Found 3 Possible Name Variants:

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We call a first-order formula one-dimensional if every maximal block of existential (or universal) quantifiers in it leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO^2. We denote the resulting formalisms, resp., GF_1, and TGF_1. We show that GF_1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF_1 we show that it is decidable, has the finite model property, and its satisfiability problem is 2-ExpTime-complete (NExpTime-complete in the absence of equality). All the above-mentioned results are obtained for signatures with no constants. We finally discuss the impact of their addition, observing that constants do not spoil the decidability but increase the complexity of the satisfiability problem.

Emanuel Kieroński. One-Dimensional Guarded Fragments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kieronski:LIPIcs.MFCS.2019.16, author = {Kiero\'{n}ski, Emanuel}, title = {{One-Dimensional Guarded Fragments}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.16}, URN = {urn:nbn:de:0030-drops-109608}, doi = {10.4230/LIPIcs.MFCS.2019.16}, annote = {Keywords: guarded fragment, two-variable logic, satisfiability, finite model property} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We show that the finite satisfiability problem for the unary negation fragment with an arbitrary number of transitive relations is decidable and 2-ExpTime-complete. Our result actually holds for a more general setting in which one can require that some binary symbols are interpreted as arbitrary transitive relations, some as partial orders and some as equivalences. We also consider finite satisfiability of various extensions of our primary logic, in particular capturing the concepts of nominals and role hierarchies known from description logic. As the unary negation fragment can express unions of conjunctive queries, our results have interesting implications for the problem of finite query answering, both in the classical scenario and in the description logics setting.

Daniel Danielski and Emanuel Kieroński. Finite Satisfiability of Unary Negation Fragment with Transitivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{danielski_et_al:LIPIcs.MFCS.2019.17, author = {Danielski, Daniel and Kiero\'{n}ski, Emanuel}, title = {{Finite Satisfiability of Unary Negation Fragment with Transitivity}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.17}, URN = {urn:nbn:de:0030-drops-109612}, doi = {10.4230/LIPIcs.MFCS.2019.17}, annote = {Keywords: unary negation fragment, transitivity, finite satisfiability, finite open-world query answering, description logics} }

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

A one-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. This fragment contains two-variable logic, and it is known that over words both formalisms have the same complexity and expressive power. Here we investigate the one-dimensional fragment over trees. We consider unranked unordered trees accessible by one or both of the descendant and child relations, as well as ordered trees equipped additionally with sibling relations. We show that over unordered trees the satisfiability problem is ExpSpace-complete when only the descendant relation is available and 2ExpTime-complete with both the descendant and child or with only the child relation. Over ordered trees the problem remains 2ExpTime-complete. Regarding expressivity, we show that over ordered trees and over unordered trees accessible by both the descendant and child the one-dimensional fragment is equivalent to the two-variable fragment with counting quantifiers.

Emanuel Kieronski and Antti Kuusisto. One-Dimensional Logic over Trees. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kieronski_et_al:LIPIcs.MFCS.2017.64, author = {Kieronski, Emanuel and Kuusisto, Antti}, title = {{One-Dimensional Logic over Trees}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {64:1--64:13}, 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.2017.64}, URN = {urn:nbn:de:0030-drops-80658}, doi = {10.4230/LIPIcs.MFCS.2017.64}, annote = {Keywords: satisfiability, expressivity, trees, fragments of first-order logic} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We consider two extensions of this logic. We show that adding either additional binary symbols or counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem. However, combining the two extensions and adding both binary symbols and counting quantifiers leads to an explosion of this complexity. We also compare the expressive power of the two-variable fragment over trees with its extension with counting quantifiers. It turns out that the two logics are equally expressive, although counting quantifiers do add expressive power in the restricted case of unordered trees.

