27 Search Results for "Zetzsche, Georg"


Document
Directed Regular and Context-Free Languages

Authors: Moses Ganardi, Irmak Sağlam, and Georg Zetzsche

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
We study the problem of deciding whether a given language is directed. A language L is directed if every pair of words in L have a common (scattered) superword in L. Deciding directedness is a fundamental problem in connection with ideal decompositions of downward closed sets. Another motivation is that deciding whether two directed context-free languages have the same downward closures can be decided in polynomial time, whereas for general context-free languages, this problem is known to be coNEXP-complete. We show that the directedness problem for regular languages, given as NFAs, belongs to AC¹, and thus polynomial time. Moreover, it is NL-complete for fixed alphabet sizes. Furthermore, we show that for context-free languages, the directedness problem is PSPACE-complete.

Cite as

Moses Ganardi, Irmak Sağlam, and Georg Zetzsche. Directed Regular and Context-Free Languages. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{ganardi_et_al:LIPIcs.STACS.2024.36,
  author =	{Ganardi, Moses and Sa\u{g}lam, Irmak and Zetzsche, Georg},
  title =	{{Directed Regular and Context-Free Languages}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.36},
  URN =		{urn:nbn:de:0030-drops-197465},
  doi =		{10.4230/LIPIcs.STACS.2024.36},
  annote =	{Keywords: Subword, ideal, language, regular, context-free, equivalence, downward closure, compression}
}
Document
Remarks on Parikh-Recognizable Omega-languages

Authors: Mario Grobler, Leif Sabellek, and Sebastian Siebertz

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)


Abstract
Several variants of Parikh automata on infinite words were recently introduced by Guha et al. [FSTTCS, 2022]. We show that one of these variants coincides with blind counter machine as introduced by Fernau and Stiebe [Fundamenta Informaticae, 2008]. Fernau and Stiebe showed that every ω-language recognized by a blind counter machine is of the form ⋃_iU_iV_i^ω for Parikh recognizable languages U_i, V_i, but blind counter machines fall short of characterizing this class of ω-languages. They posed as an open problem to find a suitable automata-based characterization. We introduce several additional variants of Parikh automata on infinite words that yield automata characterizations of classes of ω-language of the form ⋃_iU_iV_i^ω for all combinations of languages U_i, V_i being regular or Parikh-recognizable. When both U_i and V_i are regular, this coincides with Büchi’s classical theorem. We study the effect of ε-transitions in all variants of Parikh automata and show that almost all of them admit ε-elimination. Finally we study the classical decision problems with applications to model checking.

Cite as

Mario Grobler, Leif Sabellek, and Sebastian Siebertz. Remarks on Parikh-Recognizable Omega-languages. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{grobler_et_al:LIPIcs.CSL.2024.31,
  author =	{Grobler, Mario and Sabellek, Leif and Siebertz, Sebastian},
  title =	{{Remarks on Parikh-Recognizable Omega-languages}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{31:1--31:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.31},
  URN =		{urn:nbn:de:0030-drops-196743},
  doi =		{10.4230/LIPIcs.CSL.2024.31},
  annote =	{Keywords: Parikh automata, blind counter machines, infinite words, B\"{u}chi’s theorem}
}
Document
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-dev.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
Counter Machines with Infrequent Reversals

Authors: Alain Finkel, Shankara Narayanan Krishna, Khushraj Madnani, Rupak Majumdar, and Georg Zetzsche

