and {\\em events}, and correspondingly use $B$  and \n$E$ for the sets of conditions and events.\n We will often confuse conditions with places and \nevents with transitions.\n\\begin{definition}\n An \\emph{occurrence net} (\\textsc{on}) $C = \\langle B, E, F, \\mathsf{c}\\rangle$ is an acyclic, safe net \n satisfying the\n  following restrictions:\n  \\begin{itemize}\n    \\item $\\forall b\\in \\mathsf{c}$. $\\pre{b} = \\emptyset$,\n    %\n    \\item $\\forall b\\in B$. $\\exists b'\\in \\mathsf{c}$ such that $b' \\leq_C b$,\n    %\n    \\item $\\forall b\\in B$. $\\pre{b}$\n          is either empty or a singleton,\n   \n    \\item for all $e\\in E$ the set $\\hist{e} = \\setcomp{e' \\in E}{e'\\leq_C e}$ is finite,\n          and\n   \n    \\item  $\\#$ is an irreflexive and symmetric relation defined as follows:\n          \n           \\begin{itemize}\n             \\item $e\\ \\#_0\\ e'$ iff $e, e' \\in E$, $e\\neq e'$ and \n                   $\\pre{e}\\cap\\pre{e'}\\neq \\emptyset$,\n             %\n             \\item $x\\ \\#\\ x'$ iff $\\exists y, y'\\in E$ such that $y\\ \\#_0\\ y'$ and $y \\leq_C x$ and \n                   $y' \\leq_C x'$.\n             \\end{itemize}\n     \\end{itemize}\n\\end{definition}\nThe intuition behind occurrence nets is the following: each condition $b$ represents \nthe occurrence of a token,\nwhich is produced by the \\emph{unique} event in $\\pre{b}$, \nunless $b$ belongs to the initial marking,\nand it is used by only one transition (hence if $e, e'\\in\\post{b}$, then $e\\ \\#\\ e'$).\nOn an occurrence net $C$ it is natural to define a notion of \\emph{causality} among elements of the \nnet: we say that $x$ is \\emph{causally dependent} on $y$ iff $y \\leq_C x$.\n \nOccurrence nets are often the result of the \\emph{unfolding} of a (safe) net. \nIn this perspective an occurrence net is meant to describe precisely the non-sequential semantics\nof a net, and each reachable marking of the occurrence net corresponds to a reachable marking\nin the net to be unfolded. Here we focus purely on occurrence nets and not on the nets\nthey are unfoldings of.\n\n\\begin{definition}\\label{de:cn-conf}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be a \\textsc{on}\\ and \n $X\\subseteq E$ be a subset of events. Then $X$ is a \\emph{configuration} of\n $C$ whenever $\\CF{X}$ and $\\forall e\\in X$. $\\hist{e}\\subseteq X$.\n The set of configurations of the occurrence net $C$ is denoted by $\\Conf{C}{\\textsc{on}}$.\n\\end{definition}\n\nGiven an occurrence net $C = \\langle B, E, F, \\mathsf{c}\\rangle$ and a state\n$X \\in \\states{C}$, it is easy to see that it is \\emph{conflict free}, \\emph{i.e.}\n$\\forall e, e'\\in X$. $e\\neq e'\\ \\Rightarrow\\ \\neg (e\\ \\#\\ e')$, and \\emph{left closed},\n\\emph{i.e.} $\\forall e \\in X$. $\\setcomp{e'\\in E}{e'\\leq_C e}\\subseteq X$.\n\n\\begin{proposition}\\label{pr:states-are-conf}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net and $X\\in \\states{C}$. Then\n $X\\in \\Conf{C}{\\textsc{on}}$.\n\\end{proposition}\nWe now recall the connection among occurrence nets and prime event structures~\\cite{Win:ES}.\n\\begin{proposition}\\label{pr:on-to-pes}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net. Then\n $\\mathcal{P}(C) = (E, \\leq_C, \\#)$\n  is a \\textsc{pes}, and \n $\\Conf{C}{\\textsc{on}} = \\Conf{\\mathcal{P}(C)}{\\textsc{pes}}$.\n\\end{proposition}\n\n\\begin{example}\\label{ex:cn}\n\nFigure~\\ref{fig:occ-nets} illustrates some  (finite) occurrence nets. We can associate \\textsc{pes} es to them as follows.\nThe net $C_1$ has two concurrent events, which hence are neither causally ordered nor in conflict,\nconsequently both $<$ and $\\#$ are  empty. The two events $e_1$ and $e_2$ in  $C_2$ are in conflict,\nnamely $e_1\\ \\#\\ e_2$, while they are causally ordered in $C_3$, namely $e_1 < e_2$, and not in conflict.\nFinally, in $C_4$ we have $e_1< e_3$ and $e_2<e_4$ and $e_1\\ \\#\\ e_2$. Additionally, conflict inheritance \ngive us $e_1\\ \\#\\ e_4$, $e_2\\ \\#\\ e_3$ and $e_3\\ \\#\\ e_4$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\input{cn-example.tex}\n\\caption{Some occurrence nets}\n\\label{fig:occ-nets}\n\\end{center}\n\\end{figure}\n\n\\end{example}\n\nConversely, every \\textsc{pes}\\ can be associated with an occurrence net. \n\n\\begin{proposition}\\label{pr:pes-to-on}\n Let $P = (E, <, \\#)$ be a \\textsc{pes}\\ and let $\\bot\\not\\in E$ be a new symbol. \n Then $\\mathcal{E}(P) = \\langle B, E, F, \\mathsf{c}\\rangle$ \n\n defined as follows \n  \\begin{itemize}\n    \\item $B = \\setcomp{(a,A)}{a\\in E\\cup\\setenum{\\bot} \\land A \\subseteq E \\land \\CF{A} \\land \n    a \\neq \\bot\\ \\Rightarrow\\ \\forall e\\in A.\\ a < e}$,\n   \\item $F = \\setcomp{(b,e)}{b = (a,A)\\ \\land\\ e\\in A}\\cup \\setcomp{(e,b)}{b = (e,A)}$, and\n   \\item $\\mathsf{c} = \\setcomp{(\\bot,A)}{A \\subseteq E\\ \\land\\ \\CF{A}}$,                                               \n \\end{itemize}\n is an occurrence net, and $\\Conf{P}{\\textsc{pes}} = \\Conf{\\mathcal{E}(P)}{\\textsc{on}}$.\n\\end{proposition}\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n We denote with  $\\ensuremath{\\mathbb{N}}$ the set of natural numbers.\n %\n Let $A$ be a set, a {\\em multiset\\\/} of $A$ is a function $m : A\n \\rightarrow \\ensuremath{\\mathbb{N}}$.\n %\n The set of multisets of $A$ is denoted by $\\mu A$.  \n %\nWe assume the usual operations on multisets such as union $+$ and difference $-$.\n %\n \n We write $m \\subseteq m'$ if $m(a) \\leq m'(a)$ for all $a \\in A$.  \n %\n For $m\\in \\mu A$, we denote with $\\flt{m}$ the multiset defined as $\\flt{m}(a)\n = 1$ if $m(a) > 0$ and $\\flt{m}(a) = 0$ otherwise. \n %\n When a multiset $m$ of $A$ is a set, \n \\emph{i.e.} $m = \\flt{m}$, we write\n $a\\in m$ to denote that $m(a) \\neq 0$, and often confuse the\n multiset $m$ with the set $\\setcomp{a\\in A}{m(a) \\neq 0}$. \n %\n Furthermore we use the standard set operations like $\\cap$, $\\cup$ or\n $\\setminus$.\n \n Given a set $A$ and a relation $<\\ \\subseteq A\\times A$, we say that $<$ is \n an \\emph{irreflexive} partial order whenever\nit is irreflexive and transitive. \nWe shall write $\\leq$ for the reflexive closure of a  partial order $<$.\n \n\\subsection{Petri nets} \nWe review the notion of Petri net along with some auxiliary notions.\n\\begin{definition}\n   A \\emph{Petri net} is a 4-tuple \n   $N = \\langle S, T, F, \\mathsf{m}\\rangle$, where\n   $S$ is a set of {\\em places} and $T$ is a set of {\\em transitions} \n   (with $S \\cap T = \\emptyset$), \n  \n   {$F \\subseteq (S\\times T)\\cup (T\\times S)$}\n    is the \n    \\emph{flow} relation, and\n   $\\mathsf{m}\\in \\mu S$ is called the {\\em initial marking}.\n \\end{definition}\n Petri nets are depicted as usual. \n %\n Given a net $N = \\langle S, T, F, \\mathsf{m}\\rangle$ and  $x\\in S\\cup T$, we define the following\n multisets:  \n\n {$\\pre{x} = \\{y\\ |\\ (y,x)\\in F\\}$}\n  and \n\n {$\\post{x} = \\{y\\ |\\ (x,y)\\in F\\}$}.\n If $x\\in S$ then \n\n\n $\\pre{x} \\in \\mu T$ and  $\\post{x} \\in \\mu T$;\n analogously, \n if $x\\in T$ then $\\pre{x}\\in\\mu S$ and $\\post{x} \\in \\mu S$.\n %\n A multiset of transitions $A\\in \\mu T$, called \\emph{step}, \n is enabled at a marking $m\\in \\mu S$, denoted by $m\\trans{A}$,\n whenever $\\pre{A} \\subseteq m$, where $\\pre{A} = \\sum_{x\\in\\flt{A}}\\ A(x)\\cdot\\pre{x}$. \n %\n A step $A$ enabled at a marking $m$ can \\emph{fire} and its firing produces \n the marking $m' = m - \\pre{A} + \\post{A}$, where \n $\\post{A} = \\sum_{x\\in\\flt{A}}\\ A(x)\\cdot\\post{x}$.\n The firing of $A$ at a marking $m$ is denoted by $m\\trans{A}m'$.\n %\n We assume that each transition $t$ of a net $N$ is such that $\\pre{t}\\neq\\emptyset$,\n meaning that no transition may fire \\emph{spontaneously}.\n %\n Given a generic marking $m$ (not necessarily  the initial one), \n the (step) \\emph{firing sequence} ({shortened as} \\textsf{fs}) of  \n $N = \\langle S, T, F, \\mathsf{m}\\rangle$  starting at $m$ is\n defined as: \n   ($i$)  $m$ is a firing sequence (of length 0), and \n %\n  ($ii$) if $m\\trans{A_1}m_1$ $\\cdots$ $m_{n-1}\\trans{A_n}m_n$ is a firing sequence \n        and $m_n\\trans{A}m'$,  \n        then also $m\\trans{A_1}m_1$ $\\cdots$ $m_{n-1}\\trans{A_n}m_n\\trans{A}m'$\n        is a firing sequence.\n %\n Let us note that each step $A$ such that $|A| = n$ \n can be written as $A_1 + \\cdots + A_n$ where for each $1 \\leq i\\leq n$ it holds that \n $A_i = \\flt{A_i}$ and  $|A_i| = 1$, and $m\\trans{A}m'$ iff for each decomposition\n of $A$ in $A_1 + \\cdots + A_n$, we have that $m\\trans{A_1}m_1\\dots m_{n-1}\\trans{A_n}m_n = m'$.\n %\n When $A$ is a singleton, \\emph{i.e.