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Documents authored by Keskin, Eren

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.126
Separability in Büchi VASS and Singly Non-Linear Systems of Inequalities

Authors: Pascal Baumann, Eren Keskin, Roland Meyer, and Georg Zetzsche

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The ω-regular separability problem for Büchi VASS coverability languages has recently been shown to be decidable, but with an EXPSPACE lower and a non-primitive recursive upper bound - the exact complexity remained open. We close this gap and show that the problem is EXPSPACE-complete. A careful analysis of our complexity bounds additionally yields a PSPACE procedure in the case of fixed dimension ≥ 1, which matches a pre-established lower bound of PSPACE for one dimensional Büchi VASS. Our algorithm is a non-deterministic search for a witness whose size, as we show, can be suitably bounded. Part of the procedure is to decide the existence of runs in VASS that satisfy certain non-linear properties. Therefore, a key technical ingredient is to analyze a class of systems of inequalities where one variable may occur in non-linear (polynomial) expressions. These so-called singly non-linear systems (SNLS) take the form A(x)⋅ y ≥ b(x), where A(x) and b(x) are a matrix resp. a vector whose entries are polynomials in x, and y ranges over vectors in the rationals. Our main contribution on SNLS is an exponential upper bound on the size of rational solutions to singly non-linear systems. The proof consists of three steps. First, we give a tailor-made quantifier elimination to characterize all real solutions to x. Second, using the root separation theorem about the distance of real roots of polynomials, we show that if a rational solution exists, then there is one with at most polynomially many bits. Third, we insert the solution for x into the SNLS, making it linear and allowing us to invoke standard solution bounds from convex geometry. Finally, we combine the results about SNLS with several techniques from the area of VASS to devise an EXPSPACE decision procedure for ω-regular separability of Büchi VASS.
Cite as

Pascal Baumann, Eren Keskin, Roland Meyer, and Georg Zetzsche. Separability in Büchi VASS and Singly Non-Linear Systems of Inequalities. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 126:1-126:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baumann_et_al:LIPIcs.ICALP.2024.126,
  author =	{Baumann, Pascal and Keskin, Eren and Meyer, Roland and Zetzsche, Georg},
  title =	{{Separability in B\"{u}chi VASS and Singly Non-Linear Systems of Inequalities}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{126:1--126:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.126},
  URN =		{urn:nbn:de:0030-drops-202695},
  doi =		{10.4230/LIPIcs.ICALP.2024.126},
  annote =	{Keywords: Vector addition systems, infinite words, separability, inequalities, quantifier elimination, rational, polynomials}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2023.8
Separability and Non-Determinizability of WSTS

Authors: Eren Keskin and Roland Meyer

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

Abstract
There is a recent separability result for the languages of well-structured transition systems (WSTS) that is surprisingly general: disjoint WSTS languages are always separated by a regular language. The result assumes that one of the languages is accepted by a deterministic WSTS, and it is not known whether this assumption is needed. There are two ways to get rid of the assumption, none of which has led to conclusions so far: (i) show that WSTS can be determinized or (ii) generalize the separability result to non-deterministic WSTS languages. Our contribution is to show that (i) does not work but (ii) does. As for (i), we give a non-deterministic WSTS language that we prove cannot be accepted by a deterministic WSTS. The proof relies on a novel characterization of the languages accepted by deterministic WSTS. As for (ii), we show how to find finitely represented inductive invariants without having the tool of ideal decompositions at hand. Instead, we work with closures under converging sequences. Our results hold for upward- and downward-compatible WSTS.
Cite as

Eren Keskin and Roland Meyer. Separability and Non-Determinizability of WSTS. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{keskin_et_al:LIPIcs.CONCUR.2023.8,
  author =	{Keskin, Eren and Meyer, Roland},
  title =	{{Separability and Non-Determinizability of WSTS}},
  booktitle =	{34th International Conference on Concurrency Theory (CONCUR 2023)},
  pages =	{8:1--8:17},
  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.2023.8},
  URN =		{urn:nbn:de:0030-drops-190025},
  doi =		{10.4230/LIPIcs.CONCUR.2023.8},
  annote =	{Keywords: WSTS, regular separability, determinization}
}
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