Andrew Scoones, Mahsa Shirmohammadi, James Worrell
Multi-priced timed automata (MPTA) are timed automata with observer variables whose derivatives can change from one location to another. Observers are write-only variables, that is, they do not affect the control flow of the automaton; thus MPTA lie between timed and hybrid automata in expressiveness. Previous work considered observers with non-negative slope in every location. In this paper we treat observers that have both positive and negative rates. Our main result is an algorithm to decide a gap version of the reachability problem for this variant of MPTA. We translate the gap reachability problem into a gap satisfiability problem for mixed integer-real systems of nonlinear constraints. Our main technical contribution -- a result of independent interest -- is a procedure to solve such contraints via a combination of branch-and-bound and relaxation-and-rounding.
{"title":"Reachability for Multi-Priced Timed Automata with Positive and Negative Rates","authors":"Andrew Scoones, Mahsa Shirmohammadi, James Worrell","doi":"arxiv-2407.18131","DOIUrl":"https://doi.org/arxiv-2407.18131","url":null,"abstract":"Multi-priced timed automata (MPTA) are timed automata with observer variables whose derivatives can change from one location to another. Observers are write-only variables, that is, they do not affect the control flow of the automaton; thus MPTA lie between timed and hybrid automata in expressiveness. Previous work considered observers with non-negative slope in every location. In this paper we treat observers that have both positive and negative rates. Our main result is an algorithm to decide a gap version of the reachability problem for this variant of MPTA. We translate the gap reachability problem into a gap satisfiability problem for mixed integer-real systems of nonlinear constraints. Our main technical contribution -- a result of independent interest -- is a procedure to solve such contraints via a combination of branch-and-bound and relaxation-and-rounding.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771875","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}
Antonio Casares, Olivier Idir, Denis Kuperberg, Corto Mascle, Aditya Prakash
We present a polynomial-time algorithm minimising the number of states of history-deterministic generalised coB"uchi automata, building on the work of Abu Radi and Kupferman on coB"uchi automata. On the other hand, we establish that the minimisation problem for both deterministic and history-deterministic generalised B"uchi automata is NP-complete, as well as the problem of minimising at the same time the number of states and colours of history-deterministic generalised coB"uchi automata.
{"title":"On the Minimisation of Deterministic and History-Deterministic Generalised (co)Büchi Automata","authors":"Antonio Casares, Olivier Idir, Denis Kuperberg, Corto Mascle, Aditya Prakash","doi":"arxiv-2407.18090","DOIUrl":"https://doi.org/arxiv-2407.18090","url":null,"abstract":"We present a polynomial-time algorithm minimising the number of states of\u0000history-deterministic generalised coB\"uchi automata, building on the work of\u0000Abu Radi and Kupferman on coB\"uchi automata. On the other hand, we establish\u0000that the minimisation problem for both deterministic and history-deterministic\u0000generalised B\"uchi automata is NP-complete, as well as the problem of\u0000minimising at the same time the number of states and colours of\u0000history-deterministic generalised coB\"uchi automata.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785177","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}
$Omega$-automata and Wilke algebras are formalisms for characterising $omega$-regular languages via their ultimately periodic words. $Omega$-automata read finite representations of ultimately periodic words, called lassos, and they are a subclass of lasso automata. We introduce lasso semigroups as a generalisation of Wilke algebras that mirrors how lasso automata generalise $Omega$-automata, and we show that finite lasso semigroups characterise regular lasso languages. We then show a dual adjunction between lasso automata and quotients of the free lasso semigroup with a recognising set, and as our main result we show that this dual adjunction restricts to one between $Omega$-automata and quotients of the free Wilke algebra with a recognising set.