}\n $A = \\setenum{t}$, we write \n $m\\trans{t}m'$.   \n\n The set of firing sequences of a net $N$ \n starting at a marking $m$ is denoted by $\\firseq{N}{m}$ and it is ranged over by $\\sigma$.\n Given a \\textsf{fs}\\ $\\sigma = m\\trans{A_1}\\sigma'\\trans{A_n}m_n$, we denote with\n $\\start{\\sigma}$  the marking $m$, with $\\lead{\\sigma}$ the marking $m_n$ \n and with $\\remains{\\sigma}$ the \n \\textsf{fs}\\ $\\sigma'\\trans{A_n}m_n$. \n %\n Given a net $N = \\langle S, T, F, \\mathsf{m}\\rangle$, a marking $m$ is \\emph{reachable} \n iff there exists a \n \\textsf{fs}\\ $\\sigma \\in \\firseq{N}{\\mathsf{m}}$ \n such that $\\lead{\\sigma}$ is $m$. \n %\n The set of reachable markings of $N$ is\n $\\reachMark{N} = \\bigcup_{\\sigma\\in\\firseq{N}{\\mathsf{m}}} \\lead{\\sigma}$.\n %\n Given a \\textsf{fs}\\ $\\sigma = m\\trans{A_1}m_1\\cdots m_{n-1}\\trans{A_n}m'$, \n we write  \n $X_{\\sigma} = \\sum_{i=1}^{n} A_i$\n for the multiset of transitions associated to  \\textsf{fs}. \nWe call $X_{\\sigma}$ a \\emph{state} of the net and write\n \\(\n   \\states{N} = \\setcomp{X_{\\sigma}\\in \\mu T}{\\sigma\\in\\firseq{N}{\\mathsf{m}}}\n \\)\n for the set of states of $N$. \n \n \n\n\\begin{definition}  \n A net $N = \\langle S, T, F, \\mathsf{m}\\rangle$ is said to be \\emph{safe} if\n each marking $m\\in \\reachMark{N}$ is such that $m = \\flt{m}$. \n\\end{definition}\n\nIn this paper we consider safe nets $N =  \\langle S, T, F, \\mathsf{m}\\rangle$ \nwhere each transition can be fired, \\emph{i.e.}\n$\\forall t\\in T\\ \\exists m\\in \\reachMark{N}.\\ m\\trans{t}$, and each place is marked in a \ncomputation, \\emph{i.e.} $\\forall s\\in S\\ \\exists m\\in \\reachMark{N}.\\ m(s) = 1$.\n\n\\subsection{Prime event structures}\n\nWe now recall the notion of prime event structure~\\cite{Win:ES}.\n\n\\begin{definition}\\label{de:pes-winskel}\n A \\emph{prime event structure ({\\textsc{pes}})} is a triple $P = (E, <, \\#)$, where \n \\begin{itemize}\n  \\item $E$ is a  countable set of \\emph{events},\n  \\item $<\\ \\subseteq E\\times E$ is\nan irreflexive partial order called the \\emph{causality relation}, such that\n        $\\forall e\\in E$. $\\setcomp{e'\\in E}{e' < e}$ is finite, and \n  \\item $\\#\\ \\subseteq E\\times E$ is a \\emph{conflict relation}, which is irreflexive,\n    symmetric and \\emph{hereditary} relation with respect to $<$: if  $e\\ \\#\\ e' < e''$, then\n    $e\\ \\#\\ e''$ for all $e,e',e'' \\in E$.\n \\end{itemize}       \n\\end{definition}\n\nGiven an event $e\\in E$,  $\\hist{e}$  denotes the set $\\setcomp{e'\\in E}{e'\\leq e}$. \n{A subset of events $X \\subseteq E$ is left-closed if $\\forall e\\in X. \n\\hist{e}\\subseteq X$.\n} \nGiven a\n subset $X\\subseteq E$ of events, we say that $X$ is \\emph{conflict free} iff \nfor all $e, e'\\in X$ it holds that \n$e\\neq e'\\ \\Rightarrow\\ \\neg(e\\ \\#\\ e')$, and we denote it with $\\CF{X}$. \nGiven $X\\subseteq E$ such that $\\CF{X}$ and $Y\\subseteq X$, then also $\\CF{Y}$.\n\nWhen adding reversibility to \\textsc{pes} es, conflict heredity may not hold.\nTherefore, we rely on a weaker form of \\textsc{pes}\\ by following  the approach in~\\cite{PU:jlamp15}.\n\\begin{definition}\\label{de:pre-pes}\n   A \\emph{pre-\\textsc{pes}} (p\\textsc{pes}) is a triple $P = (E, <, \\#)$, where\n   \\begin{itemize}\n    \\item $E$ is a set of \\emph{events},\n    \\item $\\#\\ \\subseteq E\\times E$ is an irreflexive and symmetric relation,\n    \\item $<\\ \\subseteq E\\times E$ is an irreflexive partial order such that\nfor every $e\\in E$. $\\setcomp{e'\\in E}{e' < e}$ is finite and\n                  conflict free, and\n    \\item $\\forall e, e'\\in E$. if $e < e'$ then not $e\\ \\#\\ e'$.\n   \\end{itemize}\n  \\end{definition}\n{A p\\textsc{pes}\\ is  a prime event structure in which conflict heredity does not hold, \nand since every \\textsc{pes}\\ is also a p\\textsc{pes}\\ the notions and results stated below \nfor p\\textsc{pes} es also apply to \\textsc{pes} es.\n}\n\n\\begin{definition}\\label{de:ppes-conf-enab}\n   Let $P = (E, <, \\#)$ be a p\\textsc{pes}\\ and\n  \n   $X\\subseteq E$ such that $\\CF{X}$.\n   For $A\\subseteq E$, we say that $A$ is \\emph{enabled}\n   at $X$ if\n   \\begin{itemize}\n    \\item $A\\cap X = \\emptyset$ and $\\CF{X\\cup A}$, and\n    \\item $\\forall e\\in A$. if $e' < e$ then $e'\\in X$.\n   \\end{itemize}\n   If $A$ is enabled at $X$, then $X \\stackrel{A}{\\longrightarrow} Y$ where $Y = X\\cup A$.\n  \\end{definition}\n\\begin{definition}\\label{de:rpes-forwconf}\n   Let $P = (E, <, \\#)$ be a p\\textsc{pes}\\ and\n  \n   {$X\\subseteq E$ s.t. $\\CF{X}$}.\n   $X$ is a \\emph{forwards reachable configuration} if there exists a sequence\n  \n   {$A_1,\\ldots,A_n$}, such that\n   \\[ X_i\\stackrel{A_i}{\\longrightarrow} X_{i+1} \\ \\mathit{ for all}\\ i, \\ \\mathit{and}\\\n       X_1 = \\emptyset \\ \\mathit{and}\\ X_{n+1}=X.\n       \\]\nWe write $\\Conf{P}{p\\textsc{pes}}$ for the set of all  (forwards reachable) configurations of $P$.\n  \\end{definition}\nWhen a p\\textsc{pes}\\ is a \\textsc{pes}\\ we shall write $\\Conf{P}{\\textsc{pes}}$ instead of $\\Conf{P}{p\\textsc{pes}}$, with \n$\\Conf{P}{\\textsc{pes}}=\\Conf{P}{p\\textsc{pes}}$ holding.\nFrom a p\\textsc{pes}\\ a \\textsc{pes}\\ can be obtained.\n\n\n\\begin{definition}\\label{de:hereditary-closure}\n Let $P = (E, <, \\#)$ be a p\\textsc{pes}. Then $\\mathsf{hc}(P) = (E, <, \\sharp)$ is the \n \\emph{hereditary closure} of $P$, where $\\sharp$ is\n derived by using the following rules \n \\[ \n\\begin{array}{ccccc}\n\\infer{e\\ \\sharp\\ e'}{e\\ \\#\\ e'} & \\qquad & \\infer{e\\ \\sharp\\ e''}{e\\ \\sharp\\ e'\\ \\ \\ \\ \\ e' < e''}\n & \\qquad  & \\infer{e\\ \\sharp\\ e'}{e'\\ \\sharp\\ e}\\\\\n\\end{array}\n\\]\n\\end{definition}\nThe following proposition relates p\\textsc{pes}\\ to \\textsc{pes}\\ \\cite{PU:jlamp15}.\n\\begin{proposition}\\label{pr:ppes-prop}\n Let $P = (E, <, \\#)$ be a p\\textsc{pes}, then\n \\begin{itemize}\n   \\item $\\mathsf{hc}(P) = (E, \\leq, \\sharp)$ is a \\textsc{pes},\n   \\item if $P$ is a \\textsc{pes}, then $\\mathsf{hc}(P) = P$, and\n   \\item $\\Conf{P}{p\\textsc{pes}} = \\Conf{\\mathsf{hc}(P)}{\\textsc{pes}}$.\n \\end{itemize}\n\\end{proposition}\n\n\\subsection{Reversible prime event structures} \nWe now focus on the notion of \n\\emph{reversible prime event structure}. The definitions and the results in this subsection \nare drawn from \\cite{PU:jlamp15}. In reversible event structures some \nevents are categorised as \\emph{reversible}. The added relations\nare among events and those representing the \\emph{actual} undoing of the reversible events.\nThe undoing of events is represented by \\emph{removing} them (from a configuration), and is achieved\nby \\emph{executing} the appropriate \\emph{reversing} events.\n\n\\begin{definition}\\label{de:rpes}\n A \\emph{reversible prime event structure} (r\\textsc{pes}) is the tuple \n $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ where \n $(E, <, \\#)$ is a p\\textsc{pes}, $U\\subseteq E$ are the\n \\emph{reversible\/undoable}\n\n  events\n (with reverse events being denoted by \n {$\\underline{U} = \\setcomp{\\underline{u}}{u\\inU}$ and disjoint from $E$, i.e., $\\un U \\cap E = \\emptyset$})\nand\n { \\begin{itemize}\n   \\item $\\triangleright\\ \\subseteq E\\times \\underline{U}$ is the \\emph{prevention} relation,\n   %\n   \\item $\\prec\\ \\subseteq E\\times \\underline{U}$ is the \\emph{reverse causality} relation and\n         it is such that $u\\prec \\underline{u}$ for each $u\\inU$ and \n         $\\setcomp{e\\in E}{e\\prec\\underline{u}}$ is finite and conflict-free for every $u\\inU$, \n   %\n   \\item if $e \\prec \\underline{u}$ then not $e \\triangleright \\underline{u}$,\n   %\n   \\item the \n         \\emph{sustained causation}  $\\ll$  is a transitive relation defined such that $e \\ll e'$ if $e < e'$ and if $e\\inU$, then \n         $e' \\triangleright \\underline{e}$, and \n   %\n   \\item $\\#$ is hereditary with respect to $\\ll$: if $e\\ \\#\\ e' \\ll e''$, then $e\\ \\#\\ e''$.\n \\end{itemize}\n}\n\\end{definition}\nThe ingredients of a r\\textsc{pes}\\ partly overlap with those of a \\textsc{pes}: there is a causality\nrelation ($<$) and a conflict one ($\\#$) and the two are related by the \\emph{sustained causation} relation $\\ll$. The new ingredients are the \n\\emph{prevention} relation and the \\emph{reverse causality} relation.\nThe prevention relation states that certain events should be absent when trying to reverse an event, e.g., \n{$e\\triangleright \\underline{u}$}\nstates that  {$e$} should be absent when reversing {$u$}.  The reverse causality relation\n{$e \\prec \\underline{u}$} says that \n{$\\underline{u}$} can be executed \nonly when $e$ is present.