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


Abstract
Bounding the number of reversals in a counter machine is one of the most prominent restrictions to achieve decidability of the reachability problem. Given this success, we explore whether this notion can be relaxed while retaining decidability. To this end, we introduce the notion of an f-reversal-bounded counter machine for a monotone function f: ℕ → ℕ. In such a machine, every run of length n makes at most f(n) reversals. Our first main result is a dichotomy theorem: We show that for every monotone function f, one of the following holds: Either (i) f grows so slowly that every f-reversal bounded counter machine is already k-reversal bounded for some constant k or (ii) f belongs to Ω(log(n)) and reachability in f-reversal bounded counter machines is undecidable. This shows that classical reversal bounding already captures the decidable cases of f-reversal bounding for any monotone function f. The key technical ingredient is an analysis of the growth of small solutions of iterated compositions of Presburger-definable constraints. In our second contribution, we investigate whether imposing f-reversal boundedness improves the complexity of the reachability problem in vector addition systems with states (VASS). Here, we obtain an analogous dichotomy: We show that either (i) f grows so slowly that every f-reversal-bounded VASS is already k-reversal-bounded for some constant k or (ii) f belongs to Ω(n) and the reachability problem for f-reversal-bounded VASS remains Ackermann-complete. This result is proven using run amalgamation in VASS. Overall, our results imply that classical restriction of reversal boundedness is a robust one.

Cite as

Alain Finkel, Shankara Narayanan Krishna, Khushraj Madnani, Rupak Majumdar, and Georg Zetzsche. Counter Machines with Infrequent Reversals. 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. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{finkel_et_al:LIPIcs.FSTTCS.2023.42,
  author =	{Finkel, Alain and Krishna, Shankara Narayanan and Madnani, Khushraj and Majumdar, Rupak and Zetzsche, Georg},
  title =	{{Counter Machines with Infrequent Reversals}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{42:1--42:17},
  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.2023.42},
  URN =		{urn:nbn:de:0030-drops-194152},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.42},
  annote =	{Keywords: Counter machines, reversal-bounded, reachability, decidability, complexity}
}
Document
Monus Semantics in Vector Addition Systems with States

Authors: Pascal Baumann, Khushraj Madnani, Filip Mazowiecki, and Georg Zetzsche

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


Abstract
Vector addition systems with states (VASS) are a popular model for concurrent systems. However, many decision problems have prohibitively high complexity. Therefore, it is sometimes useful to consider overapproximating semantics in which these problems can be decided more efficiently. We study an overapproximation, called monus semantics, that slightly relaxes the semantics of decrements: A key property of a vector addition systems is that in order to decrement a counter, this counter must have a positive value. In contrast, our semantics allows decrements of zero-valued counters: If such a transition is executed, the counter just remains zero. It turns out that if only a subset of transitions is used with monus semantics (and the others with classical semantics), then reachability is undecidable. However, we show that if monus semantics is used throughout, reachability remains decidable. In particular, we show that reachability for VASS with monus semantics is as hard as that of classical VASS (i.e. Ackermann-hard), while the zero-reachability and coverability are easier (i.e. EXPSPACE-complete and NP-complete, respectively). We provide a comprehensive account of the complexity of the general reachability problem, reachability of zero configurations, and coverability under monus semantics. We study these problems in general VASS, two-dimensional VASS, and one-dimensional VASS, with unary and binary counter updates.

Cite as

Pascal Baumann, Khushraj Madnani, Filip Mazowiecki, and Georg Zetzsche. Monus Semantics in Vector Addition Systems with States. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{baumann_et_al:LIPIcs.CONCUR.2023.10,
  author =	{Baumann, Pascal and Madnani, Khushraj and Mazowiecki, Filip and Zetzsche, Georg},
  title =	{{Monus Semantics in Vector Addition Systems with States}},
  booktitle =	{34th International Conference on Concurrency Theory (CONCUR 2023)},
  pages =	{10:1--10:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.10},
  URN =		{urn:nbn:de:0030-drops-190047},
  doi =		{10.4230/LIPIcs.CONCUR.2023.10},
  annote =	{Keywords: Vector addition systems, Overapproximation, Reachability, Coverability}
}
Document
Priority Downward Closures

Authors: Ashwani Anand and Georg Zetzsche

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


Abstract
When a system sends messages through a lossy channel, then the language encoding all sequences of messages can be abstracted by its downward closure, i.e. the set of all (not necessarily contiguous) subwords. This is useful because even if the system has infinitely many states, its downward closure is a regular language. However, if the channel has congestion control based on priorities assigned to the messages, then we need a finer abstraction: The downward closure with respect to the priority embedding. As for subword-based downward closures, one can also show that these priority downward closures are always regular. While computing finite automata for the subword-based downward closure is well understood, nothing is known in the case of priorities. We initiate the study of this problem and provide algorithms to compute priority downward closures for regular languages, one-counter languages, and context-free languages.

