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Documents authored by Köcher, Chris

Document
DOI: 10.4230/LIPIcs.MFCS.2025.38
The Complexity of Separability for Semilinear Sets and Parikh Automata

Authors: Elias Rojas Collins, Chris Köcher, and Georg Zetzsche

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
In a separability problem, we are given two sets K and L from a class 𝒞, and we want to decide whether there exists a set S from a class 𝒮 such that K ⊆ S and S ∩ L = ∅. In this case, we speak of separability of sets in 𝒞 by sets in 𝒮. We study two types of separability problems. First, we consider separability of semilinear sets (i.e. subsets of ℕ^d for some d) by sets definable by quantifier-free monadic Presburger formulas (or equivalently, the recognizable subsets of ℕ^d). Here, a formula is monadic if each atom uses at most one variable. Second, we consider separability of languages of Parikh automata by regular languages. A Parikh automaton is a machine with access to counters that can only be incremented, and have to meet a semilinear constraint at the end of the run. Both of these separability problems are known to be decidable with elementary complexity. Our main results are that both problems are coNP-complete. In the case of semilinear sets, coNP-completeness holds regardless of whether the input sets are specified by existential Presburger formulas, quantifier-free formulas, or semilinear representations. Our results imply that recognizable separability of rational subsets of Σ* × ℕ^d (shown decidable by Choffrut and Grigorieff) is coNP-complete as well. Another application is that regularity of deterministic Parikh automata (where the target set is specified using a quantifier-free Presburger formula) is coNP-complete as well.
Cite as

Elias Rojas Collins, Chris Köcher, and Georg Zetzsche. The Complexity of Separability for Semilinear Sets and Parikh Automata. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{collins_et_al:LIPIcs.MFCS.2025.38,
  author =	{Collins, Elias Rojas and K\"{o}cher, Chris and Zetzsche, Georg},
  title =	{{The Complexity of Separability for Semilinear Sets and Parikh Automata}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{38:1--38:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.38},
  URN =		{urn:nbn:de:0030-drops-241457},
  doi =		{10.4230/LIPIcs.MFCS.2025.38},
  annote =	{Keywords: Vector Addition System, Separability, Regular Language}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.15
Regular Separators for VASS Coverability Languages

Authors: Chris Köcher and Georg Zetzsche

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

Abstract
We study regular separators of vector addition systems (VASS, for short) with coverability semantics. A regular language R is a regular separator of languages K and L if K ⊆ R and L ∩ R = ∅. It was shown by Czerwiński, Lasota, Meyer, Muskalla, Kumar, and Saivasan (CONCUR 2018) that it is decidable whether, for two given VASS, there exists a regular separator. In fact, they show that a regular separator exists if and only if the two VASS languages are disjoint. However, they provide a triply exponential upper bound and a doubly exponential lower bound for the size of such separators and leave open which bound is tight. We show that if two VASS have disjoint languages, then there exists a regular separator with at most doubly exponential size. Moreover, we provide tight size bounds for separators in the case of fixed dimensions and unary/binary encodings of updates and NFA/DFA separators. In particular, we settle the aforementioned question. The key ingredient in the upper bound is a structural analysis of separating automata based on the concept of basic separators, which was recently introduced by Czerwiński and the second author. This allows us to determinize (and thus complement) without the powerset construction and avoid one exponential blowup.
Cite as

Chris Köcher and Georg Zetzsche. Regular Separators for VASS Coverability Languages. 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. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kocher_et_al:LIPIcs.FSTTCS.2023.15,
  author =	{K\"{o}cher, Chris and Zetzsche, Georg},
  title =	{{Regular Separators for VASS Coverability Languages}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{15:1--15:19},
  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.2023.15},
  URN =		{urn:nbn:de:0030-drops-193883},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.15},
  annote =	{Keywords: Vector Addition System, Separability, Regular Language}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2018.9
The Cayley-Graph of the Queue Monoid: Logic and Decidability

Authors: Faried Abu Zaid and Chris Köcher

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

Abstract
We investigate the decidability of logical aspects of graphs that arise as Cayley-graphs of the so-called queue monoids. These monoids model the behavior of the classical (reliable) fifo-queues. We answer a question raised by Huschenbett, Kuske, and Zetzsche and prove the decidability of the first-order theory of these graphs with the help of an - at least for the authors - new combination of the well-known method from Ferrante and Rackoff and an automata-based approach. On the other hand, we prove that the monadic second-order of the queue monoid's Cayley-graph is undecidable.
Cite as

Faried Abu Zaid and Chris Köcher. The Cayley-Graph of the Queue Monoid: Logic and Decidability. 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. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abuzaid_et_al:LIPIcs.FSTTCS.2018.9,
  author =	{Abu Zaid, Faried and K\"{o}cher, Chris},
  title =	{{The Cayley-Graph of the Queue Monoid: Logic and Decidability}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{9:1--9:17},
  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.2018.9},
  URN =		{urn:nbn:de:0030-drops-99088},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.9},
  annote =	{Keywords: Queues, Transformation Monoid, Cayley-Graph, Logic, First-Order Theory, MSO Theory, Model Checking}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.45
Rational, Recognizable, and Aperiodic Sets in the Partially Lossy Queue Monoid

Authors: Chris Köcher

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
Partially lossy queue monoids (or plq monoids) model the behavior of queues that can forget arbitrary parts of their content. While many decision problems on recognizable subsets in the plq monoid are decidable, most of them are undecidable if the sets are rational. In particular, in this monoid the classes of rational and recognizable subsets do not coincide. By restricting multiplication and iteration in the construction of rational sets and by allowing complementation we obtain precisely the class of recognizable sets. From these special rational expressions we can obtain an MSO logic describing the recognizable subsets. Moreover, we provide similar results for the class of aperiodic subsets in the plq monoid.
Cite as

Chris Köcher. Rational, Recognizable, and Aperiodic Sets in the Partially Lossy Queue Monoid. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kocher:LIPIcs.STACS.2018.45,
  author =	{K\"{o}cher, Chris},
  title =	{{Rational, Recognizable, and Aperiodic Sets in the Partially Lossy Queue Monoid}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{45:1--45:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.45},
  URN =		{urn:nbn:de:0030-drops-84839},
  doi =		{10.4230/LIPIcs.STACS.2018.45},
  annote =	{Keywords: Partially Lossy Queues, Transformation Monoid, Rational Sets, Recognizable Sets, Aperiodic Sets, MSO logic}
}
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