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

Vector Addition Systems (VAS), aka Petri nets, are a popular model of concurrency. The reachability set of a VAS is the set of configurations reachable from the initial configuration. Leroux has studied the geometric properties of VAS reachability sets, and used them to derive decision procedures for important analysis problems. In this paper we continue the geometric study of reachability sets. We show that every reachability set admits a finite decomposition into disjoint almost hybridlinear sets enjoying nice geometric properties. Further, we prove that the decomposition of the reachability set of a given VAS is effectively computable. As a corollary, we derive a new proof of Hauschildt’s 1990 result showing the decidability of the question whether the reachability set of a given VAS is semilinear. As a second corollary, we prove that the complement of a reachability set, if it is infinite, always contains an infinite linear set.

Roland Guttenberg, Mikhail Raskin, and Javier Esparza. Geometry of Reachability Sets of Vector Addition Systems. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{guttenberg_et_al:LIPIcs.CONCUR.2023.6, author = {Guttenberg, Roland and Raskin, Mikhail and Esparza, Javier}, title = {{Geometry of Reachability Sets of Vector Addition Systems}}, booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)}, pages = {6:1--6:16}, 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.2023.6}, URN = {urn:nbn:de:0030-drops-190005}, doi = {10.4230/LIPIcs.CONCUR.2023.6}, annote = {Keywords: Vector Addition System, Petri net, Reachability Set, Almost hybridlinear, Partition, Geometry} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

Regular model checking is a technique for the verification of infinite-state systems whose configurations can be represented as finite words over a suitable alphabet. It applies to systems whose set of initial configurations is regular, and whose transition relation is captured by a length-preserving transducer. To verify safety properties, regular model checking iteratively computes automata recognizing increasingly larger regular sets of reachable configurations, and checks if they contain unsafe configurations. Since this procedure often does not terminate, acceleration, abstraction, and widening techniques have been developed to compute a regular superset of the reachable configurations.
In this paper we develop a complementary procedure. Instead of approaching the set of reachable configurations from below, we start with the set of all configurations and approach it from above. We use that the set of reachable configurations is equal to the intersection of all inductive invariants of the system. Since this intersection is non-regular in general, we introduce b-bounded invariants, defined as those representable by CNF-formulas with at most b clauses. We prove that, for every b ≥ 0, the intersection of all b-bounded inductive invariants is regular, and we construct an automaton recognizing it. We show that whether this automaton accepts some unsafe configuration is in EXPSPACE for every b ≥ 0, and PSPACE-complete for b = 1. Finally, we study how large must b be to prove safety properties of a number of benchmarks.

Javier Esparza, Mikhail Raskin, and Christoph Welzel. Regular Model Checking Upside-Down: An Invariant-Based Approach. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{esparza_et_al:LIPIcs.CONCUR.2022.23, author = {Esparza, Javier and Raskin, Mikhail and Welzel, Christoph}, title = {{Regular Model Checking Upside-Down: An Invariant-Based Approach}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir 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.CONCUR.2022.23}, URN = {urn:nbn:de:0030-drops-170862}, doi = {10.4230/LIPIcs.CONCUR.2022.23}, annote = {Keywords: parameterized verification, structural analysis, regular languages, regular model-checking, traps} }

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

In a previous paper we introduced immediate observation (IO) Petri nets, a class of interest in the study of population protocols and enzymatic chemical networks. In the first part of this paper we show that IO nets are globally flat, and so their safety properties can be checked by efficient symbolic model checking tools using acceleration techniques, like FAST. In the second part we study Branching IO nets (BIO nets), whose transitions can create tokens. BIO nets extend both IO nets and communication-free nets, also called BPP nets, a widely studied class. We show that, while BIO nets are no longer globally flat, and their sets of reachable markings may be non-semilinear, they are still locally flat. As a consequence, the coverability and reachability problem for BIO nets, and even a certain set-parameterized version of them, are in PSPACE. This makes BIO nets the first natural net class with non-semilinear reachability relation for which the reachability problem is provably simpler than for general Petri nets.

Mikhail Raskin, Chana Weil-Kennedy, and Javier Esparza. Flatness and Complexity of Immediate Observation Petri Nets. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{raskin_et_al:LIPIcs.CONCUR.2020.45, author = {Raskin, Mikhail and Weil-Kennedy, Chana and Esparza, Javier}, title = {{Flatness and Complexity of Immediate Observation Petri Nets}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {45:1--45:19}, 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.2020.45}, URN = {urn:nbn:de:0030-drops-128574}, doi = {10.4230/LIPIcs.CONCUR.2020.45}, annote = {Keywords: Petri Nets, Reachability Analysis, Parameterized Verification, Flattability} }

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

Unambiguous non-deterministic finite automata (UFA) are non-deterministic automata (over finite words) such that there is at most one accepting run over each input. Such automata are known to be potentially exponentially more succinct than deterministic automata, and non-deterministic automata can be exponentially more succinct than them.
In this paper we establish a superpolynomial lower bound for the state complexity of the translation of an UFA to a non-deterministic automaton for the complement language. This disproves the formerly conjectured polynomial upper bound for this translation. This lower bound only involves a one letter alphabet, and makes use of the random graph methods.
The same proof also shows that the translation of sweeping automata to non-deterministic automata is superpolynomial.

Mikhail Raskin. A Superpolynomial Lower Bound for the Size of Non-Deterministic Complement of an Unambiguous Automaton. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 138:1-138:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{raskin:LIPIcs.ICALP.2018.138, author = {Raskin, Mikhail}, title = {{A Superpolynomial Lower Bound for the Size of Non-Deterministic Complement of an Unambiguous Automaton}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {138:1--138:11}, 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.2018.138}, URN = {urn:nbn:de:0030-drops-91428}, doi = {10.4230/LIPIcs.ICALP.2018.138}, annote = {Keywords: unambiguous automata, language complement, lower bound} }

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

We present the first linear lower bound for the number of bits required to be accessed in the worst case to increment an integer in an arbitrary space-optimal binary representation. The best previously known lower bound was logarithmic. It is known that a logarithmic number of read bits in the worst case is enough to increment some of the integer representations that use one bit of redundancy, therefore we show an exponential gap between space-optimal and redundant counters.
Our proof is based on considering the increment procedure for a space optimal counter as a permutation and calculating its parity. For every space optimal counter, the permutation must be odd, and implementing an odd permutation requires reading at least half the bits in the worst case. The combination of these two observations explains why the worst-case space-optimal problem is substantially different from both average-case approach with constant expected number of reads and almost space optimal representations with logarithmic number of reads in the worst case.

Mikhail Raskin. A Linear Lower Bound for Incrementing a Space-Optimal Integer Representation in the Bit-Probe Model. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 88:1-88:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{raskin:LIPIcs.ICALP.2017.88, author = {Raskin, Mikhail}, title = {{A Linear Lower Bound for Incrementing a Space-Optimal Integer Representation in the Bit-Probe Model}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {88:1--88:12}, 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.2017.88}, URN = {urn:nbn:de:0030-drops-74105}, doi = {10.4230/LIPIcs.ICALP.2017.88}, annote = {Keywords: binary counter, data structure, integer representation, bit-probe model, lower bound} }

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