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Documents authored by Szykuła, Marek

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.59
Completely Reachable Automata: A Polynomial Algorithm and Quadratic Upper Bounds

Authors: Robert Ferens and Marek Szykuła

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

Abstract
A complete deterministic finite (semi)automaton (DFA) with a set of states Q is completely reachable if every non-empty subset of Q can be obtained as the image of the action of some word applied to Q. The concept of completely reachable automata appeared several times, in particular, in connection with synchronizing automata; the class contains the Černý automata and covers a few separately investigated subclasses. The notion was introduced by Bondar and Volkov (2016), who also raised the question about the complexity of deciding if an automaton is completely reachable. We develop a polynomial-time algorithm for this problem, which is based on a new complement-intersecting technique for finding an extending word for a subset of states. The algorithm works in 𝒪(|Σ|⋅ n³) time, where n = |Q| is the number of states and |Σ| is the size of the input alphabet. Finally, we prove a weak Don’s conjecture for this class of automata: a subset of size k is reachable with a word of length smaller than 2n(n-k). This implies a quadratic upper bound in n on the length of the shortest synchronizing words (reset threshold) for the class of completely reachable automata and generalizes earlier upper bounds derived for its subclasses.
Cite as

Robert Ferens and Marek Szykuła. Completely Reachable Automata: A Polynomial Algorithm and Quadratic Upper Bounds. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ferens_et_al:LIPIcs.ICALP.2023.59,
  author =	{Ferens, Robert and Szyku{\l}a, Marek},
  title =	{{Completely Reachable Automata: A Polynomial Algorithm and Quadratic Upper Bounds}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{59:1--59:17},
  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.2023.59},
  URN =		{urn:nbn:de:0030-drops-181110},
  doi =		{10.4230/LIPIcs.ICALP.2023.59},
  annote =	{Keywords: \v{C}ern\'{y} conjecture, complete reachability, DFA, extending word, reachability, reset threshold, reset word, simple idempotent, synchronizing automaton, synchronizing word}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.85
An Improved Algorithm for Finding the Shortest Synchronizing Words

Authors: Marek Szykuła and Adam Zyzik

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
A synchronizing word of a deterministic finite complete automaton is a word whose action maps every state to a single one. Finding a shortest or a short synchronizing word is a central computational problem in the theory of synchronizing automata and is applied in other areas such as model-based testing and the theory of codes. Because the problem of finding a shortest synchronizing word is computationally hard, among exact algorithms only exponential ones are known. We redesign the previously fastest known exact algorithm based on the bidirectional breadth-first search and improve it with respect to time and space in a practical sense. We develop new algorithmic enhancements and adapt the algorithm to multithreaded and GPU computing. Our experiments show that the new algorithm is multiple times faster than the previously fastest one and its advantage quickly grows with the hardness of the problem instance. Given a modest time limit, we compute the lengths of the shortest synchronizing words for random binary automata up to 570 states, significantly beating the previous record. We refine the experimental estimation of the average reset threshold of these automata. Finally, we develop a general computational package devoted to the problem, where an efficient and practical implementation of our algorithm is included, together with several well-known heuristics.
Cite as

Marek Szykuła and Adam Zyzik. An Improved Algorithm for Finding the Shortest Synchronizing Words. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{szykula_et_al:LIPIcs.ESA.2022.85,
  author =	{Szyku{\l}a, Marek and Zyzik, Adam},
  title =	{{An Improved Algorithm for Finding the Shortest Synchronizing Words}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{85:1--85:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.85},
  URN =		{urn:nbn:de:0030-drops-170237},
  doi =		{10.4230/LIPIcs.ESA.2022.85},
  annote =	{Keywords: \v{C}ern\{\'{y}\} conjecture, reset threshold, reset word, subset checking, synchronizing automaton, synchronizing word}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.46
Lower Bounds on Avoiding Thresholds