Cite as

Ashwani Anand and Georg Zetzsche. Priority Downward Closures. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{anand_et_al:LIPIcs.CONCUR.2023.39,
  author =	{Anand, Ashwani and Zetzsche, Georg},
  title =	{{Priority Downward Closures}},
  booktitle =	{34th International Conference on Concurrency Theory (CONCUR 2023)},
  pages =	{39:1--39:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.39},
  URN =		{urn:nbn:de:0030-drops-190339},
  doi =		{10.4230/LIPIcs.CONCUR.2023.39},
  annote =	{Keywords: downward closure, priority order, pushdown automata, non-deterministic finite automata, abstraction, computability}
}
Document
Invited Talk
Context-Bounded Analysis of Concurrent Programs (Invited Talk)

Authors: Pascal Baumann, Moses Ganardi, Rupak Majumdar, Ramanathan S. Thinniyam, and Georg Zetzsche

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
Context-bounded analysis of concurrent programs is a technique to compute a sequence of under-approximations of all behaviors of the program. For a fixed bound k, a context bounded analysis considers only those runs in which a single process is interrupted at most k times. As k grows, we capture more and more behaviors of the program. Practically, context-bounding has been very effective as a bug-finding tool: many bugs can be found even with small bounds. Theoretically, context-bounded analysis is decidable for a large number of programming models for which verification problems are undecidable. In this paper, we survey some recent work in context-bounded analysis of multithreaded programs. In particular, we show a general decidability result. We study context-bounded reachability in a language-theoretic setup. We fix a class of languages (satisfying some mild conditions) from which each thread is chosen. We show context-bounded safety and termination verification problems are decidable iff emptiness is decidable for the underlying class of languages and context-bounded boundedness is decidable iff finiteness is decidable for the underlying class.

Cite as

Pascal Baumann, Moses Ganardi, Rupak Majumdar, Ramanathan S. Thinniyam, and Georg Zetzsche. Context-Bounded Analysis of Concurrent Programs (Invited Talk). In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{baumann_et_al:LIPIcs.ICALP.2023.3,
  author =	{Baumann, Pascal and Ganardi, Moses and Majumdar, Rupak and Thinniyam, Ramanathan S. and Zetzsche, Georg},
  title =	{{Context-Bounded Analysis of Concurrent Programs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.3},
  URN =		{urn:nbn:de:0030-drops-180559},
  doi =		{10.4230/LIPIcs.ICALP.2023.3},
  annote =	{Keywords: Context-bounded analysis, Multi-threaded programs, Decidability}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Checking Refinement of Asynchronous Programs Against Context-Free Specifications

Authors: Pascal Baumann, Moses Ganardi, Rupak Majumdar, Ramanathan S. Thinniyam, and Georg Zetzsche

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
In the language-theoretic approach to refinement verification, we check that the language of traces of an implementation all belong to the language of a specification. We consider the refinement verification problem for asynchronous programs against specifications given by a Dyck language. We show that this problem is EXPSPACE-complete - the same complexity as that of language emptiness and for refinement verification against a regular specification. Our algorithm uses several technical ingredients. First, we show that checking if the coverability language of a succinctly described vector addition system with states (VASS) is contained in a Dyck language is EXPSPACE-complete. Second, in the more technical part of the proof, we define an ordering on words and show a downward closure construction that allows replacing the (context-free) language of each task in an asynchronous program by a regular language. Unlike downward closure operations usually considered in infinite-state verification, our ordering is not a well-quasi-ordering, and we have to construct the regular language ab initio. Once the tasks can be replaced, we show a reduction to an appropriate VASS and use our first ingredient. In addition to the inherent theoretical interest, refinement verification with Dyck specifications captures common practical resource usage patterns based on reference counting, for which few algorithmic techniques were known.

