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Track A: Algorithms, Complexity and Games

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

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.

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.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

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

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.

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.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

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

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.

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.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} }

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

A word equation with one variable in a free group is given as U = V, where both U and V are words over the alphabet of generators of the free group and X, X⁻¹, for a fixed variable X. An element of the free group is a solution when substituting it for X yields a true equality (interpreted in the free group) of left- and right-hand sides. It is known that the set of all solutions of a given word equation with one variable is a finite union of sets of the form {α wⁱ β : i ∈ ℤ}, where α, w, β are reduced words over the alphabet of generators, and a polynomial-time algorithm (of a high degree) computing this set is known. We provide a cubic time algorithm for this problem, which also shows that the set of solutions consists of at most a quadratic number of the above-mentioned sets. The algorithm uses only simple tools of word combinatorics and group theory and is simple to state. Its analysis is involved and focuses on the combinatorics of occurrences of powers of a word within a larger word.

Robert Ferens and Artur Jeż. Solving One Variable Word Equations in the Free Group in Cubic Time. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ferens_et_al:LIPIcs.STACS.2021.30, author = {Ferens, Robert and Je\.{z}, Artur}, title = {{Solving One Variable Word Equations in the Free Group in Cubic Time}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {30:1--30:17}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.30}, URN = {urn:nbn:de:0030-drops-136751}, doi = {10.4230/LIPIcs.STACS.2021.30}, annote = {Keywords: Word equations, free group, one-variable equations} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states that are mapped to a state from S by the action of w. We study the computational complexity of three problems related to the existence of words yielding certain preimages, which are especially motivated by the theory of synchronizing automata. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a word totally extending the subset (giving the whole set of states) - it is equivalent to the problem whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a word resizing the subset (giving a preimage of a different size). We also consider the variants of the problem where an upper bound on the length of the word is given in the input. Because in most cases our problems are computationally hard, we additionally consider parametrized complexity by the size of the given subset. We focus on the most interesting cases that are the subclasses of strongly connected, synchronizing, and binary automata.

Mikhail V. Berlinkov, Robert Ferens, and Marek Szykula. Complexity of Preimage Problems for Deterministic Finite Automata. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{berlinkov_et_al:LIPIcs.MFCS.2018.32, author = {Berlinkov, Mikhail V. and Ferens, Robert and Szykula, Marek}, title = {{Complexity of Preimage Problems for Deterministic Finite Automata}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.32}, URN = {urn:nbn:de:0030-drops-96149}, doi = {10.4230/LIPIcs.MFCS.2018.32}, annote = {Keywords: avoiding word, extending word, extensible subset, reset word, synchronizing automaton} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and
[(p-1)(q-1), p(q-2)+n-q+1], where 2 <= p < q <= n, p+q > n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n).
This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers.
We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case.
Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n).

Michalina Dzyga, Robert Ferens, Vladimir V. Gusev, and Marek Szykula. Attainable Values of Reset Thresholds. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dzyga_et_al:LIPIcs.MFCS.2017.40, author = {Dzyga, Michalina and Ferens, Robert and Gusev, Vladimir V. and Szykula, Marek}, title = {{Attainable Values of Reset Thresholds}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {40:1--40:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.40}, URN = {urn:nbn:de:0030-drops-81220}, doi = {10.4230/LIPIcs.MFCS.2017.40}, annote = {Keywords: Cerny conjecture, exponent, primitive digraph, reset word, synchronizing automaton} }

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