{"title":"Dual Adjunction Between $Ω$-Automata and Wilke Algebra Quotients","authors":"Anton Chernev, Helle Hvid Hansen, Clemens Kupke","doi":"arxiv-2407.14115","DOIUrl":"https://doi.org/arxiv-2407.14115","url":null,"abstract":"$Omega$-automata and Wilke algebras are formalisms for characterising\u0000$omega$-regular languages via their ultimately periodic words.\u0000$Omega$-automata read finite representations of ultimately periodic words,\u0000called lassos, and they are a subclass of lasso automata. We introduce lasso\u0000semigroups as a generalisation of Wilke algebras that mirrors how lasso\u0000automata generalise $Omega$-automata, and we show that finite lasso semigroups\u0000characterise regular lasso languages. We then show a dual adjunction between\u0000lasso automata and quotients of the free lasso semigroup with a recognising\u0000set, and as our main result we show that this dual adjunction restricts to one\u0000between $Omega$-automata and quotients of the free Wilke algebra with a\u0000recognising set.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746286","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}
Obliging games have been introduced in the context of the game perspective on reactive synthesis in order to enforce a degree of cooperation between the to-be-synthesized system and the environment. Previous approaches to the analysis of obliging games have been small-step in the sense that they have been based on a reduction to standard (non-obliging) games in which single moves correspond to single moves in the original (obliging) game. Here, we propose a novel, large-step view on obliging games, reducing them to standard games in which single moves encode long-term behaviors in the original game. This not only allows us to give a meaningful definition of the environment winning in obliging games, but also leads to significantly improved bounds on both strategy sizes and the solution runtime for obliging games.
{"title":"Faster and Smaller Solutions of Obliging Games","authors":"Daniel Hausmann, Nir Piterman","doi":"arxiv-2407.11856","DOIUrl":"https://doi.org/arxiv-2407.11856","url":null,"abstract":"Obliging games have been introduced in the context of the game perspective on\u0000reactive synthesis in order to enforce a degree of cooperation between the\u0000to-be-synthesized system and the environment. Previous approaches to the\u0000analysis of obliging games have been small-step in the sense that they have\u0000been based on a reduction to standard (non-obliging) games in which single\u0000moves correspond to single moves in the original (obliging) game. Here, we\u0000propose a novel, large-step view on obliging games, reducing them to standard\u0000games in which single moves encode long-term behaviors in the original game.\u0000This not only allows us to give a meaningful definition of the environment\u0000winning in obliging games, but also leads to significantly improved bounds on\u0000both strategy sizes and the solution runtime for obliging games.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718844","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}
This paper investigates the state complexities of subword-closed and superword-closed languages, comparing them to regular languages. We focus on the square root operator and the substitution operator. We establish an exponential lower bound for superword-closed languages for the k-th root. For subword-closed languages we analyze in detail a specific instance of the square root problem for which a quadratic complexity is proven. For the substitution operator, we show an exponential lower bound for the general substitution. We then find some conditions for which we prove a quadratic upper bound.
{"title":"On state complexity for subword-closed languages","authors":"Jérôme Guyot","doi":"arxiv-2407.10355","DOIUrl":"https://doi.org/arxiv-2407.10355","url":null,"abstract":"This paper investigates the state complexities of subword-closed and\u0000superword-closed languages, comparing them to regular languages. We focus on\u0000the square root operator and the substitution operator. We establish an\u0000exponential lower bound for superword-closed languages for the k-th root. For\u0000subword-closed languages we analyze in detail a specific instance of the square\u0000root problem for which a quadratic complexity is proven. For the substitution\u0000operator, we show an exponential lower bound for the general substitution. We\u0000then find some conditions for which we prove a quadratic upper bound.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718842","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 this paper we examine the difficulty of finding an equivalent deterministic automaton when confronted with a non-deterministic one. While for some automata the exponential blow-up in their number of states is unavoidable, we show that in general, any approximation of state complexity with polynomial precision remains PSPACE-hard. The same is true when using the subset construction to determinize the NFA, meaning that it is PSPACE-hard to predict whether subset construction will produce an exponential ''blow-up'' in the number of states or not. To give an explanation for its behaviour, we propose the notion of subset complexity, which serves as an upper bound on the size of subset construction. Due to it simple and intuitive nature it allows to identify large classes of automata which can have limited non-determinism and completely avoid the ''blow-up''. Subset complexity also remains invariant under NFA reversal and allows to predict how the introduction or removal of transitions from the NFA will affect its size.
