Document

**Published in:** LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

We study the reachability problem for one-counter automata in which transitions can carry disequality tests. A disequality test is a guard that prohibits a specified counter value. This reachability problem has been known to be NP-hard and in PSPACE, and characterising its computational complexity has been left as a challenging open question by Almagor, Cohen, Pérez, Shirmohammadi, and Worrell (2020). We reduce the complexity gap, placing the problem into the second level of the polynomial hierarchy, namely into the class coNP^NP. In the presence of both equality and disequality tests, our upper bound is at the third level, P^NP^NP.
To prove this result, we show that non-reachability can be witnessed by a pair of invariants (forward and backward). These invariants are almost inductive. They aim to over-approximate only a "core" of the reachability set instead of the entire set. The invariants are also leaky: it is possible to escape the set. We complement this with separate checks as the leaks can only occur in a controlled way.

Dmitry Chistikov, Jérôme Leroux, Henry Sinclair-Banks, and Nicolas Waldburger. Invariants for One-Counter Automata with Disequality Tests. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chistikov_et_al:LIPIcs.CONCUR.2024.17, author = {Chistikov, Dmitry and Leroux, J\'{e}r\^{o}me and Sinclair-Banks, Henry and Waldburger, Nicolas}, title = {{Invariants for One-Counter Automata with Disequality Tests}}, booktitle = {35th International Conference on Concurrency Theory (CONCUR 2024)}, pages = {17:1--17:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-339-3}, ISSN = {1868-8969}, year = {2024}, volume = {311}, editor = {Majumdar, Rupak and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2024.17}, URN = {urn:nbn:de:0030-drops-207898}, doi = {10.4230/LIPIcs.CONCUR.2024.17}, annote = {Keywords: Inductive invariant, Vector addition system, One-counter automaton} }

Document

**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

We investigate the dimension-parametric complexity of the reachability problem in vector addition systems with states (VASS) and its extension with pushdown stack (pushdown VASS). Up to now, the problem is known to be F_d-hard for VASS of dimension 3d+2 (the complexity class F_d corresponds to the kth level of the fast-growing hierarchy), and no essentially better bound is known for pushdown VASS. We provide a new construction that improves the lower bound for VASS: F_d-hardness in dimension 2d+3. Furthermore, building on our new insights we show a new lower bound for pushdown VASS: F_d-hardness in dimension d/2 + 6. This dimension-parametric lower bound is strictly stronger than the upper bound for VASS, which suggests that the (still unknown) complexity of the reachability problem in pushdown VASS is higher than in plain VASS (where it is Ackermann-complete).

Wojciech Czerwiński, Ismaël Jecker, Sławomir Lasota, Jérôme Leroux, and Łukasz Orlikowski. New Lower Bounds for Reachability in Vector Addition Systems. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 35:1-35:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{czerwinski_et_al:LIPIcs.FSTTCS.2023.35, author = {Czerwi\'{n}ski, Wojciech and Jecker, Isma\"{e}l and Lasota, S{\l}awomir and Leroux, J\'{e}r\^{o}me and Orlikowski, {\L}ukasz}, title = {{New Lower Bounds for Reachability in Vector Addition Systems}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {35:1--35:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.35}, URN = {urn:nbn:de:0030-drops-194088}, doi = {10.4230/LIPIcs.FSTTCS.2023.35}, annote = {Keywords: vector addition systems, reachability problem, pushdown vector addition system, lower bounds} }

Document

**Published in:** LIPIcs, Volume 279, 34th International Conference on Concurrency Theory (CONCUR 2023)

A set of configurations H is a home-space for a set of configurations X of a Petri net if every configuration reachable from (any configuration in) X can reach (some configuration in) H. The semilinear home-space problem for Petri nets asks, given a Petri net and semilinear sets of configurations X, H, if H is a home-space for X. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when X is a singleton and H is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. In particular, we prove that for any Petri net and any linear set of configurations L we can effectively compute a semilinear set C of configurations, called a non-reachability core for L, such that for every set X the set L is not a home-space for X if, and only if, C is reachable from X. We show that the established relation to the reachability problem yields the Ackermann-completeness of the (semilinear) home-space problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time.

