We investigate Petri nets with data, an extension of plain Petri nets where tokens carry values from an infinite data domain, and executability of transitions is conditioned by equalities between data values. We provide a decision procedure for the bi-reachability problem: given a Petri net and its two configurations, we ask if each of the configurations is reachable from the other. This pushes forward the decidability borderline, as the bi-reachability problem subsumes the coverability problem (which is known to be decidable) and is subsumed by the reachability problem (whose decidability status is unknown).
我们研究了带数据的 Petri 网，它是普通 Petri 网的扩展，其中的标记携带无限数据域中的值，且过渡的可执行性以数据值之间的相等为条件。我们提供了双可达性问题的判定过程：给定一个 Petri 网和它的两个配置，我们问每个配置是否都可以从另一个配置到达。这就推进了可判定性的边界，因为双可达性问题包含了可覆盖性问题（已知可判定）和可达性问题（其可判定性状况未知）。
{"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
Formal language theory has recently been successfully employed to unravel the power of transformer encoders. This setting is primarily applicable in Natural Languange Processing (NLP), as a token embedding function (where a bounded number of tokens is admitted) is first applied before feeding the input to the transformer. On certain kinds of data (e.g. time series), we want our transformers to be able to handle emph{arbitrary} input sequences of numbers (or tuples thereof) without emph{a priori} limiting the values of these numbers. In this paper, we initiate the study of the expressive power of transformer encoders on sequences of data (i.e. tuples of numbers). Our results indicate an increase in expressive power of hard attention transformers over data sequences, in stark contrast to the case of strings. In particular, we prove that Unique Hard Attention Transformers (UHAT) over inputs as data sequences no longer lie within the circuit complexity class $AC^0$ (even without positional encodings), unlike the case of string inputs, but are still within the complexity class $TC^0$ (even with positional encodings). Over strings, UHAT without positional encodings capture only regular languages. In contrast, we show that over data sequences UHAT can capture non-regular properties. Finally, we show that UHAT capture languages definable in an extension of linear temporal logic with unary numeric predicates and arithmetics.
{"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}
Data trees serve as an abstraction of structured data, such as XML documents. A number of specification formalisms for languages of data trees have been developed, many of them adhering to the paradigm of register automata, which is based on storing data values encountered on the tree in registers for subsequent comparison with further data values. Already on word languages, the expressiveness of such automata models typically increases with the power of control (e.g. deterministic, non-deterministic, alternating). Language inclusion is typically undecidable for non-deterministic or alternating models unless the number of registers is radically restricted, and even then often remains non-elementary. We present an automaton model for data trees that retains a reasonable level of expressiveness, in particular allows non-determinism and any number of registers, while admitting language inclusion checking in elementary complexity, in fact in parametrized exponential time. We phrase the description of our automaton model in the language of nominal sets, building on the recently introduced paradigm of explicit name allocation in nominal automata.
数据树是结构化数据（如 XML 文档）的抽象，目前已开发出许多数据树语言的规范形式，其中许多都遵循寄存器自动机范式，即把数据树上遇到的数据值存储在寄存器中，以便随后与其他数据值进行比较。就单词语言而言，这种自动机模型的可执行性通常会随着控制能力（如确定性、非确定性、交替性）的增加而增加。对于非确定性或交替模型，除非寄存器的数量受到严格限制，否则语言包容通常是不可判定的，即便如此，语言包容也常常是非基本的。我们提出了一种数据树的自动机模型，它保持了合理的表达水平，特别是允许非确定性和任意数量的寄存器，同时允许在基本复杂度下进行语言包容检验，实际上是在参数化指数时间内。我们用名义集语言来描述我们的自动机模型，以最近引入的名义自动机中的显式名称分配范式为基础。
{"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}
Jumping automata are finite automata that read their input in a non-sequential manner, by allowing a reading head to ``jump'' between positions on the input, consuming a permutation of the input word. We argue that allowing the head to jump should incur some cost. To this end, we propose three quantitative semantics for jumping automata, whereby the jumps of the head in an accepting run define the cost of the run. The three semantics correspond to different interpretations of jumps: the emph{absolute distance} semantics counts the distance the head jumps, the emph{reversal} semantics counts the number of times the head changes direction, and the emph{Hamming distance} measures the number of letter-swaps the run makes. We study these measures, with the main focus being the emph{boundedness problem}: given a jumping automaton, decide whether its (quantitative) languages is bounded by some given number $k$. We establish the decidability and complexity for this problem under several variants.
