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

We study an operator on classes of languages. For each class 𝒞, it produces a new class FO²(𝕀_𝒞) associated with a variant of two-variable first-order logic equipped with a signature 𝕀_𝒞 built from 𝒞. For 𝒞 = {∅, A*}, we obtain the usual FO²(<)} logic, equipped with linear order. For 𝒞 = {∅,{ε},A+,A*}, we get the variant FO²(<,+1), which also includes the successor predicate. If 𝒞 consists of all Boolean combinations of languages A*aA*, where a is a letter, we get the variant FO²(< ,Bet), which includes "between" relations. We prove a generic algebraic characterization of the classes FO^2(𝕀_𝒞). It elegantly generalizes those known for all the cases mentioned above. Moreover, it implies that if 𝒞 has decidable separation (plus some standard properties), then FO²2(𝕀_𝒞) has a decidable membership problem.
We actually work with an equivalent definition of FO²(𝕀_𝒞) in terms of unary temporal logic. For each class 𝒞, we consider a variant TL(𝒞) of unary temporal logic whose future/past modalities depend on 𝒞 and such that TL(𝒞) = FO²(𝕀_𝒞). Finally, we also characterize FL(𝒞) and PL(𝒞), the pure-future and pure-past restrictions of TL(𝒞). Like for TL(𝒞), these characterizations imply that if 𝒞 is a class with decidable separation, then FL(𝒞) and PL(𝒞) have decidable membership.

Thomas Place and Marc Zeitoun. A Generic Characterization of Generalized Unary Temporal Logic and Two-Variable First-Order Logic. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 45:1-45:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{place_et_al:LIPIcs.CSL.2024.45, author = {Place, Thomas and Zeitoun, Marc}, title = {{A Generic Characterization of Generalized Unary Temporal Logic and Two-Variable First-Order Logic}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {45:1--45:23}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.45}, URN = {urn:nbn:de:0030-drops-196888}, doi = {10.4230/LIPIcs.CSL.2024.45}, annote = {Keywords: Classes of regular languages, Generalized unary temporal logic, Generalized two-variable first-order logic, Generic decidable characterizations, Membership, Separation} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

We look at classes of languages associated to fragments of first-order logic BΣ₁, in which quantifier alternations are disallowed. Each such fragment is fully determined by choosing the set of predicates on positions that may be used. Equipping first-order logic with the linear ordering and possibly the successor relation as predicates yields two natural fragments, which were investigated by Simon and Knast, who proved that these two variants have decidable membership: "does an input regular language belong to the class ?". We extend their results in two orthogonal directions.
- First, instead of membership, we explore the more general separation problem: decide if two regular languages can be separated by a language from the class under study.
- Second, we use more general inputs: classes 𝒢 of group languages (i.e., recognized by a DFA in which each letter induces a permutation of the states) and extensions thereof, written 𝒢^+. We rely on a characterization of BΣ₁ by the operator BPol: given an input class 𝒞, it outputs a class BPol(𝒞) that corresponds to a variant of BΣ₁ equipped with special predicates associated to 𝒞. The classes BPol(𝒢) and BPol(𝒢^+) capture many natural variants of BΣ₁ which use predicates such as the linear ordering, the successor, the modular predicates or the alphabetic modular predicates.
We show that separation is decidable for BPol(𝒢) and BPol(𝒢^+) when this is the case for 𝒢. This was already known for BPol(𝒢) and for two particular classes of the form BPol(𝒢^+). Yet, the algorithms were indirect and relied on involved frameworks, yielding poor upper complexity bounds. In contrast, our approach is direct. We work only with elementary concepts (mainly, finite automata). Our main contribution consists in polynomial time Turing reductions from both BPol(𝒢)- and BPol(𝒢^+)-separation to 𝒢-separation. This yields polynomial time algorithms for several key variants of BΣ₁, including those equipped with the linear ordering and possibly the successor and/or the modular predicates.