\n\n\n \n\\begin{example}\\label{ex:cr not causal}\nLet $\\mathsf{P} = (E,U,<,\\#,\\prec,\\mathrel{\\triangleright})$ where\n$E = U = \\{a,b,c\\}$, $a<b$ and $a \\prec \\un a$, $b \\prec \\un b$, $c \\prec \\un c$, $c \\prec \\un a$ \nwith $b \\mathrel{\\triangleright} \\un a$ and no conflict.\nThen $a \\ll b$ because $a < b$ and $b \\mathrel{\\triangleright} \\un a$.\n$\\mathsf{P}$ states that $b$ causally depends on $a$ and that $c$ is concurrent w.r.t. both $a$ and $b$. \nNote that  every event is reversible in $\\mathsf{P}$ because $U = E$.  As expected, the reverse causality relation $\\prec$\nis defined such that every reverse event requires the presence of the corresponding reversible event, i.e., $e\\prec \\un e$ for all $e\\in E$. \nAdditionally, it also requires $c \\prec \\un a$, i.e.,  $a$ can be reversed only when $c$ is present. The prevention \nrelation states that $a$ cannot be reversed when $b$ is present, i.e., $b \\mathrel{\\triangleright} \\un a$.\n\\end{example}\n\\begin{definition}\\label{de:rpes-conf-enab}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be an r\\textsc{pes}\\ and\n  $X\\subseteq E$ be a set of events such that $\\CF{X}$. For\n $A\\subseteq E$ and $B\\subseteq U$, we say that $A\\cup \\underline{B}$ is \\emph{enabled}\n at $X$ if\n \\begin{itemize}\n  \\item $A\\cap X = \\emptyset$, $B \\subseteq X$ and $\\CF{X\\cup A}$, \n  \\item $\\forall e\\in A, e'\\in E$. if $e' < e$ then $e'\\in X\\setminus B$,\n  \\item $\\forall e\\in B, e'\\in E$. if $e' \\prec \\underline{e}$ then \n        $e' \\in X\\setminus (B\\setminus\\setenum{e})$,\n  \\item $\\forall e\\in B, e'\\in E$. if $e' \\triangleright \\underline{e}$ then $e'\\not\\in X\\cup A$.      \n \\end{itemize}\n If $A\\cup \\underline{B}$ is enabled at $X$ then \n $X \\stackrel{A\\cup\\underline{B}}{\\longrightarrow} Y$ where $Y = (X\\setminus B)\\cup A$.\n\\end{definition}\n\\begin{example}\\label{ex:rpes-red}\nConsider  the r\\textsc{pes}\\ in Example~\\ref{ex:cr not causal}.\nWe have, e.g.,  $\\emptyset \\stackrel{\\{a,c\\}}{\\longrightarrow} \\{a,c\\} \n \\stackrel{\\{\\un a\\}}{\\longrightarrow} \\{c\\}$ and $\\emptyset \\stackrel{\\{a\\}}{\\longrightarrow} \\{a\\}\n \\stackrel{\\{b\\}}{\\longrightarrow} \\{a, b\\} \n \\stackrel{\\{c, \\un b\\}}{\\longrightarrow} \\{a,c\\} \\stackrel{\\{b\\}}{\\longrightarrow} \\{a,b,c\\}$. \n \\end{example}\nReachable configurations are subsets of events which can be reached from the empty set by performing\nevents or undoing previously performed events.\n\\begin{definition}\\label{de:rpes-conf}\nLet $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a r\\textsc{pes}\\ and\n let $X\\subseteq E$ be a subset of events such that $\\CF{X}$.\n We say that $X$ is a \\emph{(reachable) configuration} if there exist two sequences\n of\n sets $A_i$ and $B_i$, for $i=1,\\ldots,n$,  such that\n \\begin{itemize}\n  \\item $A_i\\subseteq E$ and $B_i\\subseteq U$ for all $i$, and\n  \\item $X_i\\stackrel{A_i\\cup\\underline{B_i}}{\\longrightarrow} X_{i+1}$ for all $i$ with \n   $X_1 = \\emptyset$ and $X_{n+1}=X$.\n \\end{itemize}\nThe set of configurations of $\\mathsf{P}$ is denoted by $\\Conf{\\mathsf{P}}{r\\textsc{pes}}$.\n\\end{definition}\n\\begin{example} \nThe  set of configurations of $\\mathsf{P}$ defined in Example~\\ref{ex:cr not causal} is \n$\\Conf{\\mathsf{P}}{r\\textsc{pes}} = \\{\\emptyset, \\{a\\}, \\{c\\}, \\{a,b\\}, \\{a,c\\}, \\{a,b,c\\}\\}$\nas illustrated by the sequences shown in Example~\\ref{ex:rpes-red}.\n \\end{example}\n\n\nAs discussed in Section~\\ref{sec:intro}, r\\textsc{pes} es accommodate different flavours of \nreversibility. Hereafter, we  focus on \\emph{causal} reversibility \\cite{Lanese14}, \nwhich is one of the most common models of \nreversibility in distributed systems, in which an event can reversed \nonly when all the events it has caused have already been  reversed.\n  \n  \\begin{definition}\\label{de:cr-and-causal-rpes}\n   Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be an r\\textsc{pes}. Then $\\mathsf{P}$ is \n   \\emph{cause-respecting} if for any $e, e'\\in E$, if $e < e'$ then $e \\ll e'$. \n   $\\mathsf{P}$ is \\emph{causal} if for any $e\\in E$ and $u\\inU$ the following holds:\n\t $e \\prec \\underline{u}$ iff  $e = u$, and  \n    $e \\triangleright \\underline{u}$ iff $u < e$.\n  \\end{definition}\n  %\n \n\\begin{example}\\label{ex:cr not causal2}\nThe r\\textsc{pes}\\ $\\mathsf{P}$ in Example~\\ref{ex:cr not causal} is a cause-respecting r\\textsc{pes}.\nHowever $\\mathsf{P}$ is not causal because of $c \\prec \\un a$, which means that $c$ has to be present \nfor $a$ to be reversed even if $c$ does not causally depend on $a$.  If we remove $c \\prec \\un a$ then we obtain a causal r\\textsc{pes}.\n\\end{example}\n  \n\n\\begin{example}\\label{out-of-causal-order}\nAn example of out-of-causal order reversibility can be obtained from the definition of \nthe r\\textsc{pes}\\ $\\mathsf{P}$ in \nExample~\\ref{ex:cr not causal} by replacing $b \\mathrel{\\triangleright} \\un a$ by \n$a \\mathrel{\\triangleright} \\un b$. Then, we have $\\emptyset \\stackrel{\\{a\\}}{\\longrightarrow} \\{a\\}\n\\stackrel{\\{b,c\\}}{\\longrightarrow} \\{a,b,c\\}\\stackrel{\\{\\un a\\}}{\\longrightarrow} \\{b,c\\}$.\n Note that $a$ can be reversed even in the presence of the event $b$, which causally \ndepends on $a$.\n\\end{example}\n\nCause-respecting and causal r\\textsc{pes} es enjoy the following useful properties \\cite{PU:jlamp15}.\n \n\\begin{proposition}\\label{pr:rpes-causal-cause-respecting}\nLet $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a r\\textsc{pes}.\nLet $X$ be a left-closed and conflict-free subset of events in $E$ and\nlet $A, B\\subseteq U$. Then\n     \\begin{itemize}\n\\item if $\\mathsf{P}$ is cause-respecting and $X \\stackrel{A\\cup \\underline{B}}{\\longrightarrow} X'$, \n      then $X'$ is also left-closed,\n\\item if $\\mathsf{P}$ is cause-respecting and\n            $X \\stackrel{\\underline{B}}{\\longrightarrow} X'$, then\n            $X' \\stackrel{B}{\\longrightarrow} X$,\n\\item if $\\mathsf{P}$ is causal and\n            $X \\stackrel{A\\cup \\underline{B}}{\\longrightarrow} X'$, then\n            $X' \\stackrel{B\\cup\\underline{A}}{\\longrightarrow} X$.\n     \\end{itemize}\n\\end{proposition}\n\n\\begin{example}\\label{not-left-closed}\nThe above properties do not hold when a r\\textsc{pes}\\ is not cause-respecting \/ causal. Consider the \nr\\textsc{pes}\\ in Example~\\ref{out-of-causal-order}. We have that $\\{a,b,c\\}\\stackrel{\\{\\un a\\}}{\\longrightarrow} \\{b,c\\}$ but \n$\\{b,c\\}$ is not left-closed.\n\\end{example}\n\n\nA particular r\\^ole will be played by the configurations that can be reached without\nexecuting any reversible events.\n\\begin{definition}\\label{de:rpes-forwconf}\n Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a r\\textsc{pes}\\ and\n  $X\\in\\Conf{\\mathsf{P}}{r\\textsc{pes}}$ be a configuration.\n $X$ is \\emph{forwards reachable} if there exists a sequence\n of sets $A_i\\subseteq E$, for $i=1,\\ldots,n$, such that \n {$\n X_i\\stackrel{A_i}{\\longrightarrow} X_{i+1}$ for all  $i$, with  \n     $X_1 = \\emptyset$ and $X_{n+1}=X$.\n}\n\\end{definition}\nAlthough $\\{b,c\\}$ in Example~\\ref{not-left-closed} is a reachable configuration it is \nnot forwards reachable. However, the configurations of a cause-respecting r\\textsc{pes}\\ are forwards reachable\n\\cite{PU:jlamp15}.\n\\begin{proposition}\n     Let $\\mathsf{P} = (E, U, <, \\#, \\prec, \\triangleright)$ be a cause-respecting r\\textsc{pes}, and\n     let $X$ be configuration of $\\mathsf{P}$. Then $X$ is forwards reachable.\n\\end{proposition}\n\n\n\n\\section{Achieving non causal consistent reversibility}\nIn the previous section we have shown how reversible causal nets and reversible prime event\nstructures are related, and we have seen that ....\n\nto do:\n\\begin{enumerate}\n \\item define a notion of net where causality is present but each event can be undone easily,\n       using inhibitor arcs\n \\item how to enforce $\\triangleright$ to be cause-respecting also on these net? \n \\item do we have always have \\emph{single} events? \n\\end{enumerate}\n\\section{Reversible causal nets and reversible prime event structures}\\label{sec:rcn-and-rpes}\nWe now introduce\nthe notion of \\emph{reversible causal nets}.\nA similar notion has been proposed in~\\cite{MMU:coordination19} for adding\n causally-consistent reversibility to Petri nets by making  reversible every event in\nthe unfolding of the net. \nIn this work we deal with a generalised version of \nreversible causal nets in which\n transitions may be irreversible, i.e.,  \nwe do not require every transition of a net to be undoable.