{"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
We present a theoretical framework for the extraction and transformation of text documents. We propose to use a two-phase process where the first phase extracts span-tuples from a document, and the second phase maps the content of the span-tuples into new documents. We base the extraction phase on the framework of document spanners and the transformation phase on the theory of polyregular functions, the class of regular string-to-string functions with polynomial growth. For supporting practical extract-transform scenarios, we propose an extension of document spanners described by regex formulas from span-tuples to so-called multispan-tuples, where variables are mapped to sets of spans. We prove that this extension, called regex multispanners, has the same desirable properties as standard spanners described by regex formulas. In our framework, an Extract-Transform (ET) program is given by a regex multispanner followed by a polyregular function. In this paper, we study the expressibility and evaluation problem of ET programs when the transformation function is linear, called linear ET programs. We show that linear ET programs are equally expressive as non-deterministic streaming string transducers under bag semantics. Moreover, we show that linear ET programs are closed under composition. Finally, we present an enumeration algorithm for evaluating every linear ET program over a document with linear time preprocessing and constant delay.
我们提出了一个提取和转换文本文档的理论框架。我们建议使用两阶段流程，第一阶段从文档中提取跨元组，第二阶段将跨元组的内容映射到新文档中。我们将提取阶段建立在文档跨度框架的基础上，将转换阶段建立在多正则函数理论的基础上，多正则函数是一类具有多项式增长的正则字符串到字符串函数。为了支持实际的提取-转换应用场景，我们提出了一种由 regex 公式描述的文档生成器的扩展，即从 span 元组扩展到所谓的多 span 元组，其中变量被映射到 spans 集合。我们证明，这种扩展（称为 regex 多跨度）与 regex 公式描述的标准跨度具有相同的理想特性。在我们的框架中，一个提取-转换（ET）程序由一个 regex 多跨度程序和一个多正则函数给出。在本文中，我们研究了转换函数为线性时 ET 程序的可表达性和评估问题，这些程序被称为线性 ET 程序。我们证明，在 bag 语义下，线性 ET 程序与非确定性流式字符串转换器具有同等的表达能力。此外，我们还证明了线性 ET 程序在组合下是封闭的。最后，我们提出了一种枚举算法，用于以线性时间预处理和恒定延迟评估文档上的每个线性 ET 程序。
{"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
Well-structured transition systems (WSTS) are an abstract family of systems that encompasses a vast landscape of infinite-state systems. By requiring a well-quasi-ordering (wqo) on the set of states, a WSTS enables generic algorithms for classic verification tasks such as coverability and termination. However, even for systems that are WSTS like vector addition systems (VAS), the framework is notoriously ill-equipped to analyse reachability (as opposed to coverability). Moreover, some important types of infinite-state systems fall out of WSTS' scope entirely, such as pushdown systems (PDS). Inspired by recent algorithmic techniques on VAS, we propose an abstract notion of systems where the set of runs is equipped with a wqo and supports amalgamation of runs. We show that it subsumes a large class of infinite-state systems, including (reachability languages of) VAS and PDS, and even all systems from the abstract framework of valence systems, except for those already known to be Turing-complete. Moreover, this abstract setting enables simple and general algorithmic solutions to unboundedness problems, which have received much attention in recent years. We present algorithms for the (i) simultaneous unboundedness problem (which implies computability of downward closures and decidability of separability by piecewise testable languages), (ii) computing priority downward closures, (iii) deciding whether a language is bounded, meaning included in $w_1^*cdots w_k^*$ for some words $w_1,ldots,w_k$, and (iv)~effective regularity of unary languages. This leads to either drastically simpler proofs or new decidability results for a rich variety of systems.