Thomas Place and Marc Zeitoun. A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 43:1-43:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{place_et_al:LIPIcs.FSTTCS.2022.43, author = {Place, Thomas and Zeitoun, Marc}, title = {{A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {43:1--43:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.43}, URN = {urn:nbn:de:0030-drops-174356}, doi = {10.4230/LIPIcs.FSTTCS.2022.43}, annote = {Keywords: Automata, Separation, Covering, Concatenation hierarchies, Group languages} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with decidable separation (plus mild additional properties) has decidable separation. We actually show decidability of a stronger property, called covering. This generalizes many results on the subject in a unified framework. A key ingredient is that star-free closure coincides with another closure operator where Kleene stars are also allowed in restricted contexts.

Thomas Place and Marc Zeitoun. On All Things Star-Free (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 126:1-126:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{place_et_al:LIPIcs.ICALP.2019.126, author = {Place, Thomas and Zeitoun, Marc}, title = {{On All Things Star-Free}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {126:1--126:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.126}, URN = {urn:nbn:de:0030-drops-107028}, doi = {10.4230/LIPIcs.ICALP.2019.126}, annote = {Keywords: Regular languages, separation problem, star-free closure, group languages} }

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

We investigate the complexity of the separation problem associated to classes of regular languages. For a class C, C-separation takes two regular languages as input and asks whether there exists a third language in C which includes the first and is disjoint from the second. First, in contrast with the situation for the classical membership problem, we prove that for most classes C, the complexity of C-separation does not depend on how the input languages are represented: it is the same for nondeterministic finite automata and monoid morphisms. Then, we investigate specific classes belonging to finitely based concatenation hierarchies. It was recently proved that the problem is always decidable for levels 1/2 and 1 of any such hierarchy (with inefficient algorithms). Here, we build on these results to show that when the alphabet is fixed, there are polynomial time algorithms for both levels. Finally, we investigate levels 3/2 and 2 of the famous Straubing-Thérien hierarchy. We show that separation is PSpace-complete for level 3/2 and between PSpace-hard and EXPTime for level 2.

Thomas Place and Marc Zeitoun. The Complexity of Separation for Levels in Concatenation Hierarchies. 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. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{place_et_al:LIPIcs.FSTTCS.2018.47, author = {Place, Thomas and Zeitoun, Marc}, title = {{The Complexity of Separation for Levels in Concatenation Hierarchies}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {47:1--47:17}, 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.2018.47}, URN = {urn:nbn:de:0030-drops-99463}, doi = {10.4230/LIPIcs.FSTTCS.2018.47}, annote = {Keywords: Regular languages, separation, concatenation hierarchies, complexity} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We investigate a standard operator on classes of languages: unambiguous polynomial closure. We show that if C is a class of regular languages having some mild properties, the membership problem for its unambiguous polynomial closure UPol(C) reduces to the same problem for C. We give a new, self-contained and elementary proof of this result. We also show that unambiguous polynomial closure coincides with alternating left and right deterministic closure. Finally, if additionally C is finite, we show that the separation and covering problems are decidable for UPol(C).

Thomas Place and Marc Zeitoun. Separating Without Any Ambiguity. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 137:1-137:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{place_et_al:LIPIcs.ICALP.2018.137, author = {Place, Thomas and Zeitoun, Marc}, title = {{Separating Without Any Ambiguity}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {137:1--137:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.137}, URN = {urn:nbn:de:0030-drops-91419}, doi = {10.4230/LIPIcs.ICALP.2018.137}, annote = {Keywords: Regular languages, separation problem, decidable characterizations} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