Petr Jančar and Jérôme Leroux. The Semilinear Home-Space Problem Is Ackermann-Complete for Petri Nets. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jancar_et_al:LIPIcs.CONCUR.2023.36, author = {Jan\v{c}ar, Petr and Leroux, J\'{e}r\^{o}me}, title = {{The Semilinear Home-Space Problem Is Ackermann-Complete for Petri Nets}}, booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-299-0}, ISSN = {1868-8969}, year = {2023}, volume = {279}, editor = {P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.36}, URN = {urn:nbn:de:0030-drops-190300}, doi = {10.4230/LIPIcs.CONCUR.2023.36}, annote = {Keywords: Petri nets, home-space property, semilinear sets, Ackermannian complexity} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

LIPIcs, Volume 272, MFCS 2023, Complete Volume

48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 1-1302, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{leroux_et_al:LIPIcs.MFCS.2023, title = {{LIPIcs, Volume 272, MFCS 2023, Complete Volume}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {1--1302}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023}, URN = {urn:nbn:de:0030-drops-185332}, doi = {10.4230/LIPIcs.MFCS.2023}, annote = {Keywords: LIPIcs, Volume 272, MFCS 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Front Matter, Table of Contents, Preface, Conference Organization

48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{leroux_et_al:LIPIcs.MFCS.2023.0, author = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.0}, URN = {urn:nbn:de:0030-drops-185349}, doi = {10.4230/LIPIcs.MFCS.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

Vector addition system with states is an ubiquitous model of computation with extensive applications in computer science. The reachability problem for vector addition systems is central since many other problems reduce to that question. The problem is decidable and it was recently proved that the dimension of the vector addition system is an important parameter of the complexity. In fixed dimensions larger than two, the complexity is not known (with huge complexity gaps). In dimension two, the reachability problem was shown to be PSPACE-complete by Blondin et al. in 2015. We consider an extension of this model, called 2-TVASS, where the first counter can be tested for zero. This model naturally extends the classical model of one counter automata (OCA). We show that reachability is still solvable in polynomial space for 2-TVASS. As in the work Blondin et al., our approach relies on the existence of small reachability certificates obtained by concatenating polynomially many cycles.

Jérôme Leroux and Grégoire Sutre. Reachability in Two-Dimensional Vector Addition Systems with States: One Test Is for Free. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{leroux_et_al:LIPIcs.CONCUR.2020.37, author = {Leroux, J\'{e}r\^{o}me and Sutre, Gr\'{e}goire}, title = {{Reachability in Two-Dimensional Vector Addition Systems with States: One Test Is for Free}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {37:1--37:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.37}, URN = {urn:nbn:de:0030-drops-128498}, doi = {10.4230/LIPIcs.CONCUR.2020.37}, annote = {Keywords: Counter machine, Vector addition system, Reachability problem, Formal verification, Infinite-state system} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

The reachability problem is a central decision problem in verification of vector addition systems with states (VASS). In spite of recent progress, the complexity of the reachability problem remains unsettled, and it is closely related to the lengths of shortest VASS runs that witness reachability.
We obtain three main results for VASS of fixed dimension. For the first two, we assume that the integers in the input are given in unary, and that the control graph of the given VASS is flat (i.e., without nested cycles). We obtain a family of VASS in dimension 3 whose shortest runs are exponential, and we show that the reachability problem is NP-hard in dimension 7. These results resolve negatively questions that had been posed by the works of Blondin et al. in LICS 2015 and Englert et al. in LICS 2016, and contribute a first construction that distinguishes 3-dimensional flat VASS from 2-dimensional ones. Our third result, by means of a novel family of products of integer fractions, shows that 4-dimensional VASS can have doubly exponentially long shortest runs. The smallest dimension for which this was previously known is 14.

Wojciech Czerwiński, Sławomir Lasota, Ranko Lazić, Jérôme Leroux, and Filip Mazowiecki. Reachability in Fixed Dimension Vector Addition Systems with States. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{czerwinski_et_al:LIPIcs.CONCUR.2020.48, author = {Czerwi\'{n}ski, Wojciech and Lasota, S{\l}awomir and Lazi\'{c}, Ranko and Leroux, J\'{e}r\^{o}me and Mazowiecki, Filip}, title = {{Reachability in Fixed Dimension Vector Addition Systems with States}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {48:1--48:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.48}, URN = {urn:nbn:de:0030-drops-128605}, doi = {10.4230/LIPIcs.CONCUR.2020.48}, annote = {Keywords: reachability problem, vector addition systems, Petri nets} }

Document

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Petri nets are a classical model of concurrency widely used and studied in formal verification with many applications in modeling and analyzing hardware and software, data bases, and reactive systems. The reachability problem is central since many other problems reduce to reachability questions. In 2011, we proved that a variant of the reachability problem, called the reversible reachability problem is exponential-space complete. Recently, this problem found several unexpected applications in particular in the theory of population protocols. In this paper we revisit the reversible reachability problem in order to prove that the minimal distance in the reachability graph of two mutually reachable configurations is linear with respect to the Euclidean distance between those two configurations.

