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

Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon's congruence ~_k is one of the most classical approaches. Two words are ~_k-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of ~_k-classes.
For each equivalence class of ~_k, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in ~_k. We present an algorithm for computing the canonical representative of the ~_k-class of a given word w in A^* of length n. The running time of our algorithm is in O(|A| n) even if k <= n is part of the input. This is surprising since the number of possible subwords grows exponentially in k. The case k>n is not interesting since then, the equivalence class of w is a singleton. If the alphabet is fixed, the running time of our algorithm is linear in the size of the input word. Moreover, for fixed alphabet, we show that the computation of shortlex normal forms for ~_k is possible in deterministic logarithmic space.
One of the consequences of our algorithm is that one can check with the same complexity whether two words are ~_k-equivalent (with k being part of the input).

Lukas Fleischer and Manfred Kufleitner. Testing Simon's congruence. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fleischer_et_al:LIPIcs.MFCS.2018.62, author = {Fleischer, Lukas and Kufleitner, Manfred}, title = {{Testing Simon's congruence}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {62:1--62:13}, 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.2018.62}, URN = {urn:nbn:de:0030-drops-96445}, doi = {10.4230/LIPIcs.MFCS.2018.62}, annote = {Keywords: regular language, scattered subword, piecewise testability, string algorithm} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We investigate the complexity of deciding, given a multiplication table representing a semigroup S, a subset X of S and an element t of S, whether t can be expressed as a product of elements of X. It is well-known that this problem is {NL}-complete and that the more general Cayley groupoid membership problem, where the multiplication table is not required to be associative, is {P}-complete. For groups, the problem can be solved in deterministic log-space which raised the question of determining the exact complexity of this variant. Barrington, Kadau, Lange and McKenzie showed that for Abelian groups and for certain solvable groups, the problem is contained in the complexity class {FOLL} and they concluded that these variants are not hard for any complexity class containing {Parity}. The more general case of arbitrary groups remained open. In this work, we show that for both groups and for commutative semigroups, the problem is solvable in {qAC}^0 (quasi-polynomial size circuits of constant depth with unbounded fan-in) and conclude that these variants are also not hard for any class containing {Parity}. Moreover, we prove that {NL}-completeness already holds for the classes of 0-simple semigroups and nilpotent semigroups. Together with our results on groups and commutative semigroups, we prove the existence of a natural class of finite semigroups which generates a variety of finite semigroups with {NL}-complete Cayley semigroup membership, while the Cayley semigroup membership problem for the class itself is not {NL}-hard. We also discuss applications of our technique to {FOLL}.

Lukas Fleischer. On the Complexity of the Cayley Semigroup Membership Problem. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fleischer:LIPIcs.CCC.2018.25, author = {Fleischer, Lukas}, title = {{On the Complexity of the Cayley Semigroup Membership Problem}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {25:1--25:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.25}, URN = {urn:nbn:de:0030-drops-88649}, doi = {10.4230/LIPIcs.CCC.2018.25}, annote = {Keywords: subsemigroup, multiplication table, generators, completeness, quasi-polynomial-size circuits, FOLL} }

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

We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Thérien.

Lukas Fleischer and Manfred Kufleitner. The Intersection Problem for Finite Monoids. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fleischer_et_al:LIPIcs.STACS.2018.30, author = {Fleischer, Lukas and Kufleitner, Manfred}, title = {{The Intersection Problem for Finite Monoids}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {30:1--30:14}, 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.2018.30}, URN = {urn:nbn:de:0030-drops-85079}, doi = {10.4230/LIPIcs.STACS.2018.30}, annote = {Keywords: intersection problem, finite monoid, recognizing morphism, complexity} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Morphisms to finite semigroups can be used for recognizing omega-regular languages. The so-called strongly recognizing morphisms can be seen as a deterministic computation model which provides minimal objects (known as the syntactic morphism) and a trivial complementation procedure. We give a quadratic-time algorithm for computing the syntactic morphism from any given strongly recognizing morphism, thereby showing that minimization is easy as well. In addition, we give algorithms for efficiently solving various decision problems for weakly recognizing morphisms. Weakly recognizing morphism are often smaller than their strongly recognizing counterparts. Finally, we describe the language operations needed for converting formulas in monadic second-order logic (MSO) into strongly recognizing morphisms, and we give some experimental results.

Lukas Fleischer and Manfred Kufleitner. Efficient Algorithms for Morphisms over Omega-Regular Languages. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 112-124, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fleischer_et_al:LIPIcs.FSTTCS.2015.112, author = {Fleischer, Lukas and Kufleitner, Manfred}, title = {{Efficient Algorithms for Morphisms over Omega-Regular Languages}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {112--124}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.112}, URN = {urn:nbn:de:0030-drops-56200}, doi = {10.4230/LIPIcs.FSTTCS.2015.112}, annote = {Keywords: B\"{u}chi automata, omega-regular language, syntactic semigroup, recognizing morphism, MSO} }

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