Cite as

Pascal Baumann, Moses Ganardi, Rupak Majumdar, Ramanathan S. Thinniyam, and Georg Zetzsche. Checking Refinement of Asynchronous Programs Against Context-Free Specifications. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 110:1-110:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{baumann_et_al:LIPIcs.ICALP.2023.110,
  author =	{Baumann, Pascal and Ganardi, Moses and Majumdar, Rupak and Thinniyam, Ramanathan S. and Zetzsche, Georg},
  title =	{{Checking Refinement of Asynchronous Programs Against Context-Free Specifications}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{110:1--110:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.110},
  URN =		{urn:nbn:de:0030-drops-181622},
  doi =		{10.4230/LIPIcs.ICALP.2023.110},
  annote =	{Keywords: Asynchronous programs, VASS, Dyck languages, Language inclusion, Refinement verification}
}
Document
Regular Separability in Büchi VASS

Authors: Pascal Baumann, Roland Meyer, and Georg Zetzsche

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)


Abstract
We study the (ω-)regular separability problem for Büchi VASS languages: Given two Büchi VASS with languages L₁ and L₂, check whether there is a regular language that fully contains L₁ while remaining disjoint from L₂. We show that the problem is decidable in general and PSPACE-complete in the 1-dimensional case, assuming succinct counter updates. The results rely on several arguments. We characterize the set of all regular languages disjoint from L₂. Based on this, we derive a (sound and complete) notion of inseparability witnesses, non-regular subsets of L₁. Finally, we show how to symbolically represent inseparability witnesses and how to check their existence.

Cite as

Pascal Baumann, Roland Meyer, and Georg Zetzsche. Regular Separability in Büchi VASS. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{baumann_et_al:LIPIcs.STACS.2023.9,
  author =	{Baumann, Pascal and Meyer, Roland and Zetzsche, Georg},
  title =	{{Regular Separability in B\"{u}chi VASS}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.9},
  URN =		{urn:nbn:de:0030-drops-176617},
  doi =		{10.4230/LIPIcs.STACS.2023.9},
  annote =	{Keywords: Separability problem, Vector addition systems, Infinite words, Decidability}
}
Document
Membership Problems in Finite Groups

Authors: Markus Lohrey, Andreas Rosowski, and Georg Zetzsche

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)


Abstract
We show that the subset sum problem, the knapsack problem and the rational subset membership problem for permutation groups are NP-complete. Concerning the knapsack problem we obtain NP-completeness for every fixed n ≥ 3, where n is the number of permutations in the knapsack equation. In other words: membership in products of three cyclic permutation groups is NP-complete. This sharpens a result of Luks [Eugene M. Luks, 1991], which states NP-completeness of the membership problem for products of three abelian permutation groups. We also consider the context-free membership problem in permutation groups and prove that it is PSPACE-complete but NP-complete for a restricted class of context-free grammars where acyclic derivation trees must have constant Horton-Strahler number. Our upper bounds hold for black box groups. The results for context-free membership problems in permutation groups yield new complexity bounds for various intersection non-emptiness problems for DFAs and a single context-free grammar.

Cite as

Markus Lohrey, Andreas Rosowski, and Georg Zetzsche. Membership Problems in Finite Groups. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2022.71,
  author =	{Lohrey, Markus and Rosowski, Andreas and Zetzsche, Georg},
  title =	{{Membership Problems in Finite Groups}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{71:1--71:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.71},
  URN =		{urn:nbn:de:0030-drops-168694},
  doi =		{10.4230/LIPIcs.MFCS.2022.71},
  annote =	{Keywords: algorithms for finite groups, intersection non-emptiness problems, knapsack problems in groups}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Reachability in Bidirected Pushdown VASS

Authors: Moses Ganardi, Rupak Majumdar, Andreas Pavlogiannis, Lia Schütze, and Georg Zetzsche

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)