结构良好的过渡系统（WSTS）是一个抽象的系统族，它涵盖了无限状态系统的广泛领域。WSTS要求对状态集进行良好准排序（wqo），因此它能为可覆盖性和终止等经典验证任务提供通用算法。然而，即使是像向量加法系统（VAS）这样的WSTS系统，该框架在分析可达性（相对于可覆盖性）方面也是众所周知的能力不足。此外，一些重要的无穷态系统，如推倒系统（PDS），完全不属于 WSTS 的范畴。受最近关于 VAS 算法技术的启发，我们提出了一种系统抽象概念，其中运行集配备了 wqo 并支持运行的合并。我们证明，它包含了一大类无限状态系统，包括 VAS 和 PDS 的（可达性语言），甚至包括价系统抽象框架中的所有系统，除了那些已知图灵完备的系统。此外，这种抽象设置还能为近年来备受关注的无界性问题提供简单而通用的算法解决方案。我们提出了针对以下问题的算法：(i) 同步无界性问题（这意味着下向闭包的可计算性和片断可测试语言的可分性的可解性）；(ii) 计算优先级下向闭包；(iii) 判断语言是否有界，即对于某些词 $w_1,ldots,w_k$，包含在 $w_1^*cdots w_k^*$ 中；(iv) 单元语的有效规则性。这将为丰富多样的系统带来更简单的证明或新的可解性结果。
{"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}
In many complex systems that can be modeled using Petri nets time can be a very important factor which should be taken into account during creation and analysis of the model. Time data can describe starting moments of some actions or their duration before their immediate effects start to influence some other areas of the modeled system. Places in a Petri net often describe static components of the system, but they can also describe states. Such a state can have time restrictions, for example, telling how long it can influence other elements in the model. Time values describing some system may be inconsistent or incomplete, which can cause problems during the creation of the model. In this paper, a new extension of time Petri nets is proposed, which allows the creation of models with different types of time data, which previously were possible to be properly used in separate types of well-known time Petri nets. The proposed new time Petri net solves this problem by integrating different aspects of already existing time Petri nets into one unified net.
在许多可以使用 Petri 网建模的复杂系统中，时间是一个非常重要的因素，在创建和分析模型时应将其考虑在内。时间数据可以描述某些动作的起始时刻，也可以描述在这些动作的直接影响开始对建模系统的其他区域产生影响之前的持续时间。Petri 网中的位置通常描述系统的静态组件，但也可以描述状态。这种状态可以有时间限制，例如，它可以在多长时间内影响模型中的其他元素。描述某些系统的时间值可能不一致或不完整，这会在创建模型时造成问题。本文提出了时间 Petri 网的一个新扩展，它允许创建具有不同类型时间数据的模型，而这些数据以前只能在不同类型的著名时间 Petri 网中正确使用。
{"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
Periodic recurrence is a prominent behavioural of many biological phenomena, including cell cycle and circadian rhythms. Although deterministic models are commonly used to represent the dynamics of periodic phenomena, it is known that they are little appropriate in the case of systems in which stochastic noise induced by small population numbers is actually responsible for periodicity. Within the stochastic modelling settings automata-based model checking approaches have proven an effective means for the analysis of oscillatory dynamics, the main idea being that of coupling a period detector automaton with a continuous-time Markov chain model of an alleged oscillator. In this paper we address a complementary aspect, i.e. that of assessing the dependency of oscillation related measure (period and amplitude) against the parameters of a stochastic oscillator. To this aim we introduce a framework which, by combining an Approximate Bayesian Computation scheme with a hybrid automata capable of quantifying how distant an instance of a stochastic oscillator is from matching a desired (average) period, leads us to identify regions of the parameter space in which oscillation with given period are highly likely. The method is demonstrated through a couple of case studies, including a model of the popular Repressilator circuit.