Bartosz Bednarczyk, Witold Charatonik, and Emanuel Kieronski. Extending Two-Variable Logic on Trees. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bednarczyk_et_al:LIPIcs.CSL.2017.11, author = {Bednarczyk, Bartosz and Charatonik, Witold and Kieronski, Emanuel}, title = {{Extending Two-Variable Logic on Trees}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.11}, URN = {urn:nbn:de:0030-drops-76794}, doi = {10.4230/LIPIcs.CSL.2017.11}, annote = {Keywords: two-variable logic, trees, satisfiability, expressivity, counting quantifiers} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. We investigate one-dimensional fragment over words and over omega-words. We show that it is expressively equivalent to the two-variable fragment of first-order logic. We also show that its satisfiability problem is NExpTime-complete. Further, we show undecidability of some extensions, whose two-variable counterparts remain decidable.

Emanuel Kieronski. One-Dimensional Logic over Words. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kieronski:LIPIcs.CSL.2016.38, author = {Kieronski, Emanuel}, title = {{One-Dimensional Logic over Words}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {38:1--38:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.38}, URN = {urn:nbn:de:0030-drops-65782}, doi = {10.4230/LIPIcs.CSL.2016.38}, annote = {Keywords: satisfiability, expressivity, words, fragments of first-order logic} }

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**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

The uniform one-dimensional fragment U1 of first-order logic was introduced recently as a natural generalization of the two-variable fragment FO2 to contexts with relation symbols of all arities. It was shown that U1 has the exponential model property and NEXPTIME-complete satisfiability problem. In this paper we investigate two restrictions of U1 that still contain FO2. We call these logics RU1 and SU1, or the restricted and strongly restricted uniform one-dimensional fragments. We introduce Ehrenfeucht-Fraisse games for the logics and prove that while SU1 and RU1 are expressively equivalent, they are strictly contained in U1. Furthermore, we consider extensions of the logics SU1, RU1 and U1 with unrestricted use of a single built-in equivalence relation E. We prove that while all the obtained systems retain the finite model property, their complexities differ. Namely, the satisfiability problem is NEXPTIME-complete for SU1(E) and 2NEXPTIME-complete for both RU1(E) and U1(E). Finally, we show undecidability of some natural extensions of SU1.

Emanuel Kierónski and Antti Kuusisto. Uniform One-Dimensional Fragments with One Equivalence Relation. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 597-615, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kieronski_et_al:LIPIcs.CSL.2015.597, author = {Kier\'{o}nski, Emanuel and Kuusisto, Antti}, title = {{Uniform One-Dimensional Fragments with One Equivalence Relation}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {597--615}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.597}, URN = {urn:nbn:de:0030-drops-54418}, doi = {10.4230/LIPIcs.CSL.2015.597}, annote = {Keywords: two-variable logic, uniform one-dimensional fragments, complexity, expressivity, equivalence relations} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

We prove that the satisfiability problem for the two-variable, universal fragment of first-order logic with constants (or, alternatively phrased, for the Bernays-Schönfinkel class with two universally quantified variables) remains decidable after augmenting the fragment by the transitive closure of a single binary relation. We give a 2-NExpTime-upper bound and a 2-ExpTime-lower bound for the complexity of the problem. We also study the cases in which the number of constants is restricted. It appears that with two constants the considered fragment has the finite model property and NExpTime-complete satisfiability problem. Adding a third constant does not change the complexity but allows to construct infinity axioms. A fourth constant lifts the lower complexity bound to TwoExpTime. Finally, we observe that we are close to the border between decidability and undecidability: adding a third variable or the transitive closure of a second binary relation lead to undecidability.

Emanuel Kieronski and Jakub Michaliszyn. Two-Variable Universal Logic with Transitive Closure. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 396-410, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kieronski_et_al:LIPIcs.CSL.2012.396, author = {Kieronski, Emanuel and Michaliszyn, Jakub}, title = {{Two-Variable Universal Logic with Transitive Closure}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {396--410}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.396}, URN = {urn:nbn:de:0030-drops-36860}, doi = {10.4230/LIPIcs.CSL.2012.396}, annote = {Keywords: two-variable logic, transitive closure, decidability} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We consider the satisfiability problem for modal logic over classes of structures definable by universal first-order formulas with three variables. We exhibit a simple formula for which the problem is undecidable. This improves an earlier result in which nine variables were used. We also show that for classes defined by three-variable, universal Horn formulas the problem is decidable. This subsumes decidability results for many natural modal logics, including T, B, K4, S4, S5.

