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DOI: 10.4230/LIPIcs.CONCUR.2023.6

Geometry of Reachability Sets of Vector Addition Systems

Authors: Roland Guttenberg, Mikhail Raskin, and Javier Esparza

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

Abstract

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.

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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}
}

Document

DOI: 10.4230/LIPIcs.SAND.2022.11

Fast and Succinct Population Protocols for Presburger Arithmetic

Authors: Philipp Czerner, Roland Guttenberg, Martin Helfrich, and Javier Esparza

Published in: LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

Abstract

In their 2006 seminal paper in Distributed Computing, Angluin et al. present a construction that, given any Presburger predicate as input, outputs a leaderless population protocol that decides the predicate. The protocol for a predicate of size m (when expressed as a Boolean combination of threshold and remainder predicates with coefficients in binary) runs in 𝒪(m ⋅ n² log n) expected number of interactions, which is almost optimal in n, the number of interacting agents. However, the number of states of the protocol is exponential in m. This is a problem for natural computing applications, where a state corresponds to a chemical species and it is difficult to implement protocols with many states. Blondin et al. described in STACS 2020 another construction that produces protocols with a polynomial number of states, but exponential expected number of interactions. We present a construction that produces protocols with 𝒪(m) states that run in expected 𝒪(m⁷ ⋅ n²) interactions, optimal in n, for all inputs of size Ω(m). For this, we introduce population computers, a carefully crafted generalization of population protocols easier to program, and show that our computers for Presburger predicates can be translated into fast and succinct population protocols.

Cite as

Philipp Czerner, Roland Guttenberg, Martin Helfrich, and Javier Esparza. Fast and Succinct Population Protocols for Presburger Arithmetic. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{czerner_et_al:LIPIcs.SAND.2022.11,
  author =	{Czerner, Philipp and Guttenberg, Roland and Helfrich, Martin and Esparza, Javier},
  title =	{{Fast and Succinct Population Protocols for Presburger Arithmetic}},
  booktitle =	{1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-224-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{221},
  editor =	{Aspnes, James and Michail, Othon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.11},
  URN =		{urn:nbn:de:0030-drops-159535},
  doi =		{10.4230/LIPIcs.SAND.2022.11},
  annote =	{Keywords: population protocols, fast, succinct, population computers}
}

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Documents authored by Guttenberg, Roland

Geometry of Reachability Sets of Vector Addition Systems

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Fast and Succinct Population Protocols for Presburger Arithmetic

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