{"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}
There are many types of automata and grammar models that have been studied in the literature, and for these models, it is common to determine whether certain problems are decidable. One problem that has been difficult to answer throughout the history of automata and formal language theory is to decide whether a given system $M$ accepts a bounded language (whether there exist words $w_1, ldots,w_k$ such that $L(M) subseteq w_1^* cdots w_k^*$?). Boundedness was only known to be decidable for regular and context-free languages until recently when it was shown to also be decidable for finite automata and pushdown automata augmented with reversal-bounded counters, and for vector addition systems with states. However, decidability of this problem has still gone unanswered for the majority of automata/grammar models with a decidable emptiness problem that have been studied in the literature. In this paper, we develop new techniques to show that the boundedness problem is decidable for larger classes of one-way nondeterministic automata and grammar models by reducing the problem to the decidability of boundedness for simpler classes of automata. One technique involves characterizing the models in terms of multi-tape automata. We give new characterizations of finite-turn Turing machines, finite-turn Turing machines augmented with various storage structures (like a pushdown, multiple reversal-bounded counters, partially-blind counters, etc.), and simple matrix grammars. The characterizations are then used to show that the boundedness problem for these models is decidable. Another technique uses the concept of the store language of an automaton. This is used to show that the boundedness problem is decidable for pushdown automata that can "flip" their pushdown a bounded number of times. Boundedness remains decidable even if we augment this device with additional stores.
{"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
A transducer is finite-valued if for some bound k, it maps any given input to at most k outputs. For classical, one-way transducers, it is known since the 80s that finite valuedness entails decidability of the equivalence problem. This decidability result is in contrast to the general case, which makes finite-valued transducers very attractive. For classical transducers, it is also known that finite valuedness is decidable and that any k-valued finite transducer can be decomposed as a union of k single-valued finite transducers. In this paper, we extend the above results to copyless streaming string transducers (SSTs), answering questions raised by Alur and Deshmukh in 2011. SSTs strictly extend the expressiveness of one-way transducers via additional variables that store partial outputs. We prove that any k-valued SST can be effectively decomposed as a union of k (single-valued) deterministic SSTs. As a corollary, we obtain equivalence of SSTs and two-way transducers in the finite-valued case (those two models are incomparable in general). Another corollary is an elementary upper bound for checking equivalence of finite-valued SSTs. The latter problem was already known to be decidable, but the proof complexity was unknown (it relied on Ehrenfeucht's conjecture). Finally, our main result is that finite valuedness of SSTs is decidable. The complexity is PSpace, and even PTime when the number of variables is fixed.
如果对于某个界值 k，一个转换器最多能将任何给定输入映射到 k 个输出，那么这个转换器就是有限值的。对于经典的单向变换器，自 20 世纪 80 年代以来，人们就知道有限有值性意味着等价问题的可解性。对于经典变换器，人们也知道有限有值性是可解的，而且任何 k 值有限变换器都可以分解为 k 个单值有限变换器的联合。在本文中，我们将上述结果扩展到了无副本流串变换器（SST），回答了阿卢尔和德什穆克在 2011 年提出的问题。我们证明，任何 k 值 SST 都可以有效地分解为 k 个（单值）确定性 SST 的联合。作为推论，我们得到了 SST 和双向变换器在有限值情况下的等价性（这两种模型在一般情况下是不可比的）。另一个推论是检查无穷值 SST 等价性的基本上界。最后，我们的主要结果是 SST 的有限有值性是可解的。最后，我们的主要结果是，SST 的有限值性是可解的。其复杂度是 PS 空间，当变量数固定时甚至是 PT 时间。
{"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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