Emanuel Kieronski, Jakub Michaliszyn, and Jan Otop. Modal Logics Definable by Universal Three-Variable Formulas. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 264-275, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kieronski_et_al:LIPIcs.FSTTCS.2011.264, author = {Kieronski, Emanuel and Michaliszyn, Jakub and Otop, Jan}, title = {{Modal Logics Definable by Universal Three-Variable Formulas}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {264--275}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.264}, URN = {urn:nbn:de:0030-drops-33495}, doi = {10.4230/LIPIcs.FSTTCS.2011.264}, annote = {Keywords: modal logic, decidability} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

We show that the satisfiability and the finite satisfiability problems for two-variable logic, FO2, over the class of structures with three linear orders, are undecidable. This sharpens an earlier result that FO2 with eight linear orders is undecidable. The theorem holds for a restricted case in which linear orders are the only non-unary relations. Recently, a contrasting result has been shown, that the finite satisfiability problem for FO2 with two linear orders and with no additional non-unary relations is decidable. We observe that our proof can be adapted to some interesting fragments of FO2, in particular it works for the two-variable guarded fragment, GF2, even if the order relations are used only as guards. Finally, we show that GF2 with an arbitrary number of linear orders which can be used only as guards becomes decidable if except linear orders only unary relations are allowed.

Emanuel Kieronski. Decidability Issues for Two-Variable Logics with Several Linear Orders. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 337-351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kieronski:LIPIcs.CSL.2011.337, author = {Kieronski, Emanuel}, title = {{Decidability Issues for Two-Variable Logics with Several Linear Orders}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {337--351}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.337}, URN = {urn:nbn:de:0030-drops-32415}, doi = {10.4230/LIPIcs.CSL.2011.337}, annote = {Keywords: two-variable logic, linear orders, guarded fragment, decidability} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We call a first-order formula one-dimensional if every maximal block of existential (or universal) quantifiers in it leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO^2. We denote the resulting formalisms, resp., GF_1, and TGF_1. We show that GF_1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF_1 we show that it is decidable, has the finite model property, and its satisfiability problem is 2-ExpTime-complete (NExpTime-complete in the absence of equality). All the above-mentioned results are obtained for signatures with no constants. We finally discuss the impact of their addition, observing that constants do not spoil the decidability but increase the complexity of the satisfiability problem.

Emanuel Kieroński. One-Dimensional Guarded Fragments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kieronski:LIPIcs.MFCS.2019.16, author = {Kiero\'{n}ski, Emanuel}, title = {{One-Dimensional Guarded Fragments}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.16}, URN = {urn:nbn:de:0030-drops-109608}, doi = {10.4230/LIPIcs.MFCS.2019.16}, annote = {Keywords: guarded fragment, two-variable logic, satisfiability, finite model property} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We show that the finite satisfiability problem for the unary negation fragment with an arbitrary number of transitive relations is decidable and 2-ExpTime-complete. Our result actually holds for a more general setting in which one can require that some binary symbols are interpreted as arbitrary transitive relations, some as partial orders and some as equivalences. We also consider finite satisfiability of various extensions of our primary logic, in particular capturing the concepts of nominals and role hierarchies known from description logic. As the unary negation fragment can express unions of conjunctive queries, our results have interesting implications for the problem of finite query answering, both in the classical scenario and in the description logics setting.