Abstract
A pushdown vector addition system with states (PVASS) extends the model of vector addition systems with a pushdown store. A PVASS is said to be bidirected if every transition (pushing/popping a symbol or modifying a counter) has an accompanying opposite transition that reverses the effect. Bidirectedness arises naturally in many models; it can also be seen as a overapproximation of reachability. We show that the reachability problem for bidirected PVASS is decidable in Ackermann time and primitive recursive for any fixed dimension. For the special case of one-dimensional bidirected PVASS, we show reachability is in PSPACE, and in fact in polynomial time if the stack is polynomially bounded. Our results are in contrast to the directed setting, where decidability of reachability is a long-standing open problem already for one dimensional PVASS, and there is a PSPACE-lower bound already for one-dimensional PVASS with bounded stack. The reachability relation in the bidirected (stateless) case is a congruence over ℕ^d. Our upper bounds exploit saturation techniques over congruences. In particular, we show novel elementary-time constructions of semilinear representations of congruences generated by finitely many vector pairs. In the case of one-dimensional PVASS, we employ a saturation procedure over bounded-size counters. We complement our upper bound with a TOWER-hardness result for arbitrary dimension and k-EXPSPACE hardness in dimension 2k+6 using a technique by Lazić and Totzke to implement iterative exponentiations.

Cite as

Moses Ganardi, Rupak Majumdar, Andreas Pavlogiannis, Lia Schütze, and Georg Zetzsche. Reachability in Bidirected Pushdown VASS. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 124:1-124:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ganardi_et_al:LIPIcs.ICALP.2022.124,
  author =	{Ganardi, Moses and Majumdar, Rupak and Pavlogiannis, Andreas and Sch\"{u}tze, Lia and Zetzsche, Georg},
  title =	{{Reachability in Bidirected Pushdown VASS}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{124:1--124:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.124},
  URN =		{urn:nbn:de:0030-drops-164651},
  doi =		{10.4230/LIPIcs.ICALP.2022.124},
  annote =	{Keywords: Vector addition systems, Pushdown, Reachability, Decidability, Complexity}
}
Document
Existential Definability over the Subword Ordering

Authors: Pascal Baumann, Moses Ganardi, Ramanathan S. Thinniyam, and Georg Zetzsche

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)


Abstract
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the Σ₁ (i.e., existential) fragment is undecidable, already for binary alphabets A. However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable. We show that if |A| ≥ 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments Σ_i: If |A| ≥ 3, then a relation is definable in Σ_i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the Σ_i-fragments for i ≥ 2 of the pure logic, where the words of A^* are not available as constants.

Cite as

Pascal Baumann, Moses Ganardi, Ramanathan S. Thinniyam, and Georg Zetzsche. Existential Definability over the Subword Ordering. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{baumann_et_al:LIPIcs.STACS.2022.7,
  author =	{Baumann, Pascal and Ganardi, Moses and Thinniyam, Ramanathan S. and Zetzsche, Georg},
  title =	{{Existential Definability over the Subword Ordering}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.7},
  URN =		{urn:nbn:de:0030-drops-158178},
  doi =		{10.4230/LIPIcs.STACS.2022.7},
  annote =	{Keywords: subword, subsequence, definability, expressiveness, first order logic, existential fragment, quantifier alternation}
}
Document
Scope-Bounded Reachability in Valence Systems

Authors: Aneesh K. Shetty, S. Krishna, and Georg Zetzsche

Published in: LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)


Abstract
Multi-pushdown systems are a standard model for concurrent recursive programs, but they have an undecidable reachability problem. Therefore, there have been several proposals to underapproximate their sets of runs so that reachability in this underapproximation becomes decidable. One such underapproximation that covers a relatively high portion of runs is scope boundedness. In such a run, after each push to stack i, the corresponding pop operation must come within a bounded number of visits to stack i. In this work, we generalize this approach to a large class of infinite-state systems. For this, we consider the model of valence systems, which consist of a finite-state control and an infinite-state storage mechanism that is specified by a finite undirected graph. This framework captures pushdowns, vector addition systems, integer vector addition systems, and combinations thereof. For this framework, we propose a notion of scope boundedness that coincides with the classical notion when the storage mechanism happens to be a multi-pushdown. We show that with this notion, reachability can be decided in PSPACE for every storage mechanism in the framework. Moreover, we describe the full complexity landscape of this problem across all storage mechanisms, both in the case of (i) the scope bound being given as input and (ii) for fixed scope bounds. Finally, we provide an almost complete description of the complexity landscape if even a description of the storage mechanism is part of the input.