{"title":"Blow-up in Non-Deterministic Automata","authors":"Ivan Baburin, Ryan Cotterell","doi":"arxiv-2407.09891","DOIUrl":"https://doi.org/arxiv-2407.09891","url":null,"abstract":"In this paper we examine the difficulty of finding an equivalent\u0000deterministic automaton when confronted with a non-deterministic one. While for\u0000some automata the exponential blow-up in their number of states is unavoidable,\u0000we show that in general, any approximation of state complexity with polynomial\u0000precision remains PSPACE-hard. The same is true when using the subset\u0000construction to determinize the NFA, meaning that it is PSPACE-hard to predict\u0000whether subset construction will produce an exponential ''blow-up'' in the\u0000number of states or not. To give an explanation for its behaviour, we propose\u0000the notion of subset complexity, which serves as an upper bound on the size of\u0000subset construction. Due to it simple and intuitive nature it allows to\u0000identify large classes of automata which can have limited non-determinism and\u0000completely avoid the ''blow-up''. Subset complexity also remains invariant\u0000under NFA reversal and allows to predict how the introduction or removal of\u0000transitions from the NFA will affect its size.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718843","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}
Active learning of finite automata has been vigorously pursued for the purposes of analysis and explanation of black-box systems. In this paper, we study an L*-style learning algorithm for weighted automata over the max-plus semiring. The max-plus setting exposes a "consistency" issue in the previously studied semiring-generic extension of L*: we show that it can fail to maintain consistency of tables, and can thus make equivalence queries on obviously wrong hypothesis automata. We present a theoretical fix by a mathematically clean notion of column-closedness. We also present a nontrivial and reasonably broad class of weighted languages over the max-plus semiring in which our algorithm terminates.
{"title":"Learning Weighted Finite Automata over the Max-Plus Semiring and its Termination","authors":"Takamasa Okudono, Masaki Waga, Taro Sekiyama, Ichiro Hasuo","doi":"arxiv-2407.09775","DOIUrl":"https://doi.org/arxiv-2407.09775","url":null,"abstract":"Active learning of finite automata has been vigorously pursued for the\u0000purposes of analysis and explanation of black-box systems. In this paper, we\u0000study an L*-style learning algorithm for weighted automata over the max-plus\u0000semiring. The max-plus setting exposes a \"consistency\" issue in the previously\u0000studied semiring-generic extension of L*: we show that it can fail to maintain\u0000consistency of tables, and can thus make equivalence queries on obviously wrong\u0000hypothesis automata. We present a theoretical fix by a mathematically clean\u0000notion of column-closedness. We also present a nontrivial and reasonably broad\u0000class of weighted languages over the max-plus semiring in which our algorithm\u0000terminates.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718846","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}
Rida Ait El Manssour, Vincent Cheval, Mahsa Shirmohammadi, James Worrell
In this paper we introduce holonomic tree automata: a common extension of weighted tree automata and holonomic recurrences. We show that the generating function of the tree series represented by such an automaton is differentially algebraic. Conversely, we give an algorithm that inputs a differentially algebraic power series, represented as a solution of a rational dynamical system, and outputs an automaton whose generating function is the given series. Such an automaton yields a recurrence that can be used to compute the terms of the power series. We use the algorithm to obtain automaton representations of exponential generating functions of families of combinatorial objects given as combinatorial species. Using techniques from differential algebra, we show that it is decidable both whether two automata represent the same formal tree series and whether they have the same generating function.