Daniel Danielski and Emanuel Kieroński. Finite Satisfiability of Unary Negation Fragment with Transitivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{danielski_et_al:LIPIcs.MFCS.2019.17, author = {Danielski, Daniel and Kiero\'{n}ski, Emanuel}, title = {{Finite Satisfiability of Unary Negation Fragment with Transitivity}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.17}, URN = {urn:nbn:de:0030-drops-109612}, doi = {10.4230/LIPIcs.MFCS.2019.17}, annote = {Keywords: unary negation fragment, transitivity, finite satisfiability, finite open-world query answering, description logics} }

Document

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

A one-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. This fragment contains two-variable logic, and it is known that over words both formalisms have the same complexity and expressive power. Here we investigate the one-dimensional fragment over trees. We consider unranked unordered trees accessible by one or both of the descendant and child relations, as well as ordered trees equipped additionally with sibling relations. We show that over unordered trees the satisfiability problem is ExpSpace-complete when only the descendant relation is available and 2ExpTime-complete with both the descendant and child or with only the child relation. Over ordered trees the problem remains 2ExpTime-complete. Regarding expressivity, we show that over ordered trees and over unordered trees accessible by both the descendant and child the one-dimensional fragment is equivalent to the two-variable fragment with counting quantifiers.

Emanuel Kieronski and Antti Kuusisto. One-Dimensional Logic over Trees. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kieronski_et_al:LIPIcs.MFCS.2017.64, author = {Kieronski, Emanuel and Kuusisto, Antti}, title = {{One-Dimensional Logic over Trees}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {64:1--64:13}, 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.2017.64}, URN = {urn:nbn:de:0030-drops-80658}, doi = {10.4230/LIPIcs.MFCS.2017.64}, annote = {Keywords: satisfiability, expressivity, trees, fragments of first-order logic} }

Document

**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We consider two extensions of this logic. We show that adding either additional binary symbols or counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem. However, combining the two extensions and adding both binary symbols and counting quantifiers leads to an explosion of this complexity. We also compare the expressive power of the two-variable fragment over trees with its extension with counting quantifiers. It turns out that the two logics are equally expressive, although counting quantifiers do add expressive power in the restricted case of unordered trees.

Bartosz Bednarczyk, Witold Charatonik, and Emanuel Kieronski. Extending Two-Variable Logic on Trees. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bednarczyk_et_al:LIPIcs.CSL.2017.11, author = {Bednarczyk, Bartosz and Charatonik, Witold and Kieronski, Emanuel}, title = {{Extending Two-Variable Logic on Trees}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.11}, URN = {urn:nbn:de:0030-drops-76794}, doi = {10.4230/LIPIcs.CSL.2017.11}, annote = {Keywords: two-variable logic, trees, satisfiability, expressivity, counting quantifiers} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. We investigate one-dimensional fragment over words and over omega-words. We show that it is expressively equivalent to the two-variable fragment of first-order logic. We also show that its satisfiability problem is NExpTime-complete. Further, we show undecidability of some extensions, whose two-variable counterparts remain decidable.

Emanuel Kieronski. One-Dimensional Logic over Words. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kieronski:LIPIcs.CSL.2016.38, author = {Kieronski, Emanuel}, title = {{One-Dimensional Logic over Words}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {38:1--38:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.38}, URN = {urn:nbn:de:0030-drops-65782}, doi = {10.4230/LIPIcs.CSL.2016.38}, annote = {Keywords: satisfiability, expressivity, words, fragments of first-order logic} }

Document

**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

The uniform one-dimensional fragment U1 of first-order logic was introduced recently as a natural generalization of the two-variable fragment FO2 to contexts with relation symbols of all arities. It was shown that U1 has the exponential model property and NEXPTIME-complete satisfiability problem. In this paper we investigate two restrictions of U1 that still contain FO2. We call these logics RU1 and SU1, or the restricted and strongly restricted uniform one-dimensional fragments. We introduce Ehrenfeucht-Fraisse games for the logics and prove that while SU1 and RU1 are expressively equivalent, they are strictly contained in U1. Furthermore, we consider extensions of the logics SU1, RU1 and U1 with unrestricted use of a single built-in equivalence relation E. We prove that while all the obtained systems retain the finite model property, their complexities differ. Namely, the satisfiability problem is NEXPTIME-complete for SU1(E) and 2NEXPTIME-complete for both RU1(E) and U1(E). Finally, we show undecidability of some natural extensions of SU1.

