{"title":"Bi-reachability in Petri nets with data","authors":"Łukasz Kamiński, Sławomir Lasota","doi":"arxiv-2405.16176","DOIUrl":"https://doi.org/arxiv-2405.16176","url":null,"abstract":"We investigate Petri nets with data, an extension of plain Petri nets where\u0000tokens carry values from an infinite data domain, and executability of\u0000transitions is conditioned by equalities between data values. We provide a\u0000decision procedure for the bi-reachability problem: given a Petri net and its\u0000two configurations, we ask if each of the configurations is reachable from the\u0000other. This pushes forward the decidability borderline, as the bi-reachability\u0000problem subsumes the coverability problem (which is known to be decidable) and\u0000is subsumed by the reachability problem (whose decidability status is unknown).","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pascal Bergsträßer, Chris Köcher, Anthony Widjaja Lin, Georg Zetzsche
{"title":"The Power of Hard Attention Transformers on Data Sequences: A Formal Language Theoretic Perspective","authors":"Pascal Bergsträßer, Chris Köcher, Anthony Widjaja Lin, Georg Zetzsche","doi":"arxiv-2405.16166","DOIUrl":"https://doi.org/arxiv-2405.16166","url":null,"abstract":"Formal language theory has recently been successfully employed to unravel the\u0000power of transformer encoders. This setting is primarily applicable in Natural\u0000Languange Processing (NLP), as a token embedding function (where a bounded\u0000number of tokens is admitted) is first applied before feeding the input to the\u0000transformer. On certain kinds of data (e.g. time series), we want our\u0000transformers to be able to handle emph{arbitrary} input sequences of numbers\u0000(or tuples thereof) without emph{a priori} limiting the values of these\u0000numbers. In this paper, we initiate the study of the expressive power of\u0000transformer encoders on sequences of data (i.e. tuples of numbers). Our results\u0000indicate an increase in expressive power of hard attention transformers over\u0000data sequences, in stark contrast to the case of strings. In particular, we\u0000prove that Unique Hard Attention Transformers (UHAT) over inputs as data\u0000sequences no longer lie within the circuit complexity class $AC^0$ (even\u0000without positional encodings), unlike the case of string inputs, but are still\u0000within the complexity class $TC^0$ (even with positional encodings). Over\u0000strings, UHAT without positional encodings capture only regular languages. In\u0000contrast, we show that over data sequences UHAT can capture non-regular\u0000properties. Finally, we show that UHAT capture languages definable in an\u0000extension of linear temporal logic with unary numeric predicates and\u0000arithmetics.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nominal Tree Automata With Name Allocation","authors":"Simon Prucker, Lutz Schröder","doi":"arxiv-2405.14272","DOIUrl":"https://doi.org/arxiv-2405.14272","url":null,"abstract":"Data trees serve as an abstraction of structured data, such as XML documents.\u0000A number of specification formalisms for languages of data trees have been\u0000developed, many of them adhering to the paradigm of register automata, which is\u0000based on storing data values encountered on the tree in registers for\u0000subsequent comparison with further data values. Already on word languages, the\u0000expressiveness of such automata models typically increases with the power of\u0000control (e.g. deterministic, non-deterministic, alternating). Language\u0000inclusion is typically undecidable for non-deterministic or alternating models\u0000unless the number of registers is radically restricted, and even then often\u0000remains non-elementary. We present an automaton model for data trees that\u0000retains a reasonable level of expressiveness, in particular allows\u0000non-determinism and any number of registers, while admitting language inclusion\u0000checking in elementary complexity, in fact in parametrized exponential time. We\u0000phrase the description of our automaton model in the language of nominal sets,\u0000building on the recently introduced paradigm of explicit name allocation in\u0000nominal automata.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Jumping Automata Must Pay","authors":"Shaull Almagor, Ishai Salgado","doi":"arxiv-2405.11849","DOIUrl":"https://doi.org/arxiv-2405.11849","url":null,"abstract":"Jumping automata are finite automata that read their input in a\u0000non-sequential manner, by allowing a reading head to ``jump'' between positions\u0000on the input, consuming a permutation of the input word. We argue that allowing\u0000the head to jump should incur some cost. To this end, we propose three\u0000quantitative semantics for jumping automata, whereby the jumps of the head in\u0000an accepting run define the cost of the run. The three semantics correspond to\u0000different interpretations of jumps: the emph{absolute distance} semantics\u0000counts the distance the head jumps, the emph{reversal} semantics counts the\u0000number of times the head changes direction, and the emph{Hamming distance}\u0000measures the number of letter-swaps the run makes. We study these measures, with the main focus being the emph{boundedness\u0000problem}: given a jumping automaton, decide whether its (quantitative)\u0000languages is bounded by some given number $k$. We establish the decidability\u0000and complexity for this problem under several variants.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cristian Riveros, Markus L. Schmid, Nicole Schweikardt