Cite as

Aneesh K. Shetty, S. Krishna, and Georg Zetzsche. Scope-Bounded Reachability in Valence Systems. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{shetty_et_al:LIPIcs.CONCUR.2021.29,
  author =	{Shetty, Aneesh K. and Krishna, S. and Zetzsche, Georg},
  title =	{{Scope-Bounded Reachability in Valence Systems}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{29:1--29:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.29},
  URN =		{urn:nbn:de:0030-drops-144069},
  doi =		{10.4230/LIPIcs.CONCUR.2021.29},
  annote =	{Keywords: multi-pushdown systems, underapproximations, valence systems, reachability}
}
Document
A Characterization of Wreath Products Where Knapsack Is Decidable

Authors: Pascal Bergsträßer, Moses Ganardi, and Georg Zetzsche

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)


Abstract
The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group G and takes as input group elements g_1,…,g_n,g ∈ G and asks whether there are x_1,…,x_n ≥ 0 with g_1^{x_1}⋯ g_n^{x_n} = g. We study the knapsack problem for wreath products G≀H of groups G and H. Our main result is a characterization of those wreath products G≀H for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors G and H. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem. Moreover, we apply our main result to H₃(ℤ), the discrete Heisenberg group, and to Baumslag-Solitar groups BS(1,q) for q ≥ 1. First, we show that the knapsack problem is undecidable for G≀H₃(ℤ) for any G ≠ 1. This implies that for G ≠ 1 and for infinite and virtually nilpotent groups H, the knapsack problem for G≀H is decidable if and only if H is virtually abelian and solvability of systems of exponent equations is decidable for G. Second, we show that the knapsack problem is decidable for G≀BS(1,q) if and only if solvability of systems of exponent equations is decidable for G.

Cite as

Pascal Bergsträßer, Moses Ganardi, and Georg Zetzsche. A Characterization of Wreath Products Where Knapsack Is Decidable. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bergstraer_et_al:LIPIcs.STACS.2021.11,
  author =	{Bergstr\"{a}{\ss}er, Pascal and Ganardi, Moses and Zetzsche, Georg},
  title =	{{A Characterization of Wreath Products Where Knapsack Is Decidable}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.11},
  URN =		{urn:nbn:de:0030-drops-136566},
  doi =		{10.4230/LIPIcs.STACS.2021.11},
  annote =	{Keywords: knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, Baumslag-Solitar groups}
}
Document
Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups

Authors: Markus Lohrey and Georg Zetzsche

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)


Abstract
We prove that the power word problem for the solvable Baumslag-Solitar groups BS(1,q) = ⟨ a,t ∣ t a t^{-1} = a^q ⟩ can be solved in TC⁰. In the power word problem, the input consists of group elements g₁, …, g_d and binary encoded integers n₁, …, n_d and it is asked whether g₁^{n₁} ⋯ g_d^{n_d} = 1 holds. Moreover, we prove that the knapsack problem for BS(1,q) is NP-complete. In the knapsack problem, the input consists of group elements g₁, …, g_d,h and it is asked whether the equation g₁^{x₁} ⋯ g_d^{x_d} = h has a solution in ℕ^d.

Cite as

Markus Lohrey and Georg Zetzsche. Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{lohrey_et_al:LIPIcs.MFCS.2020.67,
  author =	{Lohrey, Markus and Zetzsche, Georg},
  title =	{{Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{67:1--67:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.67},
  URN =		{urn:nbn:de:0030-drops-127364},
  doi =		{10.4230/LIPIcs.MFCS.2020.67},
  annote =	{Keywords: computational group theory, matrix problems, Baumslag-Solitar groups}
}
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