An important endeavor in computer science is to precisely understand the expressive power of logical formalisms over discrete structures, such as words. Naturally, "understanding" is not a mathematical notion. Therefore, this investigation requires a concrete objective to capture such a notion. In the literature, the standard choice for this objective is the membership problem, whose aim is to find a procedure deciding whether an input regular language can be defined in the logic under study. This approach was cemented as the "right" one by the seminal work of Schuetzenberger, McNaughton and Papert on first-order logic and has been in use since then.
However, membership questions are hard: for several important fragments, researchers have failed in this endeavor despite decades of investigation. In view of recent results on one of the most famous open questions, namely the quantifier alternation hierarchy of first-order logic, an explanation may be that membership is too restrictive as a setting. These new results were indeed obtained by considering more general problems than membership, taking advantage of the increased flexibility of the enriched mathematical setting. This opens a promising avenue of research and efforts have been devoted at identifying and solving such problems for natural fragments. However, until now, these problems have been ad hoc, most fragments relying on a specific one. A unique new problem replacing membership as the right one is still missing.
The main contribution of this paper is a suitable candidate to play this role: the Covering Problem. We motivate this problem with three arguments. First, it admits an elementary set theoretic formulation, similar to membership. Second, we are able to reexplain or generalize all known results with this problem. Third, we develop a mathematical framework as well as a methodology tailored to the investigation of this problem.

Thomas Place and Marc Zeitoun. The Covering Problem: A Unified Approach for Investigating the Expressive Power of Logics. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{place_et_al:LIPIcs.MFCS.2016.77, author = {Place, Thomas and Zeitoun, Marc}, title = {{The Covering Problem: A Unified Approach for Investigating the Expressive Power of Logics}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {77:1--77:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.77}, URN = {urn:nbn:de:0030-drops-64857}, doi = {10.4230/LIPIcs.MFCS.2016.77}, annote = {Keywords: membership problem, separation problem, covering problem, regular languages, logics, decidable characterizations} }

Document

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We investigate two problems for a class C of regular word languages. The
C-membership problem asks for an algorithm to decide whether an input language belongs to C. The C-separation problem asks for an algorithm that, given as input two regular languages, decides whether there exists a third language in C containing the first language, while being disjoint from the second. These problems are considered as means to obtain a deep understanding of the class C.
It is usual for such classes to be defined by logical formalisms. Logics are often built on top of each other, by adding new predicates. A natural construction is to enrich a logic with the successor relation. In this paper, we obtain new and simple proofs of two transfer results: we show that for suitable logically defined classes, the membership, resp. the separation problem for a class enriched with the successor relation reduces to the same problem for the original class.
Our reductions work both for languages of finite words and infinite words. The proofs are mostly self-contained, and only require a basic background on regular languages. This paper therefore gives simple proofs of results that were considered as difficult, such as the decidability of the membership problem for the levels 1, 3/2, 2 and 5/2 of the dot-depth hierarchy.

Thomas Place and Marc Zeitoun. Separation and the Successor Relation. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 662-675, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{place_et_al:LIPIcs.STACS.2015.662, author = {Place, Thomas and Zeitoun, Marc}, title = {{Separation and the Successor Relation}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {662--675}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.662}, URN = {urn:nbn:de:0030-drops-49499}, doi = {10.4230/LIPIcs.STACS.2015.662}, annote = {Keywords: separation problem, regular word languages, logics, decidable characterizations, semidirect product} }

Document

**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

A separator for two languages is a third language containing the first one and disjoint from the second one. We investigate the following decision problem: given two regular input languages, decide whether there exists a locally testable (resp. a locally threshold testable) separator. In both cases, we design a decision procedure based on the occurrence of special patterns in automata accepting the input languages. We prove that the problem is computationally harder than deciding membership. The correctness proof of the algorithm yields a stronger result, namely a description of a possible separator. Finally, we discuss the same problem for context-free input languages.

Thomas Place, Lorijn van Rooijen, and Marc Zeitoun. Separating Regular Languages by Locally Testable and Locally Threshold Testable Languages. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 363-375, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{place_et_al:LIPIcs.FSTTCS.2013.363, author = {Place, Thomas and van Rooijen, Lorijn and Zeitoun, Marc}, title = {{Separating Regular Languages by Locally Testable and Locally Threshold Testable Languages}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {363--375}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.363}, URN = {urn:nbn:de:0030-drops-43867}, doi = {10.4230/LIPIcs.FSTTCS.2013.363}, annote = {Keywords: Automata, Logics, Monoids, Locally testable, Separation, Context-free.} }

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