Authors: Robert Ferens, Marek Szykuła, and Vojtěch Vorel

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
For a DFA, a word avoids a subset of states, if after reading that word the automaton cannot be in any state from the subset regardless of its initial state. A subset that admits an avoiding word is avoidable. The k-avoiding threshold of a DFA is the smallest number such that every avoidable subset of size k can be avoided with a word no longer than that number. We study the problem of determining the maximum possible k-avoiding thresholds. For every fixed k ≥ 1, we show a general construction of strongly connected DFAs with n states and the k-avoiding threshold in Θ(n^k). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in 𝒪(n²) and in 𝒪(n³). For k = 1, we show that 2n-3 is attainable for every number of states n in the class of strongly connected synchronizing binary DFAs, which is supposed to be the best possible in the class of all DFAs for n ≥ 8. For k = 2, we show that the conjectured solution for k = 1 (an upper bound in 𝒪(n)) also implies a tight upper bound in 𝒪(n²) on 2-avoiding threshold. Finally, we discuss the possibility of using k-avoiding thresholds of synchronizing automata to improve upper bounds on the length of the shortest reset words.
Cite as

Robert Ferens, Marek Szykuła, and Vojtěch Vorel. Lower Bounds on Avoiding Thresholds. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ferens_et_al:LIPIcs.MFCS.2021.46,
  author =	{Ferens, Robert and Szyku{\l}a, Marek and Vorel, Vojt\v{e}ch},
  title =	{{Lower Bounds on Avoiding Thresholds}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.46},
  URN =		{urn:nbn:de:0030-drops-144869},
  doi =		{10.4230/LIPIcs.MFCS.2021.46},
  annote =	{Keywords: avoiding word, \v{C}ern\'{y} conjecture, rank conjecture, reset threshold, reset word, synchronizing automaton, synchronizing word}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.12
Synchronizing Strongly Connected Partial DFAs

Authors: Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, and Marek Szykuła

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

Abstract
We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a reset word) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs with the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.
Cite as

Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, and Marek Szykuła. Synchronizing Strongly Connected Partial DFAs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{berlinkov_et_al:LIPIcs.STACS.2021.12,
  author =	{Berlinkov, Mikhail V. and Ferens, Robert and Ryzhikov, Andrew and Szyku{\l}a, Marek},
  title =	{{Synchronizing Strongly Connected Partial DFAs}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{12:1--12:16},
  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.2021.12},
  URN =		{urn:nbn:de:0030-drops-136579},
  doi =		{10.4230/LIPIcs.STACS.2021.12},
  annote =	{Keywords: \v{C}ern\'{y} conjecture, literal automaton, partial automaton, prefix code, rank conjecture, reset threshold, reset word, synchronizing automaton, synchronizing word}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.16
Existential Length Universality

Authors: Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, and Marek Szykuła

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
We study the following natural variation on the classical universality problem: given a language L(M) represented by M (e.g., a DFA/RE/NFA/PDA), does there exist an integer ? ≥ 0 such that Σ^? ⊆ L(M)? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such ? can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such ? is not bounded by any computable function of the number of states. In the case of a DFA, we show that the problem is NP-complete, and e^{√{n log n} (1+o(1))} is an asymptotically tight upper bound for the smallest such ?, where n is the number of states. Finally, we prove that in all these cases, the problem becomes computationally easier when the length ? is also given in binary in the input: it is polynomially solvable for a DFA, PSPACE-complete for an NFA, and co-NEXPTIME-complete for a PDA.
Cite as

Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, and Marek Szykuła. Existential Length Universality. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gawrychowski_et_al:LIPIcs.STACS.2020.16,
  author =	{Gawrychowski, Pawe{\l} and Lange, Martin and Rampersad, Narad and Shallit, Jeffrey and Szyku{\l}a, Marek},
  title =	{{Existential Length Universality}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.16},
  URN =		{urn:nbn:de:0030-drops-118770},
  doi =		{10.4230/LIPIcs.STACS.2020.16},
  annote =	{Keywords: decision problem, deterministic automaton, nondeterministic automaton, pushdown automaton, regular expression, regular language, universality}
}
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