{"title":"A framework for extraction and transformation of documents","authors":"Cristian Riveros, Markus L. Schmid, Nicole Schweikardt","doi":"arxiv-2405.12350","DOIUrl":"https://doi.org/arxiv-2405.12350","url":null,"abstract":"We present a theoretical framework for the extraction and transformation of\u0000text documents. We propose to use a two-phase process where the first phase\u0000extracts span-tuples from a document, and the second phase maps the content of\u0000the span-tuples into new documents. We base the extraction phase on the\u0000framework of document spanners and the transformation phase on the theory of\u0000polyregular functions, the class of regular string-to-string functions with\u0000polynomial growth. For supporting practical extract-transform scenarios, we propose an extension\u0000of document spanners described by regex formulas from span-tuples to so-called\u0000multispan-tuples, where variables are mapped to sets of spans. We prove that\u0000this extension, called regex multispanners, has the same desirable properties\u0000as standard spanners described by regex formulas. In our framework, an\u0000Extract-Transform (ET) program is given by a regex multispanner followed by a\u0000polyregular function. In this paper, we study the expressibility and evaluation problem of ET\u0000programs when the transformation function is linear, called linear ET programs.\u0000We show that linear ET programs are equally expressive as non-deterministic\u0000streaming string transducers under bag semantics. Moreover, we show that linear\u0000ET programs are closed under composition. Finally, we present an enumeration\u0000algorithm for evaluating every linear ET program over a document with linear\u0000time preprocessing and constant delay.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"222 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ashwani Anand, Sylvain Schmitz, Lia Schütze, Georg Zetzsche
{"title":"Verifying Unboundedness via Amalgamation","authors":"Ashwani Anand, Sylvain Schmitz, Lia Schütze, Georg Zetzsche","doi":"arxiv-2405.10296","DOIUrl":"https://doi.org/arxiv-2405.10296","url":null,"abstract":"Well-structured transition systems (WSTS) are an abstract family of systems\u0000that encompasses a vast landscape of infinite-state systems. By requiring a\u0000well-quasi-ordering (wqo) on the set of states, a WSTS enables generic\u0000algorithms for classic verification tasks such as coverability and termination.\u0000However, even for systems that are WSTS like vector addition systems (VAS), the\u0000framework is notoriously ill-equipped to analyse reachability (as opposed to\u0000coverability). Moreover, some important types of infinite-state systems fall\u0000out of WSTS' scope entirely, such as pushdown systems (PDS). Inspired by recent algorithmic techniques on VAS, we propose an abstract\u0000notion of systems where the set of runs is equipped with a wqo and supports\u0000amalgamation of runs. We show that it subsumes a large class of infinite-state\u0000systems, including (reachability languages of) VAS and PDS, and even all\u0000systems from the abstract framework of valence systems, except for those\u0000already known to be Turing-complete. Moreover, this abstract setting enables simple and general algorithmic\u0000solutions to unboundedness problems, which have received much attention in\u0000recent years. We present algorithms for the (i) simultaneous unboundedness\u0000problem (which implies computability of downward closures and decidability of\u0000separability by piecewise testable languages), (ii) computing priority downward\u0000closures, (iii) deciding whether a language is bounded, meaning included in\u0000$w_1^*cdots w_k^*$ for some words $w_1,ldots,w_k$, and (iv)~effective\u0000regularity of unary languages. This leads to either drastically simpler proofs\u0000or new decidability results for a rich variety of systems.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extended time Petri nets","authors":"Marcin Radom, Piotr Formanowicz","doi":"arxiv-2405.09208","DOIUrl":"https://doi.org/arxiv-2405.09208","url":null,"abstract":"In many complex systems that can be modeled using Petri nets time can be a\u0000very important factor which should be taken into account during creation and\u0000analysis of the model. Time data can describe starting moments of some actions\u0000or their duration before their immediate effects start to influence some other\u0000areas of the modeled system. Places in a Petri net often describe static\u0000components of the system, but they can also describe states. Such a state can\u0000have time restrictions, for example, telling how long it can influence other\u0000elements in the model. Time values describing some system may be inconsistent\u0000or incomplete, which can cause problems during the creation of the model. In\u0000this paper, a new extension of time Petri nets is proposed, which allows the\u0000creation of models with different types of time data, which previously were\u0000possible to be properly used in separate types of well-known time Petri nets.\u0000The proposed new time Petri net solves this problem by integrating different\u0000aspects of already existing time Petri nets into one unified net.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"131 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paolo BallariniMICS, Mahmoud Bentriou, Paul-Henry Cournède