Emanuel Kierónski and Antti Kuusisto. Uniform One-Dimensional Fragments with One Equivalence Relation. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 597-615, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kieronski_et_al:LIPIcs.CSL.2015.597, author = {Kier\'{o}nski, Emanuel and Kuusisto, Antti}, title = {{Uniform One-Dimensional Fragments with One Equivalence Relation}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {597--615}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.597}, URN = {urn:nbn:de:0030-drops-54418}, doi = {10.4230/LIPIcs.CSL.2015.597}, annote = {Keywords: two-variable logic, uniform one-dimensional fragments, complexity, expressivity, equivalence relations} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

We prove that the satisfiability problem for the two-variable, universal fragment of first-order logic with constants (or, alternatively phrased, for the Bernays-Schönfinkel class with two universally quantified variables) remains decidable after augmenting the fragment by the transitive closure of a single binary relation. We give a 2-NExpTime-upper bound and a 2-ExpTime-lower bound for the complexity of the problem. We also study the cases in which the number of constants is restricted. It appears that with two constants the considered fragment has the finite model property and NExpTime-complete satisfiability problem. Adding a third constant does not change the complexity but allows to construct infinity axioms. A fourth constant lifts the lower complexity bound to TwoExpTime. Finally, we observe that we are close to the border between decidability and undecidability: adding a third variable or the transitive closure of a second binary relation lead to undecidability.

Emanuel Kieronski and Jakub Michaliszyn. Two-Variable Universal Logic with Transitive Closure. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 396-410, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kieronski_et_al:LIPIcs.CSL.2012.396, author = {Kieronski, Emanuel and Michaliszyn, Jakub}, title = {{Two-Variable Universal Logic with Transitive Closure}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {396--410}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.396}, URN = {urn:nbn:de:0030-drops-36860}, doi = {10.4230/LIPIcs.CSL.2012.396}, annote = {Keywords: two-variable logic, transitive closure, decidability} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We consider the satisfiability problem for modal logic over classes of structures definable by universal first-order formulas with three variables. We exhibit a simple formula for which the problem is undecidable. This improves an earlier result in which nine variables were used. We also show that for classes defined by three-variable, universal Horn formulas the problem is decidable. This subsumes decidability results for many natural modal logics, including T, B, K4, S4, S5.

Emanuel Kieronski, Jakub Michaliszyn, and Jan Otop. Modal Logics Definable by Universal Three-Variable Formulas. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 264-275, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kieronski_et_al:LIPIcs.FSTTCS.2011.264, author = {Kieronski, Emanuel and Michaliszyn, Jakub and Otop, Jan}, title = {{Modal Logics Definable by Universal Three-Variable Formulas}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {264--275}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.264}, URN = {urn:nbn:de:0030-drops-33495}, doi = {10.4230/LIPIcs.FSTTCS.2011.264}, annote = {Keywords: modal logic, decidability} }

Document

**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

We show that the satisfiability and the finite satisfiability problems for two-variable logic, FO2, over the class of structures with three linear orders, are undecidable. This sharpens an earlier result that FO2 with eight linear orders is undecidable. The theorem holds for a restricted case in which linear orders are the only non-unary relations. Recently, a contrasting result has been shown, that the finite satisfiability problem for FO2 with two linear orders and with no additional non-unary relations is decidable. We observe that our proof can be adapted to some interesting fragments of FO2, in particular it works for the two-variable guarded fragment, GF2, even if the order relations are used only as guards. Finally, we show that GF2 with an arbitrary number of linear orders which can be used only as guards becomes decidable if except linear orders only unary relations are allowed.