\n\nThe intuition behind reversible causal nets is the following: we add special\ntransitions (events in the classical occurrence net terminology) to an occurrence net which, \nwhen executed, \\emph{undo} \nthe execution of other (standard) transitions. When we remove these special transitions from a\nreversible causal net we obtain a standard occurrence net.\n\n\n\\begin{definition}\\label{def:rcn}\n A \\emph{reversible causal net} (\\textsc{rcn})  is a tuple $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ \nwhere $\\langle B, E, F, \\mathsf{c}\\rangle$ is a safe net such that \n \\begin{itemize}\n   \\item $U\\subseteq E$ and $\\forall u\\in U$. $\\exists!\\ e\\in E\\setminusU$ such that\n         $\\pre{u} = \\post{e}$ and $\\post{u} = \\pre{e}$, \n   \\item $\\forall e, e'\\in E$. $\\pre{e} = \\pre{e'}\\ \\land\\ \\post{e} = \\post{e'}\\ \n         \\Rightarrow\\ e = e'$, \n        \n  \n   \\item $\\bigcup_{e\\in E} (\\pre{e}\\cup\\post{e}) = B$, and        \n   \\item $C_{E\\setminus U} = \\langle B, E\\setminus U, F', \\mathsf{c}\\rangle$ is an occurrence net, \n         where $F'$ is the restriction of $F$ to the transitions in $E\\setminus U$.\n \\end{itemize}        \n\\end{definition}\nThe events in $U$ are the reversing ones and\nwe often say that a reversible causal net $R$ is reversible \\emph{with respect to} $U$. \nWe  write $\\overline{E}$ \n for the set of events $E\\setminus U$ and\n$C_{\\overline{E}}$ instead of $C_{E\\setminus U}$.\nThe first condition in Definition~\\ref{def:rcn} implies that \n each reversing event $u\\inU$ is associated with  a unique event $e$ that \ncauses the effects that  $u$ is intended to \\emph{undo}; hence $e$ here is a\n\\emph{reversible} event. Moreover, \nthe second condition ensures that there is an injective\nmapping $h : U \\to E$ that associates  each event  ${u\\inU}$ with a different \nevent $e\\in E$ such that $\\pre{e} = \\post{u}$ and $\\post{e} = \\pre{u}$, in other words, each reversible event has exactly one reversing event.\nThe third requirement guarantees that all conditions (places) of the net appear at least in the pre or the postset of some event  (transitions), i.e., \nthere are no isolated conditions. \nThe last condition ensures that the net obtained by deleting all reversing events is an\noccurrence net.\n\n\\begin{example}\\label{ex:rcn}\n We present some reversible causal nets in Figure~\\ref{fig:rcn}. The reversing events are drawn in red, and\n their names are underlined.\n  The events $e_1$ and $e_2$ in $R_1$ are both reversible, while \n  $e_1$ is the only  reversible event in $R_2$. In $R_3$ the events $e_1$, $e_3$ and $e_4$ are the reversible ones.\n\n \\begin{figure}[t]\n \\begin{center}\n  \\input{rcn-example.tex}\n \\end{center}\n \\caption{Some reversible causal nets}\n \\label{fig:rcn}\n \\end{figure}\n\\end{example}\n\nWe prove that the set of reachable markings of a reversible causal net is not influenced \nby performing reversing events.\n\\begin{proposition}\\label{pr:reachable-markings-of-rcn}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}.\n Then $\\reachMark{R} = \\reachMark{C_{\\overline{E}}}$.\n\\end{proposition}\n\nA consequence of the above proposition is the following corollary, which \nestablishes\nthat each marking can be reached by using just \\emph{forward events}.\n\\begin{corollary}\\label{cor:a}\n Let $C = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}\\\n and $\\sigma$ be a \\textsf{fs}. Then, there exists a \\textsf{fs}\\ $\\sigma'$ such that\n $X_{\\sigma'}\\subseteq \\overline{E}$ and $\\lead{\\sigma} = \\lead{\\sigma'}$.\n\n\\end{corollary}\n\n\\begin{definition}\\label{de:conf-rcn}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}, and\n  $X \\subseteq \\overline{E}$ be a subset of forward events. \n Then,  $X$ is a configuration of $R$ iff $X$ is a configuration of $C_{\\overline{E}}$.\n The set of configurations of $R$ is as usually denoted with $\\Conf{R}{\\textsc{rcn}}$.\n\\end{definition}\nA configuration of a reversible causal net $R$ with respect to $U$ is a subset of  \n$E\\setminus U$;\nconsequently, the reversing events (i.e., the ones in $U$) that may have been executed to reach a particular \nmarking are not considered as part of the configuration.\nObserve that, differently from \noccurrence net, $\\states{R} \\neq \\Conf{R}{\\textsc{rcn}}$ because the former may contain also reversing events.\nHowever, as a consequence of Corollary \\ref{cor:a}, there is no loss of information.\n\nWe show how to construct a reversible causal net from an occurrence net, once we have\nidentified the events to be \\emph{reversed}.\n\n\\begin{definition}\\label{de:constructing-a-reversible-causal-net}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net and $U \\subseteq E$ be a set \n of reversible events. Define $\\mathsf{R}(C) = \\langle B, \\hat{E}, U\\times\\setenum{r}, \\hat{F}, \\mathsf{c}\\rangle$ \n be the net where $\\hat{E}$ and $\\hat{F}$ are defined as follows:\n  \\begin{itemize}\n   \\item $\\hat{E} = E\\times\\setenum{\\mathtt{f}} \\ \\cup\\  U\\times\\setenum{\\mathtt{r}}$, and\n   %\n   \\item \n$\\hat{F} = \n\t\t\\setcomp{(b, (e,\\mathtt{f}))}{(b,e)\\in F} \\quad\\cup\\quad \\setcomp{ ((e,\\mathtt{f}),b)}{(e,b)\\in F}\\quad\\cup$ \\phantom{asdfasdfas} \\phantom{asdfa}\n\t\t$\\setcomp{(b, (e,\\mathtt{r}))}{(e,b)\\in F}\\quad\\cup\\quad \\setcomp{ ((e,\\mathtt{r}),b)}{(b,e)\\in F}$\n  \\end{itemize}\n  The mapping $h : U\\times\\setenum{\\mathtt{r}} \\to E\\times\\setenum{\\mathtt{f}}$ is defined as\n  $h(e,\\mathtt{r}) = (e,\\mathtt{f})$. \n\\end{definition}\nThe construction above simply adds as many events (transitions) as those to\nbe reversed. The preset of each added event is the postset of the\ncorresponding event to be reversed, and its postset is defined as the preset of the event to \nbe reversed.\nThe events in $U\\times\\setenum{\\mathtt{r}}$ are the reversing events.\n\n\\begin{proposition}\\label{prop:rev_net}\n Let $C = \\langle B, E, F, \\mathsf{c}\\rangle$ be an occurrence net, $U\\subseteq E$ be the subset of\n reversible events, and \n $\\mathsf{R}(C) = \\langle B, \\hat{E}, U\\times\\setenum{\\mathtt{r}}, \\hat{F}, \\mathsf{c}\\rangle$ be the net \n  in Definition \\ref{de:constructing-a-reversible-causal-net}. Then, $\\mathsf{R}(C)$ is a reversible causal net with respect to\n $U\\times\\setenum{\\mathtt{r}}$.\n\\end{proposition}\n\\begin{example}\\label{ex:reversing}\n Consider the occurrence net $C_1$ in Figure~\\ref{fig:occ-nets}, and assume that both events are reversible. The \n net $R_1$ in Figure~\\ref{fig:rcn} is $\\mathsf{R}(C_1)$ (after renaming events with\n the convention that $(e,\\mathtt{f})$ is named as $e$ and $(e,\\mathtt{r})$ as $\\underline{e}$). The \\textsc{rcn}\\ $R_3$ in Figure~\\ref{fig:rcn} is \n $\\mathsf{R}(C_4)$, with $C_4$ in Figure~\\ref{fig:occ-nets} and the set of reversible events  $U = \\{e_1,e_2,e_4\\}$.\n\\end{example}\n\n\\subsection{From \\textsc{rcn}\\ to r\\textsc{pes}}\nAs it is usually done for causal nets,\nwe now associate each reversible causal net with a reversible prime event structure.\nGiven an \\textsc{rcn}\\ $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$, we denote the\nset of events $\\setcomp{e'}{e  <_R e'}$ by $\\future{e}$. Observe that this set is not necessarily conflict-free.\n\n\\begin{proposition}\\label{pr:rce-to-rpes}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net with respect to $U$, \n then $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ is its associated r\\textsc{pes}, where\n \\begin{itemize}\n  \\item $E' = \\overline{E}$ and $U'= h(U)$,\n  \\item $<$ is\n       \n         $<_{C_{\\overline{E}}}$.\n  \\item $\\#$ is the conflict relation defined on the occurrence net $C_{\\overline{E}}$,\n  \\item $e\\ \\triangleright\\ \\underline{e'}$ whenever $e\\in \\future{e'}$, \n  \\item $e\\ \\prec\\ \\underline{e'}$ whenever $e = e'$, and\n  \\item $\\ll = <$.         \n \n \\end{itemize}\n\\end{proposition}\n\n\n\\begin{example}\n Consider the reversible causal net $R_3$ in Figure~\\ref{fig:rcn}. The associated r\\textsc{pes}\\ has\n the events $\\setenum{e_1,e_2,e_3,e_4}$ and the reversible events $\\setenum{e_1,e_3,e_4}$. \n The causality relation of the associated p\\textsc{pes}\\ \n is $e_1 < e_3$, $e_2 < e_4$, the \n conflict relation is \\emph{generated} by $e_1 \\# e_2$, and it is inherited along $\\ll$, which \n coincides with $<$. The reverse causality stipulates that $e_1 \\prec \\underline{e}_1$, \n $e_3 \\prec \\underline{e}_3$ and $e_4 \\prec \\underline{e}_4$ and finally\n $e_3 \\triangleright \\underline{e}_1$, as to be allowed to undo $e_1$ it is necessary to\n undo $e_3$ first.