Jérôme Leroux. Distance Between Mutually Reachable Petri Net Configurations. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{leroux:LIPIcs.FSTTCS.2019.47, author = {Leroux, J\'{e}r\^{o}me}, title = {{Distance Between Mutually Reachable Petri Net Configurations}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.47}, URN = {urn:nbn:de:0030-drops-116094}, doi = {10.4230/LIPIcs.FSTTCS.2019.47}, annote = {Keywords: Petri nets, Vector addition systems, Formal verification, Reachability problem} }

Document

Invited Talk

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

Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. In this presentation, we overview decidability and complexity results over the last fifty years about the Petri net reachability problem.

Jérôme Leroux. Petri Net Reachability Problem (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 5:1-5:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{leroux:LIPIcs.MFCS.2019.5, author = {Leroux, J\'{e}r\^{o}me}, title = {{Petri Net Reachability Problem}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {5:1--5:3}, 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.2019.5}, URN = {urn:nbn:de:0030-drops-109493}, doi = {10.4230/LIPIcs.MFCS.2019.5}, annote = {Keywords: Petri net, Reachability problem, Formal verification, Concurrency} }

Document

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

We prove that the reachability relation of two-counter machines with one zero-test and one reset is Presburger-definable and effectively computable. Our proof is based on the introduction of two classes of Presburger-definable relations effectively stable by transitive closure. This approach generalizes and simplifies the existing different proofs and it solves an open problem introduced by Finkel and Sutre in 2000.

Alain Finkel, Jérôme Leroux, and Grégoire Sutre. Reachability for Two-Counter Machines with One Test and One Reset. 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. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{finkel_et_al:LIPIcs.FSTTCS.2018.31, author = {Finkel, Alain and Leroux, J\'{e}r\^{o}me and Sutre, Gr\'{e}goire}, title = {{Reachability for Two-Counter Machines with One Test and One Reset}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {31:1--31:14}, 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.2018.31}, URN = {urn:nbn:de:0030-drops-99305}, doi = {10.4230/LIPIcs.FSTTCS.2018.31}, annote = {Keywords: Counter machine, Vector addition system, Reachability problem, Formal verification, Presburger arithmetic, Infinite-state system} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The reachability problem for vector addition systems is one of the most difficult and central problems in theoretical computer science. The problem is known to be decidable, but despite intense investigation during the last four decades, the exact complexity is still open. For some sub-classes, the complexity of the reachability problem is known. Structurally bounded vector addition systems, the class of vector addition systems with finite reachability sets from any initial configuration, is one of those classes. In fact, the reachability problem was shown to be polynomial-space complete for that class by Praveen and Lodaya in 2008. Surprisingly, extending this property to vector addition systems with states is open. In fact, there exist vector addition systems with states that are structurally bounded but with Ackermannian large sets of reachable configurations. It follows that the reachability problem for that class is between exponential space and Ackermannian. In this paper we introduce the class of polynomial vector addition systems with states, defined as the class of vector addition systems with states with size of reachable configurations bounded polynomially in the size of the initial ones. We prove that the reachability problem for polynomial vector addition systems is exponential-space complete. Additionally, we show that we can decide in polynomial time if a vector addition system with states is polynomial. This characterization introduces the notion of iteration scheme with potential applications to the reachability problem for general vector addition systems.

Jérôme Leroux. Polynomial Vector Addition Systems With States. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 134:1-134:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{leroux:LIPIcs.ICALP.2018.134, author = {Leroux, J\'{e}r\^{o}me}, title = {{Polynomial Vector Addition Systems With States}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {134:1--134:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.134}, URN = {urn:nbn:de:0030-drops-91387}, doi = {10.4230/LIPIcs.ICALP.2018.134}, annote = {Keywords: Vector additions system with states, Reachability problem, Formal verification, Infinite state system, Linear algebra, Kosaraju-Sullivan algorithm} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Whether the reachability problem for branching vector addition systems, or equivalently the provability problem for multiplicative exponential linear logic, is decidable has been a long-standing open question. The one-dimensional case is a generalisation of the extensively studied one-counter nets, and it was recently established polynomial-time complete provided counter updates are given in unary. Our main contribution is to determine the complexity when the encoding is binary: polynomial-space complete.