{"title":"A Formal Approach for Tuning Stochastic Oscillators","authors":"Paolo BallariniMICS, Mahmoud Bentriou, Paul-Henry Cournède","doi":"arxiv-2405.09183","DOIUrl":"https://doi.org/arxiv-2405.09183","url":null,"abstract":"Periodic recurrence is a prominent behavioural of many biological phenomena,\u0000including cell cycle and circadian rhythms. Although deterministic models are\u0000commonly used to represent the dynamics of periodic phenomena, it is known that\u0000they are little appropriate in the case of systems in which stochastic noise\u0000induced by small population numbers is actually responsible for periodicity.\u0000Within the stochastic modelling settings automata-based model checking\u0000approaches have proven an effective means for the analysis of oscillatory\u0000dynamics, the main idea being that of coupling a period detector automaton with\u0000a continuous-time Markov chain model of an alleged oscillator. In this paper we\u0000address a complementary aspect, i.e. that of assessing the dependency of\u0000oscillation related measure (period and amplitude) against the parameters of a\u0000stochastic oscillator. To this aim we introduce a framework which, by combining\u0000an Approximate Bayesian Computation scheme with a hybrid automata capable of\u0000quantifying how distant an instance of a stochastic oscillator is from matching\u0000a desired (average) period, leads us to identify regions of the parameter space\u0000in which oscillation with given period are highly likely. The method is\u0000demonstrated through a couple of case studies, including a model of the popular\u0000Repressilator circuit.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Containment Problem for Deterministic Multicounter Machine Models","authors":"Oscar H. Ibarra, Ian McQuillan","doi":"arxiv-2405.08988","DOIUrl":"https://doi.org/arxiv-2405.08988","url":null,"abstract":"There are many types of automata and grammar models that have been studied in\u0000the literature, and for these models, it is common to determine whether certain\u0000problems are decidable. One problem that has been difficult to answer\u0000throughout the history of automata and formal language theory is to decide\u0000whether a given system $M$ accepts a bounded language (whether there exist\u0000words $w_1, ldots,w_k$ such that $L(M) subseteq w_1^* cdots w_k^*$?).\u0000Boundedness was only known to be decidable for regular and context-free\u0000languages until recently when it was shown to also be decidable for finite\u0000automata and pushdown automata augmented with reversal-bounded counters, and\u0000for vector addition systems with states. However, decidability of this problem\u0000has still gone unanswered for the majority of automata/grammar models with a\u0000decidable emptiness problem that have been studied in the literature. In this paper, we develop new techniques to show that the boundedness problem\u0000is decidable for larger classes of one-way nondeterministic automata and\u0000grammar models by reducing the problem to the decidability of boundedness for\u0000simpler classes of automata. One technique involves characterizing the models\u0000in terms of multi-tape automata. We give new characterizations of finite-turn\u0000Turing machines, finite-turn Turing machines augmented with various storage\u0000structures (like a pushdown, multiple reversal-bounded counters,\u0000partially-blind counters, etc.), and simple matrix grammars. The\u0000characterizations are then used to show that the boundedness problem for these\u0000models is decidable. Another technique uses the concept of the store language\u0000of an automaton. This is used to show that the boundedness problem is decidable\u0000for pushdown automata that can \"flip\" their pushdown a bounded number of times.\u0000Boundedness remains decidable even if we augment this device with additional\u0000stores.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emmanuel Filiot, Ismaël Jecker, Gabriele Puppis, Christof Löding, Anca Muscholl, Sarah Winter
{"title":"Finite-valued Streaming String Transducers","authors":"Emmanuel Filiot, Ismaël Jecker, Gabriele Puppis, Christof Löding, Anca Muscholl, Sarah Winter","doi":"arxiv-2405.08171","DOIUrl":"https://doi.org/arxiv-2405.08171","url":null,"abstract":"A transducer is finite-valued if for some bound k, it maps any given input to\u0000at most k outputs. For classical, one-way transducers, it is known since the\u000080s that finite valuedness entails decidability of the equivalence problem.\u0000This decidability result is in contrast to the general case, which makes\u0000finite-valued transducers very attractive. For classical transducers, it is\u0000also known that finite valuedness is decidable and that any k-valued finite\u0000transducer can be decomposed as a union of k single-valued finite transducers. In this paper, we extend the above results to copyless streaming string\u0000transducers (SSTs), answering questions raised by Alur and Deshmukh in 2011.\u0000SSTs strictly extend the expressiveness of one-way transducers via additional\u0000variables that store partial outputs. We prove that any k-valued SST can be\u0000effectively decomposed as a union of k (single-valued) deterministic SSTs. As a\u0000corollary, we obtain equivalence of SSTs and two-way transducers in the\u0000finite-valued case (those two models are incomparable in general). Another\u0000corollary is an elementary upper bound for checking equivalence of\u0000finite-valued SSTs. The latter problem was already known to be decidable, but\u0000the proof complexity was unknown (it relied on Ehrenfeucht's conjecture).\u0000Finally, our main result is that finite valuedness of SSTs is decidable. The\u0000complexity is PSpace, and even PTime when the number of variables is fixed.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"2010 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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