\n\\end{example}\n\nThe following result states that the r\\textsc{pes}\\ associated to a reversible causal net is \ncausal, hence cause-respecting.\n\\begin{proposition}\\label{prp:rpes_respect}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net with respect to $U$ and \n $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ be the associated r\\textsc{pes}. \n Then $\\mathcal{C}_{r}(R)$ is a\n causal r\\textsc{pes}.\n\\end{proposition}\n\nWe show that each configuration of a \\textsc{rcn}\\ is a configuration of the corresponding\nr\\textsc{pes},  and vice versa.\n\\begin{theorem}\\label{th:rnctorpes-conf-correspond}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net with respect to $U$ and \n $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ be the associated r\\textsc{pes}. \n Then $X \\subseteq E'$ is a configuration of $R$ iff $X$ is a configuration of $\\mathcal{C}_{r}(R)$.\n\\end{theorem}\n\n\nWe stress that a reversing event in a reversible causal net is enabled at a marking when\nthe conditions in the postset of the event to be reversed are marked. \nThis may happen only\nwhen all the events that causally depend on the event to be reversed have either been\nexecuted and reversed or not been executed at all.\nThus every \\textsc{rcn}\\ enjoys \\emph{causally consistent} reversibility~\\cite{rccs,rhotcs}, \nand consequently cannot implement the\nso called \\textit{out-of-causal order} reversibility~\\cite{PhiUliYuen12,UK16}.\nContrastingly,  r\\textsc{pes} es are able to model \\textit{out-of-causal order} reversibility (as illustrated in  Example~\\ref{out-of-causal-order}).\n\n\n\n\nThe proposition below establishes a  correspondence between the steps in a reversible causal net\nand the sequences of reachable configurations of the r\\textsc{pes}\\  associated to that net. \nProposition~\\ref{prop:mixed_step} below formalises what is called \\emph{mixed-reverse} transitions in~\\cite{EKM:petri_net}. \nWe now introduce some auxiliary notation. Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a \\textsc{rcn}, and $X\\subseteq E$ be a \nconfiguration of $R$, we write  $\\mathsf{mark}(X)$ to denote the marking reached \nafter executing the events\nin $X$; this marking can be expressed as $(\\mathsf{c}\\cup\\post{X})\\setminus \\pre{X}$.\n\n\\begin{proposition}\\label{prop:mixed_step}\n Let $R = \\langle B, E, U, F, \\mathsf{c}\\rangle$ be a reversible causal net and  \n $\\mathcal{C}_{r}(R) = (E', U', <, \\#, \\prec, \\triangleright)$ be its associated r\\textsc{pes}. \n Let $X\\in\\Conf{R}{\\textsc{rcn}}$ and  $A\\subseteq E$ be a set of events such that \n $\\mathsf{mark}(X)\\trans{A}$. Then $\\hat{A}\\cup \\underline{B}$ is enabled at $X$ in\n $\\mathcal{C}_{r}(R)$, where $\\hat{A} = \\setcomp{e\\in A}{e\\not\\in U}$ and\n $\\underline{B} = \\setcomp{e\\in A}{e\\in U}$.\n\\end{proposition}\n\n\n\\endinput \n\\michelenote{to be explained better that with \\textsc{rcn}\\ the only possible associated event structure\nis a cause-respecting and causal r\\textsc{pes}, as we have to take into account that a reverse event to be\nexecuted in the net needs that its preset is marked...}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMultiple photoionization of atoms and molecules belongs to the most sensitive probes of electron correlations \\cite{SPDI-Review}. This is distinctly exemplified by the process of single-photon double ionization (SPDI) which would not exist in the absence of electron correlations. Since photoabsorption is induced by a one-body operator, the ejection of the second electron relies exclusively on the interaction among the electrons in the target.\\footnote{We note that SPDI may also proceed through a shake-off mechanism \\cite{SPDI-Review} which relies on the sudden termination of electron-electron interaction present in the initial bound state.} SPDI has most comprehensively been studied in helium, which represents the prototype atom for this process.\n\nApart from the direct channel of SPDI, double electron emission can also occur after resonant photoexcitation of an autoionizing state which lies in the double continuum \\cite{resonant-SPDI}. This channel requires a more complex atomic structure than in helium. It has been studied thoroughly in alkaline-earth metal atoms (such as Be, Mg, Ca) which exhibit a heliumlike $ns^2$ electron configuration in the outer shell \\cite{SPDI-alkali, SPDI-Mg-I, SPDI-Mg-II, SPDI-Ca-I}. Since the latter is well separated from the inner shells both spatially and energetically, these atoms may be considered as quasi-two-electron systems. Resonant SPDI has been measured, for example, around the $3p\\to 3d$ resonance \nin Ca \\cite{SPDI-Ca-exp} and the $2p\\to 3d$ resonance in Mg \\cite{SPDI-Mg-exp}.\n\nAutoionization processes can also occur when an atom is not isolated in space but in close vicinity to another atom, so that the electronic structures at both centers are coupled by long-range interactions. \nFor example, two-center electron correlations can drive radiationless decay of an inner-valence vacancy in one of the atoms, in case when a single-center Auger decay of this vacancy is energetically forbidden. In such a situation, the vacancy may decay by transferring excitation energy to the neighboring atom. This process of interatomic Coulombic decay (ICD) \\cite{ICD, ICDres, ICDrev} can be much faster than a single-center radiative decay. It has been observed in a  variety of systems, comprising noble gas dimers \\cite{dimers} and clusters \\cite{clusters}. The closely related process of two-center photoionization (2CPI) was also studied, both theoretically \\cite{2CPI, Perina} and experimentally in He-Ne dimers \\cite{2CPIexp} and Ne-Ar clusters \\cite{Hergenhahn}. A similar energy transfer process was identified in the photoionization of helium droplets doped with rare gas atoms \\cite{droplets}.\n\nInteratomic electron correlations may also lead to double ionization after single-photon absorption. Double ionization of helium dimers was observed after irratiation with $\\approx 64$\\,eV synchrotron photons \\cite{SPDI-He2}. The process was shown to rely on a knock-off reaction: the first electron, which is photoejected from one of the helium atoms, kicks out the second electron from the other helium atom in an ($e$, $2e$) collision. Double ionization of magnesium in Mg-He clusters was demonstrated to be largely enhanced due to electron-transfer-mediated decay for photon energies above the helium ionization threshold \\cite{SPDI-ETMD-theo, SPDI-ETMD-exp}. Also here, the photoabsorption first produces a He$^+$ ion which, subsequently, is neutralized via electron transfer from a Mg atom. Simultaneously, a second electron is ejected from Mg to keep the energy balance. Finally, double ICD has been predicted to occur in endohedral fullerenes \\cite{DICD-theo}. For example, an excited Mg$^+$ ($2p^53s^2$) ion embedded in C$_{60}$ can decay into its ground state, forming a Mg$^+$($2p^63s)$@C$_{60}^{2+}$ complex. A related experiment has been conducted very recently on alkali dimers such as K-Rb attached to helium droplets \\cite{DICD-exp}. Excitation energy from the helium 1$s$2$s$ state was transfered to the dimer, leading to double ionization with subsequent dissociation into K$^+$ and Rb$^+$.\n\n\\begin{figure}[t]  \n\\vspace{-0.25cm}\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{2C_SPDI.eps}\n\\end{center}\n\\vspace{-0.5cm} \n\\caption{Scheme of two-center single-photon double ionization. First, atom $B$ is resonantly photoexcited. Afterwards, upon radiationless energy transfer to atom $A$ via a two-center Auger decay, the latter is doubly ionized. The process involves three active electrons and relies on both interatomic and intraatomic electron correlations, as indicated by the dashed arrows.}\n\\label{figure1}\n\\end{figure}\n\nIn the present paper, we study another interatomic mechanism of SPDI, which may proceed in two-center heteroatomic systems. The process leads to double ionization of an atom $A$ in the presence of a neighboring atom $B$. It is schematically illustrated in Fig.~\\ref{figure1}. The energy of the incident photon is assumed to be resonant with a dipole-allowed bound-bound transition in atom $B$ and to exceed the double ionization threshold of atom $A$. In a first step, atom $B$ is resonantly excited by photoabsorption, creating an autoionizing resonance state in the double continuum of the two-atomic system. Afterwards, the excitation energy is transfered via interatomic electron correlations to atom $A$ where it leads to double ionization due to intraatomic correlations between the two active electrons at center $A$.\\footnote{In the context of ICD, the second step in two-center SPDI has been termed participator double resonant interatomic Coulombic decay (pDRICD); see Fig.