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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

We introduce an algorithmic framework to decide whether inclusion holds between languages of infinite words over a finite alphabet. Our approach falls within the class of Ramsey-based methods and relies on a least fixpoint characterization of ω-languages leveraging ultimately periodic infinite words of type uv^ω, with u a finite prefix and v a finite period of an infinite word. We put forward an inclusion checking algorithm between Büchi automata, called BAInc, designed as a complete abstract interpretation using a pair of well-quasiorders on finite words. BAInc is quite simple: it consists of two least fixpoint computations (one for prefixes and the other for periods) manipulating finite sets (of pairs) of states compared by set inclusion, so that language inclusion holds when the sets (of pairs) of states of the fixpoints satisfy some basic conditions. We implemented BAInc in a tool called BAIT that we experimentally evaluated against the state-of-the-art. We gathered, in addition to existing benchmarks, a large number of new case studies stemming from program verification and word combinatorics, thereby significantly expanding both the scope and size of the available benchmark set. Our experimental results show that BAIT advances the state-of-the-art on an overwhelming majority of these benchmarks. Finally, we demonstrate the generality of our algorithmic framework by instantiating it to the inclusion problem of Büchi pushdown automata into Büchi automata.

Kyveli Doveri, Pierre Ganty, Francesco Parolini, and Francesco Ranzato. Inclusion Testing of Büchi Automata Based on Well-Quasiorders. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{doveri_et_al:LIPIcs.CONCUR.2021.3, author = {Doveri, Kyveli and Ganty, Pierre and Parolini, Francesco and Ranzato, Francesco}, title = {{Inclusion Testing of B\"{u}chi Automata Based on Well-Quasiorders}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {3:1--3:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.3}, URN = {urn:nbn:de:0030-drops-143802}, doi = {10.4230/LIPIcs.CONCUR.2021.3}, annote = {Keywords: B\"{u}chi (Pushdown) Automata, \omega-Language Inclusion, Well-quasiorders} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

In this work, we define a framework of automata constructions based on quasiorders over words to provide new insights on the class of residual automata. We present a new residualization operation and a generalized double-reversal method for building the canonical residual automaton for a given language. Finally, we use our framework to offer a quasiorder-based perspective on NL^*, an online learning algorithm for residual automata. We conclude that quasiorders are fundamental to residual automata as congruences are to deterministic automata.

Pierre Ganty, Elena Gutiérrez, and Pedro Valero. A Quasiorder-Based Perspective on Residual Automata. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ganty_et_al:LIPIcs.MFCS.2020.40, author = {Ganty, Pierre and Guti\'{e}rrez, Elena and Valero, Pedro}, title = {{A Quasiorder-Based Perspective on Residual Automata}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {40:1--40:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.40}, URN = {urn:nbn:de:0030-drops-127071}, doi = {10.4230/LIPIcs.MFCS.2020.40}, annote = {Keywords: Residual Automata, Quasiorders, Double-Reversal Method, Canonical RFA, Regular Languages} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

In this work we use a framework of finite-state automata constructions based on equivalences over words to provide new insights on the relation between well-known methods for computing the minimal deterministic automaton of a language.

Pierre Ganty, Elena Gutiérrez, and Pedro Valero. A Congruence-based Perspective on Automata Minimization Algorithms. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ganty_et_al:LIPIcs.MFCS.2019.77, author = {Ganty, Pierre and Guti\'{e}rrez, Elena and Valero, Pedro}, title = {{A Congruence-based Perspective on Automata Minimization Algorithms}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {77:1--77:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.77}, URN = {urn:nbn:de:0030-drops-110214}, doi = {10.4230/LIPIcs.MFCS.2019.77}, annote = {Keywords: Double-Reversal Method, Minimization, Automata, Congruences, Regular Languages} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Parikh's Theorem states that every context-free grammar (CFG) is equivalent to some regular CFG when the ordering of symbols in the words is ignored. The same is not true for the so-called weighted CFGs, which additionally assign a weight to each grammar rule. If the result holds for a given weighted CFG G, we say that G satisfies the Parikh property. We prove constructively that the Parikh property holds for every weighted nonexpansive CFG. We also give a decision procedure for the property when the weights are over the rationals.

Pierre Ganty and Elena Gutiérrez. The Parikh Property for Weighted Context-Free Grammars. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ganty_et_al:LIPIcs.FSTTCS.2018.32, author = {Ganty, Pierre and Guti\'{e}rrez, Elena}, title = {{The Parikh Property for Weighted Context-Free Grammars}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {32:1--32:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.32}, URN = {urn:nbn:de:0030-drops-99315}, doi = {10.4230/LIPIcs.FSTTCS.2018.32}, annote = {Keywords: Weighted Context-Free Grammars, Algebraic Language Theory, Parikh Image} }

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**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

Population protocols (Angluin et al., PODC, 2004) are a formal model of sensor networks consisting of identical mobile devices. Two devices can interact and thereby change their states. Computations are infinite sequences of interactions satisfying a strong fairness constraint.
A population protocol is well-specified if for every initial configuration C of devices, and every computation starting at C, all devices eventually agree on a consensus value depending only on C. If a protocol is well-specified, then it is said to compute the predicate that assigns to each initial configuration its consensus value.
In a previous paper we have shown that the problem whether a given protocol is well-specified and the problem whether it computes a given predicate are decidable. However, in the same paper we prove that both problems are at least as hard as the reachability problem for Petri nets. Since all known algorithms for Petri net reachability have non-primitive recursive complexity, in this paper we restrict attention to immediate observation (IO) population protocols, a class introduced and studied in (Angluin et al., PODC, 2006). We show that both problems are solvable in exponential space for IO protocols. This is the first syntactically defined, interesting class of protocols for which an algorithm not requiring Petri net reachability is found.