{"title":"On Tree Automata, Generating Functions, and Differential Equations","authors":"Rida Ait El Manssour, Vincent Cheval, Mahsa Shirmohammadi, James Worrell","doi":"arxiv-2407.08218","DOIUrl":"https://doi.org/arxiv-2407.08218","url":null,"abstract":"In this paper we introduce holonomic tree automata: a common extension of\u0000weighted tree automata and holonomic recurrences. We show that the generating\u0000function of the tree series represented by such an automaton is differentially\u0000algebraic. Conversely, we give an algorithm that inputs a differentially\u0000algebraic power series, represented as a solution of a rational dynamical\u0000system, and outputs an automaton whose generating function is the given series.\u0000Such an automaton yields a recurrence that can be used to compute the terms of\u0000the power series. We use the algorithm to obtain automaton representations of\u0000exponential generating functions of families of combinatorial objects given as\u0000combinatorial species. Using techniques from differential algebra, we show that\u0000it is decidable both whether two automata represent the same formal tree series\u0000and whether they have the same generating function.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614280","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}
The translation of Metric Interval Temporal Logic (MITL) to timed automata is a topic that has been extensively studied. A key challenge here is the conversion of future modalities into equivalent automata. Typical conversions equip the automata with a guess-and-check mechanism to ascertain the truth of future modalities. Guess-and-check can be naturally implemented via alternation. However, since timed automata tools do not handle alternation, existing methods perform an additional step of converting the alternating timed automata into timed automata. This de-alternation step proceeds by an intricate finite abstraction of the space of configurations of the alternating automaton. Recently, a model of generalized timed automata (GTA) has been proposed. The model comes with several powerful additional features, and yet, the best known zone-based reachability algorithms for timed automata have been extended to the GTA model, with the same complexity for all the zone operations. We provide a new concise translation from MITL to GTA. In particular, for the timed until modality, our translation offers an exponential improvement w.r.t. the state-of-the-art. Thanks to this conversion, MITL model checking reduces to checking liveness for GTAs. However, no liveness algorithm is known for GTAs. Due to the presence of future clocks, there is no finite time-abstract bisimulation (region equivalence) for GTAs, whereas liveness algorithms for timed automata crucially rely on the presence of the finite region equivalence. As our second contribution, we provide a new zone-based algorithm for checking Buchi non-emptiness in GTAs, which circumvents this fundamental challenge.
{"title":"MITL Model Checking via Generalized Timed Automata and a New Liveness Algorithm","authors":"S. Akshay, Paul Gastin, R. Govind, B. Srivathsan","doi":"arxiv-2407.08452","DOIUrl":"https://doi.org/arxiv-2407.08452","url":null,"abstract":"The translation of Metric Interval Temporal Logic (MITL) to timed automata is\u0000a topic that has been extensively studied. A key challenge here is the\u0000conversion of future modalities into equivalent automata. Typical conversions\u0000equip the automata with a guess-and-check mechanism to ascertain the truth of\u0000future modalities. Guess-and-check can be naturally implemented via\u0000alternation. However, since timed automata tools do not handle alternation,\u0000existing methods perform an additional step of converting the alternating timed\u0000automata into timed automata. This de-alternation step proceeds by an intricate\u0000finite abstraction of the space of configurations of the alternating automaton. Recently, a model of generalized timed automata (GTA) has been proposed. The\u0000model comes with several powerful additional features, and yet, the best known\u0000zone-based reachability algorithms for timed automata have been extended to the\u0000GTA model, with the same complexity for all the zone operations. We provide a\u0000new concise translation from MITL to GTA. In particular, for the timed until\u0000modality, our translation offers an exponential improvement w.r.t. the\u0000state-of-the-art. Thanks to this conversion, MITL model checking reduces to checking liveness\u0000for GTAs. However, no liveness algorithm is known for GTAs. Due to the presence\u0000of future clocks, there is no finite time-abstract bisimulation (region\u0000equivalence) for GTAs, whereas liveness algorithms for timed automata crucially\u0000rely on the presence of the finite region equivalence. As our second\u0000contribution, we provide a new zone-based algorithm for checking Buchi\u0000non-emptiness in GTAs, which circumvents this fundamental challenge.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614301","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}
For any synchronizing $n$-state deterministic automaton, v{C}ern'{y} conjectures the existence of a synchronizing word of length at most $(n-1)^2$. We prove that there exists a synchronizing word of length at most $2n^2 - 7n + 7$ for every synchronizing $n$-state deterministic automaton that satisfies the following two properties: 1. The image of the action of each letter contains at least $n-1$ states; 2. The actions of bijective letters generate a transitive permutation group on the state set.
{"title":"A quadratic upper bound on the reset thresholds of synchronizing automata containing a transitive permutation group","authors":"Yinfeng Zhu","doi":"arxiv-2407.08135","DOIUrl":"https://doi.org/arxiv-2407.08135","url":null,"abstract":"For any synchronizing $n$-state deterministic automaton, v{C}ern'{y}\u0000conjectures the existence of a synchronizing word of length at most $(n-1)^2$.\u0000We prove that there exists a synchronizing word of length at most $2n^2 - 7n +\u00007$ for every synchronizing $n$-state deterministic automaton that satisfies the\u0000following two properties: 1. The image of the action of each letter contains at\u0000least $n-1$ states; 2. The actions of bijective letters generate a transitive\u0000permutation group on the state set.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614302","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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