~2\\,a) in Ref.~\\cite{ICDres}.} We show that the resonant two-center SPDI can dominate over the direct SPDI of atom $A$ by several orders of magnitude. It can even be substantially stronger than the direct single ionization of atom $A$. \n\nOur paper is organized as follows. In Sec.~II we present our theoretical considerations of two-center SPDI. After formulating the general framework, an analytical expression for the double-to-single ionization ratio will be obtained. In Sec.~IV we illustrate our findings by some numerical examples and discuss their physical implications.  Concluding remarks are given in Sec.~IV. Atomic units (a.u.) will be used throughout unless otherwise stated.\n\n\n\\section{Theory of two-center single-photon double ionization}\n\nLet us consider a system consisting of two atoms, $A$ and $B$, separated \nby a sufficiently large distance $R$ such that their individuality \nis basically preserved. The atoms, which are initially in their ground states,  \nare exposed to a resonant electromagnetic field. The latter \nwill be treated as a classical electromagnetic wave of linear polarization, \nwhose electric field component reads \n\\begin{eqnarray}\n{\\bf F}({\\bf r},t)= {\\bf F}_0 \\cos\\left(\\omega t - {\\bf k} \\cdot {\\bf r}\\right).\n\\label{field}\n\\end{eqnarray}\nHere $\\omega = c k $ and ${\\bf k}$ \nare the angular frequency and wave vector,  \nand ${\\bf F}_0=F_0\\,{\\bf e}_z$ denotes the field strength vector\nwhich is chosen to define the $z$ direction.\n\nAssuming the atoms to be at rest,  \nwe take the position of the nucleus of atom $A$  \nas the origin and denote the coordinates \nof the nucleus of atom $B$, the two (active) electrons of atom $A$ \nand that of atom $B$ by ${\\bf R}$, ${\\bf r}_j$ ($j\\in\\{1,2\\}$) \nand ${\\bf r}_3 ={\\bf R} + \\boldsymbol{\\xi}$, \nrespectively, where $\\boldsymbol{\\xi}$ \nis the position of the electron of atom $B$ \nwith respect to its nucleus. \nLet atom $B$ have an excited state $\\chi_e$\nreachable from the ground state $\\chi_g$ by a dipole-allowed transition. \n\nThe total Hamiltonian describing the\ntwo atoms in the external electromagnetic field reads \n\\begin{eqnarray} \nH =  \\hat{H}_0 + \\hat{V}_{AB} + \\hat{W},  \n\\label{hamiltonian}  \n\\end{eqnarray} \nwhere $ \\hat{H}_0 $ is the sum of the Hamiltonians \nfor the noninteracting atoms $A$ and $B$,  \n$\\hat{V}_{AB}$ the interaction between the atoms \nand $\\hat{W} = \\hat{W}_A + \\hat{W}_B$ the interaction \nof the atoms with the electromagnetic field.\nWithin the dipole approximation and length gauge, the interaction $\\hat{W}$ reads   \n\\begin{eqnarray} \n\\hat{W} = \\sum_{j=1,2,3} {\\bf F}({\\bf r}_j={\\bf 0},t) \\cdot {\\bf r}_j\\ .\n\\label{W} \n\\end{eqnarray} \nThe two terms with $j\\in\\{1,2\\}$ compose the interaction $\\hat{W}_A$ with the electrons in atom $A$,\nwhereas the term with $j=3$ the interaction $\\hat{W}_B$ with the electron in atom $B$.\nFor electrons undergoing electric dipole transitions, the interatomic \ninteraction reads \n\\begin{eqnarray} \n\\hat{V}_{AB} &=& \\sum_{j=1,2}\\left( \\frac{{\\bf r}_j\\cdot \\boldsymbol{\\xi}}{R^3} \n- \\frac{ 3 ({\\bf r}_j\\cdot{\\bf R})(\\boldsymbol{\\xi}\\cdot{\\bf R})}{R^5} \\right)\\ .\n\\label{V_AB} \n\\end{eqnarray} \nIt is assumed that $\\omega_{ge} R \/c  \\ll 1$, where $\\omega_{ge}$ \nis the atomic transition frequency and $c$ the speed of light, such that \nretardation effects can be neglected\n\\footnote{Retardation effects in interatomic energy-transfer processes have \nrecently been studied in Refs.~\\cite{Buhmann,Salam}; see also Ref.~\\cite{2CPI}.}\n\nIn the process of two-center SPDI one has essentially three different basic two-electron configurations, which are schematically illustrated in Fig.~\\ref{figure1}:\n(I) $\\Psi_{g,g} = \\Phi_{g}({\\bf r}_1, {\\bf r}_2) \\chi_g(\\boldsymbol{\\xi})$ with total energy $E_{g,g} = \\varepsilon_0 + \\epsilon_1$, where both atoms are in the corresponding ground states $\\Phi_g$ and $\\chi_g$;\n(II) $\\Psi_{g,e} = \\Phi_{g}({\\bf r}_1, {\\bf r}_2) \\chi_e(\\boldsymbol{\\xi})$ with total energy $E_{g,e} = \\varepsilon_0 + \\epsilon_1$, in which atom $A$ is in the ground state while atom $B$ is in the excited state $\\chi_e$;\n(III) $\\Psi_{{\\bf p}_1,{\\bf p}_2,g} = \\Phi_{{\\bf p}_1,{\\bf p}_2}({\\bf r}_1,{\\bf r}_2) \\chi_g(\\boldsymbol{\\xi})$ with total energy $E_{{\\bf p}_1, {\\bf p}_2, g} = \\varepsilon_{p_1} + \\varepsilon_{p_2} + \\epsilon_0$, where both electrons of atom $A$ have been emitted into the continuum with asymptotic momenta ${\\bf p}_1$ and ${\\bf p}_2$, while the electron of atom $B$ has returned to the ground state. \n\nWithin the second order of time-dependent perturbation theory, the probability amplitude for two-center SPDI can be written as\n\\begin{eqnarray}\nS^{(2)}_{{\\bf p}_1,{\\bf p}_2}\\! &=&\\! -\\int_{-\\infty}^{\\infty} dt\\, \\langle \\Psi_{{\\bf p}_1,{\\bf p}_2,g}|\\hat{V}_{AB}| \\Psi_{g,e} \\rangle\\, e^{-i(E_{g,e}-E_{{\\bf p}_1,{\\bf p}_2,g})t}\\nonumber\\\\\n& & \\times \\int_{-\\infty}^{t} dt'\\, \\langle \\Psi_{g,e}|\\hat{W}_{B}| \\Psi_{g,g} \\rangle\\, e^{-i(E_{g,g}-E_{g,e})t'}\n\\label{S1}\n\\end{eqnarray}\n\nBy performing the inner time integral, we obtain\n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& -i\\int_{-\\infty}^{\\infty} dt\\, \\langle \\Psi_{{\\bf p}_1,{\\bf p}_2,g}|\\hat{V}_{AB}| \\Psi_{g,e} \\rangle\\,\\frac{F_0}{2} \\nonumber\\\\\n& & \\times\\,\\frac{\\langle \\chi_e|\\xi_z| \\chi_g \\rangle}{\\epsilon_0+\\omega-\\epsilon_1+\\frac{i}{2}\\Gamma}\\,e^{-i(E_{g,g}+\\omega-E_{{\\bf p}_1,{\\bf p}_2,g})t}\\ .\\nonumber\\\\\n\\label{S2}\n\\end{eqnarray}\nHere we have applied the rotating wave approximation and inserted the total width $\\Gamma=\\Gamma_r+\\Gamma_a$ of the excited state $\\chi_e$ in atom $B$. It accounts for the finite lifetime of this state and consists of the radiative width $\\Gamma_r$ and the two-center Auger (ICD) width $\\Gamma_a$. Taking also the outer time integral, we arrive at\n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& -i\\pi\\, \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|({\\bf r}_1 + {\\bf r}_2)| \\Phi_g \\rangle\\cdot\\left( {\\bf e}_z - \\frac{3R_z}{R^2}\\,{\\bf R}\\right) \\nonumber\\\\\n& & \\times\\ \\frac{F_0}{R^3}\\, \\frac{\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^2}{\\Delta+\\frac{i}{2}\\Gamma}\\ \\delta(\\varepsilon_{p_1} + \\varepsilon_{p_2} - \\varepsilon_0 - \\omega)\\ .\\nonumber\\\\\n\\label{S3}\n\\end{eqnarray}\nwhere the detuning from the resonance $\\Delta = \\epsilon_0+\\omega-\\epsilon_1$ has been introduced.\nThe delta function in Eq.~\\eqref{S3} displays the law of energy conservation in the process.\n\nFrom the transition amplitude we can obtain the fully differential ionization cross section in the usual way by taking the absolute square and dividing it by the interaction time $\\tau$ and the incident flux $j=\\frac{cF_0^2}{8\\pi \\omega}$, that is\n\\begin{eqnarray} \n\\frac{{\\rm d}^6\\sigma^{(2)}}{{\\rm d}^3p_1\\,{\\rm d}^3p_2} = \\frac{1}{(2\\pi)^6 j\\tau}\\,\\left|S^{(2)}_{{\\bf p}_1,{\\bf p}_2}\\right|^2\\ .\n\\label{CS}\n\\end{eqnarray}\nThe factor $(2\\pi)^{-6}$ arises from the fact that the continuum states in our calculations are normalized to a quantization volume of unity. Performing the integration over $\\varepsilon_{p_2}$ with the help of the $\\delta$-function in Eq.~\\eqref{S3}, we obtain the five-fold differential cross section\n\\begin{eqnarray} \n\\frac{{\\rm d}^5\\sigma^{(2)}}{{\\rm d}\\varepsilon_{p_1}{\\rm d}^2\\Omega_{p_1}{\\rm d}^2\\Omega_{p_2}} &=& \\frac{\\omega\\,p_1\\,p_2}{(2\\pi)^4 c R^6}\\ \\frac{\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^4}{\\Delta^2+\\frac{1}{4}\\Gamma^2}\\nonumber\\\\\n& & \\hspace{-2cm} \\times\\, \\big| \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|({\\bf r}_1 + {\\bf r}_2)| \\Phi_g \\rangle \\cdot\\left( {\\bf e}_z - 3\\cos\\theta_R\\,{\\bf e}_R\\right) \\big|^2\\ ,\\nonumber\\\\\n\\label{5CS}\n\\end{eqnarray}\nwhere we have introduced the unit vector ${\\bf e}_R={\\bf R}\/R$ along the internuclear separation and the angle $\\theta_R$ between ${\\bf R}$ and the field direction.\n\nWe can draw a comparison with the direct SPDI of atom $A$ by the electromagnetic field. The corresponding probability amplitude in the first order of perturbation theory is given by \n\\begin{eqnarray}\nS^{(1)}_{{\\bf p}_1,{\\bf p}_2} &=& -i\\int_{-\\infty}^{\\infty} dt\\, \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|\\hat{W}_A| \\Phi_g \\rangle\\, e^{-i(\\varepsilon_g-\\varepsilon_{{\\bf p}_1,{\\bf p}_2})t}\\nonumber\\\\\n&=& -i\\,\\frac{F_0}{2}\\, \\langle \\Phi_{{\\bf p}_1,{\\bf p}_2}|({\\bf r}_1 + {\\bf r}_2)\\cdot{\\bf e}_z| \\Phi_g \\rangle \\nonumber\\\\\n& & \\times\\ 2\\pi\\delta(\\varepsilon_{p_1} + \\varepsilon_{p_2} - \\varepsilon_0 - \\omega)\\ .