Diego Figueira, Ranko Lazic, Jérôme Leroux, Filip Mazowiecki, and Grégoire Sutre. Polynomial-Space Completeness of Reachability for Succinct Branching VASS in Dimension One. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 119:1-119:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{figueira_et_al:LIPIcs.ICALP.2017.119, author = {Figueira, Diego and Lazic, Ranko and Leroux, J\'{e}r\^{o}me and Mazowiecki, Filip and Sutre, Gr\'{e}goire}, title = {{Polynomial-Space Completeness of Reachability for Succinct Branching VASS in Dimension One}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {119:1--119:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.119}, URN = {urn:nbn:de:0030-drops-74374}, doi = {10.4230/LIPIcs.ICALP.2017.119}, annote = {Keywords: branching vector addition systems, reachability problem} }

Document

**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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Invited Talk

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Vector addition systems, or equivalently Petri nets, are one of the most popular formal models for the representation and the analysis of parallel processes. Many problems for vector addition systems are known to be decidable thanks to the theory of well-structured transition systems. Indeed, vector addition systems with configurations equipped with the classical point-wise ordering are well-structured transition systems. Based on this observation, problems like coverability or termination can be proven decidable.
However, the theory of well-structured transition systems does not explain the decidability of the reachability problem. In this presentation, we show that runs of vector addition systems can also be equipped with a well quasi-order. This observation provides a unified understanding of the data structures involved in solving many problems for vector addition systems, including the central reachability problem.

Jérôme Leroux and Sylvain Schmitz. Ideal Decompositions for Vector Addition Systems (Invited Talk). In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{leroux_et_al:LIPIcs.STACS.2016.1, author = {Leroux, J\'{e}r\^{o}me and Schmitz, Sylvain}, title = {{Ideal Decompositions for Vector Addition Systems}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {1:1--1:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.1}, URN = {urn:nbn:de:0030-drops-57024}, doi = {10.4230/LIPIcs.STACS.2016.1}, annote = {Keywords: Petri net, ideal, well-quasi-order, reachability, verification} }

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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 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

Reachability and boundedness problems have been shown decidable for Vector Addition Systems with one zero-test. Surprisingly, place-boundedness remained open. We provide here a variation of the Karp-Miller algorithm to compute a basis of the downward closure of the reachability set which allows to decide place-boundedness. This forward algorithm is able to pass the zero-tests thanks to a finer cover, hybrid between the reachability and cover sets, reclaiming accuracy on one component. We show that this filtered cover is still recursive, but that equality of two such filtered covers, even for usual Vector Addition Systems (with no zero-test), is undecidable.

Rémi Bonnet, Alain Finkel, Jérôme Leroux, and Marc Zeitoun. Place-Boundedness for Vector Addition Systems with one zero-test. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 192-203, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{bonnet_et_al:LIPIcs.FSTTCS.2010.192, author = {Bonnet, R\'{e}mi and Finkel, Alain and Leroux, J\'{e}r\^{o}me and Zeitoun, Marc}, title = {{Place-Boundedness for Vector Addition Systems with one zero-test}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {192--203}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.192}, URN = {urn:nbn:de:0030-drops-28638}, doi = {10.4230/LIPIcs.FSTTCS.2010.192}, annote = {Keywords: Place-boundedness, vector addition system with one zero-test, Karp-Miller algorithm} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6081, Software Verification: Infinite-State Model Checking and Static Program Analysis (2006)

This paper argues that flatness appears as a central notion in the
verification of counter automata. A counter automaton is called flat
when its control graph can be ``replaced'', equivalently w.r.t.
reachability, by another one with no nested loops.
From a practical view point, we show that flatness is a necessary and
sufficient condition for termination of accelerated symbolic model
checking, a generic semi-algorithmic technique implemented in
successful tools like FAST, LASH or TReX.
From a theoretical view point, we prove that many known semilinear
subclasses of counter automata are flat: reversal bounded counter
machines, lossy vector addition systems with states, reversible Petri nets,
persistent and conflict-free Petri nets, etc. Hence, for these subclasses,
the semilinear reachability set can be computed using a emph{uniform}
accelerated symbolic procedure (whereas previous algorithms were
specifically designed for each subclass).

Jérôme Leroux and Grégoire Sutre. Flat counter automata almost everywhere!. In Software Verification: Infinite-State Model Checking and Static Program Analysis. Dagstuhl Seminar Proceedings, Volume 6081, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{leroux_et_al:DagSemProc.06081.4, author = {Leroux, J\'{e}r\^{o}me and Sutre, Gr\'{e}goire}, title = {{Flat counter automata almost everywhere!}}, booktitle = {Software Verification: Infinite-State Model Checking and Static Program Analysis}, pages = {1--19}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6081}, editor = {Parosh Aziz Abdulla and Ahmed Bouajjani and Markus M\"{u}ller-Olm}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06081.4}, URN = {urn:nbn:de:0030-drops-7297}, doi = {10.4230/DagSemProc.06081.4}, annote = {Keywords: Symbolic representation, Infinite state system, Acceleration, Meta-transition} }

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