Javier Esparza, Pierre Ganty, Rupak Majumdar, and Chana Weil-Kennedy. Verification of Immediate Observation Population Protocols. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{esparza_et_al:LIPIcs.CONCUR.2018.31, author = {Esparza, Javier and Ganty, Pierre and Majumdar, Rupak and Weil-Kennedy, Chana}, title = {{Verification of Immediate Observation Population Protocols}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.31}, URN = {urn:nbn:de:0030-drops-95695}, doi = {10.4230/LIPIcs.CONCUR.2018.31}, annote = {Keywords: Population protocols, Immediate Observation, Parametrized verification} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Population protocols are a model for parameterized systems in which a set of identical, anonymous, finite-state processes interact pairwise through rendezvous synchronization. In each step, the pair of interacting processes is chosen by a random scheduler. Angluin et al. (PODC 2004) studied population protocols as a distributed computation model. They characterized the computational power in the limit (semi-linear predicates) of a subclass of protocols (the well-specified ones). However, the modeling power of protocols go beyond computation of semi-linear predicates and they can be used to study a wide range of distributed protocols, such as asynchronous leader election or consensus, stochastic evolutionary processes, or chemical reaction networks. Correspondingly, one is interested in checking specifications on these protocols that go beyond the well-specified computation of predicates.
In this paper, we characterize the decidability frontier for the model checking problem for population protocols against probabilistic linear-time specifications. We show that the model checking problem is decidable for qualitative objectives, but as hard as the reachability problem for Petri nets - a well-known hard problem without known elementary algorithms. On the other hand, model checking is undecidable for quantitative properties.

Javier Esparza, Pierre Ganty, Jérôme Leroux, and Rupak Majumdar. Model Checking Population Protocols. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{esparza_et_al:LIPIcs.FSTTCS.2016.27, author = {Esparza, Javier and Ganty, Pierre and Leroux, J\'{e}r\^{o}me and Majumdar, Rupak}, title = {{Model Checking Population Protocols}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.27}, URN = {urn:nbn:de:0030-drops-68628}, doi = {10.4230/LIPIcs.FSTTCS.2016.27}, annote = {Keywords: parameterized systems, population protocols, probabilistic model checking, probabilistic linear-time specifications, decidability} }

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**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

Population protocols [Angluin et al., PODC, 2004] are a formal model of sensor networks consisting of identical mobile devices. Two devices can interact and thereby change their states. Computations are infinite sequences of interactions satisfying a strong fairness constraint.
A population protocol is well-specified if for every initial configuration C of devices, and every computation starting at C, all devices eventually agree on a consensus value depending only on C. If a protocol is well-specified, then it is said to compute the predicate that assigns to each initial configuration its consensus value.
While the predicates computable by well-specified protocols have been extensively studied, the two basic verification problems remain open: is a given protocol well-specified? Does a protocol compute a given predicate? We prove that both problems are decidable. Our results also prove decidability of a natural question about home spaces of Petri nets.

Javier Esparza, Pierre Ganty, Jérôme Leroux, and Rupak Majumdar. Verification of Population Protocols. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 470-482, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{esparza_et_al:LIPIcs.CONCUR.2015.470, author = {Esparza, Javier and Ganty, Pierre and Leroux, J\'{e}r\^{o}me and Majumdar, Rupak}, title = {{Verification of Population Protocols}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {470--482}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.470}, URN = {urn:nbn:de:0030-drops-53770}, doi = {10.4230/LIPIcs.CONCUR.2015.470}, annote = {Keywords: Population protocols, Petri nets, parametrized verification} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We investigate the problem asking whether the intersection of a context-free language (CFL) and a Petri net language (PNL) (with reachability as acceptance condition) is empty. Our contribution to solve this long-standing problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data domain, is to identify a class of CFLs called the finite-index CFLs for which the problem is decidable. The k-index approximation of a CFL can be obtained by discarding all the words that cannot be derived within a budget k on the number of occurrences of non-terminals. A finite-index CFL is thus a CFL which coincides with its k-index approximation for some k. We decide whether the intersection of a finite-index CFL and a PNL is empty by reducing it to the reachability problem of Petri nets with weak inhibitor arcs, a class of systems with infinitely many states for which reachability is known to be decidable. Conversely, we show that the reachability problem for a Petri net with weak inhibitor arcs reduces to the emptiness problem of a finite-index CFL intersected with a PNL.

Mohamed Faouzi Atig and Pierre Ganty. Approximating Petri Net Reachability Along Context-free Traces. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 152-163, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{atig_et_al:LIPIcs.FSTTCS.2011.152, author = {Atig, Mohamed Faouzi and Ganty, Pierre}, title = {{Approximating Petri Net Reachability Along Context-free Traces}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {152--163}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.152}, URN = {urn:nbn:de:0030-drops-33448}, doi = {10.4230/LIPIcs.FSTTCS.2011.152}, annote = {Keywords: Petri nets, Context-free Grammars, Reachability Problem} }