\n\\label{S-1C}\n\\end{eqnarray}\nFor the special cases, when the separation vector ${\\bf R}$ between the atoms $A$ and $B$ is oriented either along the field direction or perpendicular to it, we can cast Eq.\\,\\eqref{S3} into the form\n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& \\frac{\\alpha}{R^3}\\,\\frac{\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^2}{\\Delta+\\frac{i}{2}\\Gamma}\\,S^{(1)}_{{\\bf p}_1,{\\bf p}_2}\n\\label{S4}\n\\end{eqnarray}\nwhere $\\alpha=-2$ for ${\\bf R}\\parallel {\\bf F}_0$ and $\\alpha=1$ for ${\\bf R}\\perp {\\bf F}_0$. When the field is exactly resonant with the transition $\\chi_g\\to\\chi_e$ in atom $B$ and the interatomic distance is sufficiently large, so that $\\Gamma_a\\ll\\Gamma_r$, this expression becomes \n\\begin{eqnarray} \nS^{(2)}_{{\\bf p}_1,{\\bf p}_2} &=& \\frac{3\\alpha}{2i} \\left(\\frac{c}{\\omega R}\\right)^3\\,S^{(1)}_{{\\bf p}_1,{\\bf p}_2}\\ .\n\\label{S5}\n\\end{eqnarray}\nHere we have used that the radiative width is given by $\\Gamma_r=\\frac{4\\omega_{ge}^3}{3c^3}\\left| \\langle \\chi_e|\\xi_z| \\chi_g \\rangle \\right|^2$ with $\\omega_{ge}=\\epsilon_1-\\epsilon_0=\\omega$. \n\nFrom Eq.\\,\\eqref{S5} we can infer that the cross section for two-center SPDI can be largely enhanced by a factor $[c\/(\\omega R)]^6\\gg 1$ as compared with the usual one-center process of SPDI, where the neighboring atom $B$ is not involved. For example, assuming $\\omega\\approx 21.2$\\,eV (corresponding to the first excitation energy in helium) and $R=10$\\,\\AA, an enormous enhancement by 6 orders of magnitude results. This implies further that two-center SPDI can even exceed the direct single ionization of atom $A$ considerably. Because the double-to-single ionization ratio is typically of the order $10^{-2}$, an enhancement by four orders remains. We point out that, nevertheless, two-center SPDI is not the dominant ionization channel because {\\it single} ionization of atom $A$ via 2CPI is considerably stronger, being enhanced over direct single photoionization by a factor $[c\/(\\omega R)]^6$ as well \\cite{2CPI}. The ratio of two-center SPDI--to--2CPI is thus of the order $10^{-2}$, just like the ratio between the corresponding one-center processes.\n\nThe processes of two-center SPDI and direct SPDI lead to the same final state, since atom $B$ eventually returns to its ground state and thus serves as a catalyzer. Therefore, the corresponding probability amplitudes  \\eqref{S3} and \\eqref{S-1C} are generally subject to quantum interference. However, for parameters where the two-center channel strongly dominates, the interference is of minor importance and may be neglected.\n\n\n\\section{Numerical Results and Discussion}\n\nNext we illustrate some characteristic properties of two-center SPDI by numerical examples. Our focus lies on the angular distribution of the two electrons emitted from atom $A$, which strongly depends on the position ${\\bf R}$ of the neighboring atom $B$, as we will show. \n\n\\begin{figure*}[htb]\n\t\\begin{center}\n\t\\includegraphics[width=15cm]{Bild1.eps}\n\t\t\\caption{Five-fold differential cross section \\eqref{5CS} of two-center SPDI in a He-Li$^{2+}$ system (in units of 10$^{-3}$\\,a.u.). The photon energy, $\\omega=91.8$\\,eV, is resonant with the $1s\\to2p$ transition in Li$^{2+}$ and the field strength is $F_0=10^{-5}$\\,a.u. ($\\approx 5\\times 10^4$\\,V\/cm). The atoms are separated along the field direction by $R=10$\\,a.u. From (a)\\,-\\,(c) in the upper row the polar emission angle $\\theta_1$ of the first electron is varied from $0^{\\circ}$ to $45^{\\circ}$ in steps of $15^{\\circ}$, as indicated by the red arrows. Similarly, from (d)\\,-\\,(f) in the lower row it is varied further from $60^{\\circ}$ to $90^{\\circ}$. The blue line shows a polar plot of the angular distribution of the second electron. Both electrons have the same energies $\\varepsilon_{p_1}=\\varepsilon_{p_2}\\approx 6.8$\\,eV and azimuthal angles $\\varphi_1 = \\varphi_2 = 0^{\\circ}$.}\n\t\t\\label{Fig2}\n\t\\end{center}\n\\end{figure*}\n\nA simple diatomic system is used as generic model to illustrate the effects. Since most of the studies on SPDI have considered helium atoms, we also use helium as atom $A$ in our model system. For the helium ground state, we apply a radially correlated wave function of the form\n\\begin{eqnarray}\n\\Phi_g({\\bf r}_1,{\\bf r}_2) = N\\left( e^{-\\alpha r_1}e^{-\\beta r_2} + e^{-\\alpha r_2}e^{-\\beta r_1}\\right)\\ ,\n\\end{eqnarray}\nwith $\\alpha=1.18853$, $\\beta=2.18317$ determined from a variational calculation \\cite{Keller} and $N$ denoting the normalization constant. The corresponding ground state energy amounts to $\\varepsilon_0\\approx-2.876$\\,a.u. which differs from the exact value by 1\\,\\%. To describe the double continuum of helium we employ 2C wave functions, consisting of a symmetrized product of two Coulomb waves $\\psi^{(-)}_{{\\bf p}_j}({\\bf r}_j)$ \\cite{LandauQM} with nuclear charge $Z_A=2$. Final-state correlations are accounted for by inclusion of a Gamov factor $G({\\bf p}_1-{\\bf p}_2)$ \\cite{LandauQM} which depends on the relative momentum between the electrons:\n\\begin{eqnarray}\n\\Psi_{{\\bf p}_1,{\\bf p}_2}({\\bf r}_1,{\\bf r}_2)\\!\\! &=&\\!\\! \\frac{1}{\\sqrt{2}}\\Big[ \\psi^{(-)}_{{\\bf p}_1}({\\bf r}_1)\\psi^{(-)}_{{\\bf p}_2}({\\bf r}_2)+\\psi^{(-)}_{{\\bf p}_1}({\\bf r}_2)\\psi^{(-)}_{{\\bf p}_2}({\\bf r}_1) \\Big]\\nonumber\\\\\n& & \\times\\ G({\\bf p}_1-{\\bf p}_2)\\ .\n\\end{eqnarray}\nInto the fully differential ionization cross section, the absolute square $|G({\\bf p})|^2=\\frac{2\\pi}{p}\\left( {\\rm e}^{2\\pi\/p}-1\\right)^{-1}$ of the Gamov factor enters.\nWe note that calculations of one-center double ionization of helium using these wave functions show good agreement with more sophisticated theories in terms of the shape of the photoelectron angular distributions (see, e.g., \\cite{Keller, Maulbetsch1993, Macri}; the {\\it absolute} value of the differential cross section is not well reproduced, though).\nApplication of these wave functions offers the advantage that the matrix elements in Eq.~\\eqref{S3} can be evaluated analytically by standard means.\n\n\\begin{figure*}[htb]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.95]{Bild2.eps}\n\t\t\\caption{Same as Fig.~\\ref{Fig2} but this time the polar emission angle of the first electron is kept fixed ($\\theta_1=0^\\circ$) while the orientation of the Li$^{2+}$ ion relative to the He atom varies: (a) $\\theta_R=22.5^\\circ$, (b) $\\theta_R=45^\\circ$, (c) $\\theta_R=67.5^\\circ$, (d) $\\theta_R=90^\\circ$. The straight grey lines show the resulting effective polarization axis [see Eq.~\\eqref{eff-pol}]. }\n\t\t\\label{Fig3}\n\t\\end{center}\n\\end{figure*}\n\nIn order to provide sufficient energy to overcome the double ionization threshold of helium, a Li$^{2+}$ ion is assumed to constitute the neighboring atomic center $B$ in our generic model system. The latter is hydrogenlike with nuclear charge $Z_B=3$. The photon energy $\\omega=\\epsilon_1-\\epsilon_0=3.375$\\,a.u. is assumed to be resonant with the $1s\\to2p$ transition in Li$^{2+}$. For the internuclear distance the value $R=10$\\,a.u. is taken throughout. At this distance, the radiative width largely exceeds the two-center Auger width, so that $\\Gamma\\approx \\Gamma_r$. The relative enhancement between two-center SPDI in He-Li$^{2+}$ versus ordinary one-center SPDI of helium amounts to $[c\/(\\omega R)]^6\\sim 4.4\\times 10^3$.\n \nWe have calculated photoelectron angular distributions in the case of coplanar emission, with the azimuthal angles of both ejected electrons set to $\\varphi_1 = \\varphi_2 = 0$. Their polar angles $\\theta_1$ and $\\theta_2$ are measured with respect to the polarization direction of the external field \\eqref{field}, which was chosen to define the $z$ direction. We assume equal energy sharing between the electrons. \n\nFigure 2 shows polar plots of the five-fold differential cross section \\eqref{5CS} when the interatomic separation vector lies parallel to the field axis (${\\bf R}\\parallel {\\bf F}_0$). In panels (a)\\,-\\,(f) the polar emission angle of the first electron is varied stepwise from $0^{\\circ}$ to $90^{\\circ}$. We observe that the angular distributions closely agree with those known from one-center SPDI of helium, as a comparison with results from the literature reveals (see, e.g., Fig.~1 in \\cite{Brauning} for $\\theta_1\\in\\{0^{\\circ}, 30^{\\circ}, 60^{\\circ}, 90^{\\circ}\\}$ and Fig.