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DOI: 10.4230/LIPIcs.FSTTCS.2023.10

Bandwidth of Timed Automata: 3 Classes

Authors: Eugene Asarin, Aldric Degorre, Cătălin Dima, and Bernardo Jacobo Inclán

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

Abstract

Timed languages contain sequences of discrete events ("letters") separated by real-valued delays, they can be recognized by timed automata, and represent behaviors of various real-time systems. The notion of bandwidth of a timed language defined in [Jacobo Inclán et al., 2022] characterizes the amount of information per time unit, encoded in words of the language observed with some precision ε. In this paper, we identify three classes of timed automata according to the asymptotics of the bandwidth of their languages with respect to this precision ε: automata are either meager, with an O(1) bandwidth, normal, with a Θ(log(1/ε)) bandwidth, or obese, with Θ(1/ε) bandwidth. We define two structural criteria and prove that they partition timed automata into these 3 classes of bandwidth, implying that there are no intermediate asymptotic classes. The classification problem of a timed automaton is PSPACE-complete. Both criteria are formulated using morphisms from paths of the timed automaton to some finite monoids extending Puri’s orbit graphs; the proofs are based on Simon’s factorization forest theorem.

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Eugene Asarin, Aldric Degorre, Cătălin Dima, and Bernardo Jacobo Inclán. Bandwidth of Timed Automata: 3 Classes. 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. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{asarin_et_al:LIPIcs.FSTTCS.2023.10,
  author =	{Asarin, Eugene and Degorre, Aldric and Dima, C\u{a}t\u{a}lin and Jacobo Incl\'{a}n, Bernardo},
  title =	{{Bandwidth of Timed Automata: 3 Classes}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{10:1--10:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.10},
  URN =		{urn:nbn:de:0030-drops-193838},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.10},
  annote =	{Keywords: timed automata, information theory, bandwidth, entropy, orbit graphs, factorization forests}
}

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DOI: 10.4230/LIPIcs.IPEC.2017.2

Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

Authors: Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Abstract

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X_1=(V,E_1) and X_2=(V,E_2) and a subset R\subseteq V\times V of forbidden pairs whether there is an isomorphism \pi from X_1 to X_2 such that i^\pi\ne j for all (i,j)\in R. We prove that this problem and several of its generalizations are in fact in \FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of R as parameters. The problem remains in FPT even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the number of moved vertices is unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism π between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that \pi can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT. In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [ESA 2011].

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Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán. Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arvind_et_al:LIPIcs.IPEC.2017.2,
  author =	{Arvind, Vikraman and K\"{o}bler, Johannes and Kuhnert, Sebastian and Tor\'{a}n, Jacobo},
  title =	{{Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{2:1--2:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.2},
  URN =		{urn:nbn:de:0030-drops-85690},
  doi =		{10.4230/LIPIcs.IPEC.2017.2},
  annote =	{Keywords: parameterized algorithms, hypergraph isomorphism, mislabeled graphs}
}

Document

DOI: 10.4230/DagRep.6.10.13

Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

Authors: Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, and Christopher Umans

Published in: Dagstuhl Reports, Volume 6, Issue 10 (2017)

Abstract

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress.

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Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, and Christopher Umans. Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411). In Dagstuhl Reports, Volume 6, Issue 10, pp. 13-32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{kabanets_et_al:DagRep.6.10.13,
  author =	{Kabanets, Valentine and Thierauf, Thomas and T\'{o}ran, Jacobo and Umans, Christopher},
  title =	{{Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)}},
  pages =	{13--32},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{6},
  number =	{10},
  editor =	{Kabanets, Valentine and Thierauf, Thomas and T\'{o}ran, Jacobo and Umans, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.6.10.13},
  URN =		{urn:nbn:de:0030-drops-69504},
  doi =		{10.4230/DagRep.6.10.13},
  annote =	{Keywords: Computational Complexity, lower bounds, approximation, pseudo-randomness, derandomization, circuits}
}

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Bandwidth of Timed Automata: 3 Classes

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Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

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Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

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