~4\\,(c) in \\cite{Thumm} for $\\theta_1\\approx \\pm75^{\\circ}$). With the rotation of the emission angle $\\theta_1$ of the first electron, the angular distribution of the second electron changes accordingly. The repulsion of the two electrons becomes apparent this way, but it does not lead to back-to-back emission. Instead a preferred relative angle between $\\theta_1$ and $\\theta_2$ of about $100^\\circ$--$140^\\circ$ appears. \nFor symmetry reasons, during ionization along one of the coordinate axes ($\\theta_1=0^\\circ$ or 90$^\\circ$), two emission directions are equally probable for the second electron. For emission angles $\\theta_1$ in between, one direction is preferred. \n\nThe angular emission patterns are modified when the relative orientation between the atoms is changed. This is displayed in Fig.~\\ref{Fig3} where the first electron is always ejected under $\\theta_1=0^\\circ$ and the internuclear angle $\\theta_R$ is varied. A very strong sensitivity on the position of the Li ion is found.\n\n\\begin{figure*}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.97]{Bild3.eps}\n\t\t\\caption{Density plots of the five-fold differential cross section \\eqref{5CS} for two-center SPDI in a He-Li$^{2+}$ system, depending on the polar emission angles of both electrons. From (a)\\,-\\,(e) the relative interatomic orientation is varied from $\\theta_R=0^\\circ$ to $90^\\circ$ in steps of $22.5^\\circ$. Panel (f) shows the joint angular distribution when an average over the interatomic orientation is taken. The other parameters are as in Fig.~\\ref{Fig2}.}\n\t\t\\label{Fig4}\n\t\\end{center}\n\\end{figure*}\n\nThe shapes of the photoelectron angular distributions can be understood by inspection of Eqs.~\\eqref{S3} and \\eqref{S-1C}. In case of one-center SPDI, the interaction operator $({\\bf r}_1 + {\\bf r}_2)\\cdot{\\bf e}_z$ appears, with the field polarization vector ${\\bf e}_z$, which leads to the emission patterns known for this process. In case of two-center SPDI, a similar interaction operator arises, but the field polarization is replaced by an effective polarization vector\n\\begin{eqnarray}\n\t{\\bf n}_{\\text{eff}}={\\bf e}_z - 3\\cos\\theta_R\\,{\\bf e}_R\n\t\\label{eff-pol}\n\\end{eqnarray}\nwhich represents a linear combination of the field polarization and the interatomic orientation vector.\nNote that ${\\bf n}_{\\text{eff}}$ is generally not normalized but has length $|{\\bf n}_{\\text{eff}}|=(1+3\\cos^2\\theta_R)^{1\/2}$. This effective polarization vector encodes the impact of the relative position of atom $B$ on the angular distribution of two-center SPDI. \n\nThe angular spectra in Fig.~\\ref{Fig2} refer to $\\theta_R=0^\\circ$, where ${\\bf n}_{\\text{eff}}=-2\\,{\\bf e}_z$. Thus, the effective polarization is (anti)parallel to the field polarization which explains, why the photoelectron emission patterns for two-center and one-center SPDI have identical shapes. \n\nConversely, in Fig.~\\ref{Fig3}\\,(a)\\,-\\,(c) the effective polarization direction differs from the field polarization and, thus, from the emission direction of the first electron, as indicated by the straight grey lines. The angle between ${\\bf e}_z$ and the effective polarization {\\it axis} amounts to about 30$^\\circ$ in panel (a), 75$^\\circ$ in panel (b), and 60$^\\circ$ in panel (c). Accordingly, Fig.~\\ref{Fig3}\\,(a) resembles Fig.~\\ref{Fig2}\\,(c), Fig.~\\ref{Fig3}\\,(b) resembles Fig.~\\ref{Fig2}\\,(e), and Fig.~\\ref{Fig3}\\,(c) resembles Fig.~\\ref{Fig2}\\,(d), provided a proper rotating reflection is applied. [In fact, if $\\theta_R$ is changed and at the same time $\\theta_1$ is laid along the corresponding effective polarization direction, an angular distribution equivalent to Fig.~2\\,(a) results, which is only rotated by the angle $\\theta_R$ (not shown).]\nFor $\\theta_R=90^\\circ$ [see Fig.~\\ref{Fig3}\\,d)], one has ${\\bf n}_{\\text{eff}}={\\bf e}_z$ and the emission pattern coincides with the one in Fig.~\\ref{Fig2}\\,(a). Only the absolute values of the five-fold differential cross sections differ by a factor of 4, because the effective polarization vector in Fig.~\\ref{Fig2}\\,(a) has length $|{\\bf n}_{\\text{eff}}|=2$.\n\nOur findings are confirmed by Fig.~\\ref{Fig4} which shows two-dimensional density plots of the joint angular distribution of both electrons. In panels (a)\\,-\\,(e) the interatomic orientation is stepwise changed from $\\theta_R=0^\\circ$ to $90^\\circ$. The emission pattern in Fig.~\\ref{Fig4}\\,(a), where ${\\bf R}\\parallel {\\bf F}_0$ and thus ${\\bf n}_{\\text{eff}}=-2\\,{\\bf e}_z$, agrees with the corresponding result for one-center SPDI (see Fig.~4\\,(a) in \\cite{Thumm}). \nFour pronounced emission peaks appear which lie symmetrically to the vertical line defined by $\\theta_1 + \\theta_2 = 360^\\circ$. These four peaks shift gradually as $\\theta_R$ increases in panels (b)\\,-(d) and eventually reach their initial positions in Fig.~\\ref{Fig4}\\,(e), where ${\\bf R}\\perp {\\bf F}_0$ and thus ${\\bf n}_{\\text{eff}}={\\bf e}_z$ holds. Consequently, the distributions in panels (a) and (e) are identical in shape and only differ in absolute value by a factor of 4 because of the different values of $|{\\bf n}_{\\text{eff}}|$.\n\nIn an experimental situation it can be difficult to fix the interatomic orientation between atoms $A$ and $B$. If, for example, a gas of van-der-Waals dimers $A$-$B$ was used, the internuclear axis would be randomly oriented. Therefore, we have averaged our results over the interatomic orientation, which results in the angular distribution in Fig.~\\ref{Fig4}\\,(f). The four emission peaks are smeared out into four parallel emission stripes which have the same slope and now extend over the complete plot range. \n\nOur calculations in this section were performed considering a He-Li$^{2+}$ model system in order to demonstrate some general features of two-center SPDI angular distributions. One should note, however, that it is very difficult to prepare such a system for conducting a dedicated experiment. Besides, the presence of the doubly ionized lithium ion will cause energy shifts in the neighbouring helium atom, which were not taken into account here.\n\nIn view of a potential experimental observation of two-center SPDI one could employ an earth-alkaline metal, such as calcium or magnesium, as atom $A$ in conjunction with helium as atom $B$. Using photon energies $\\omega=21.2$\\,eV (23.1\\,eV), which are resonant to the $1s\\to 2p$ ($1s\\to 3p$) transition in helium, double ionization of Ca (Mg) could be probed. The corresponding thresholds are 18.0\\,eV and 22.7\\,eV, respectively \\cite{NIST}. These target systems could be realized, for example, by Ca (or Mg) atoms attached to helium droplets (see, e.g., \\cite{Ca-He} for a related experiment). Due to the smaller energy transfers and interatomic distances involved, the relative enhancement between two-center and one-center SPDI would be much larger here than in the He-Li$^{2+}$ model system considered above.\n\n\n\\section{Conclusion}\nSingle-photon double ionization in two-center systems of atoms $A$ and $B$ was studied, where resonant photoexcitation of $B$ leads to double ionization of $A$ via the combined action of interatomic and intraatomic electron-electron correlations. It was shown that, due to its resonant character, two-center SPDI can largely dominate over the ordinary one-center SPDI of atom $A$ for interatomic distances up to few nanometers. \n\nThe influence of atom $B$ was also revealed in the angular distributions of emitted photoelectrons. They depend sensitively on the direction of an effective polarization vector, which arises as linear combination of the external field polarization and the interatomic separation vector. This effective polarization direction determines the shape of angular distributions from two-center SPDI in the same way as the field polarization vector does in case of one-center SPDI. The corresponding effects were illustrated by considering a simple two-center model system.\n\nOur predictions could be tested, for example, in Ca-He or Mg-He systems, which possess low double-ionization thresholds. When the interatomic orientation is not fixed but randomly distributed in the target, the correspondingly averaged photoelectron angular distribution of two-center SPDI looks qualitatively different from the one-center case. This characteristic feature could help to experimentally identify the two-center process, in addition to its strong resonant enhancement over one-center SPDI in terms of the total cross section.\n\n\\section*{Acknowledgement}\nThis work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No. 349581371 (MU 3149\/4-1 and VO 1278\/4-1). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}