Emanuel Kieronski. Decidability Issues for Two-Variable Logics with Several Linear Orders. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 337-351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kieronski:LIPIcs.CSL.2011.337, author = {Kieronski, Emanuel}, title = {{Decidability Issues for Two-Variable Logics with Several Linear Orders}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {337--351}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.337}, URN = {urn:nbn:de:0030-drops-32415}, doi = {10.4230/LIPIcs.CSL.2011.337}, annote = {Keywords: two-variable logic, linear orders, guarded fragment, decidability} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We call a first-order formula one-dimensional if every maximal block of existential (or universal) quantifiers in it leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO^2. We denote the resulting formalisms, resp., GF_1, and TGF_1. We show that GF_1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF_1 we show that it is decidable, has the finite model property, and its satisfiability problem is 2-ExpTime-complete (NExpTime-complete in the absence of equality). All the above-mentioned results are obtained for signatures with no constants. We finally discuss the impact of their addition, observing that constants do not spoil the decidability but increase the complexity of the satisfiability problem.

Emanuel Kieroński. One-Dimensional Guarded Fragments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kieronski:LIPIcs.MFCS.2019.16, author = {Kiero\'{n}ski, Emanuel}, title = {{One-Dimensional Guarded Fragments}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.16}, URN = {urn:nbn:de:0030-drops-109608}, doi = {10.4230/LIPIcs.MFCS.2019.16}, annote = {Keywords: guarded fragment, two-variable logic, satisfiability, finite model property} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We show that the finite satisfiability problem for the unary negation fragment with an arbitrary number of transitive relations is decidable and 2-ExpTime-complete. Our result actually holds for a more general setting in which one can require that some binary symbols are interpreted as arbitrary transitive relations, some as partial orders and some as equivalences. We also consider finite satisfiability of various extensions of our primary logic, in particular capturing the concepts of nominals and role hierarchies known from description logic. As the unary negation fragment can express unions of conjunctive queries, our results have interesting implications for the problem of finite query answering, both in the classical scenario and in the description logics setting.

Daniel Danielski and Emanuel Kieroński. Finite Satisfiability of Unary Negation Fragment with Transitivity. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{danielski_et_al:LIPIcs.MFCS.2019.17, author = {Danielski, Daniel and Kiero\'{n}ski, Emanuel}, title = {{Finite Satisfiability of Unary Negation Fragment with Transitivity}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.17}, URN = {urn:nbn:de:0030-drops-109612}, doi = {10.4230/LIPIcs.MFCS.2019.17}, annote = {Keywords: unary negation fragment, transitivity, finite satisfiability, finite open-world query answering, description logics} }

Document

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

A one-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. This fragment contains two-variable logic, and it is known that over words both formalisms have the same complexity and expressive power. Here we investigate the one-dimensional fragment over trees. We consider unranked unordered trees accessible by one or both of the descendant and child relations, as well as ordered trees equipped additionally with sibling relations. We show that over unordered trees the satisfiability problem is ExpSpace-complete when only the descendant relation is available and 2ExpTime-complete with both the descendant and child or with only the child relation. Over ordered trees the problem remains 2ExpTime-complete. Regarding expressivity, we show that over ordered trees and over unordered trees accessible by both the descendant and child the one-dimensional fragment is equivalent to the two-variable fragment with counting quantifiers.

Emanuel Kieronski and Antti Kuusisto. One-Dimensional Logic over Trees. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kieronski_et_al:LIPIcs.MFCS.2017.64, author = {Kieronski, Emanuel and Kuusisto, Antti}, title = {{One-Dimensional Logic over Trees}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {64:1--64:13}, 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.2017.64}, URN = {urn:nbn:de:0030-drops-80658}, doi = {10.4230/LIPIcs.MFCS.2017.64}, annote = {Keywords: satisfiability, expressivity, trees, fragments of first-order logic} }

Document

**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We consider two extensions of this logic. We show that adding either additional binary symbols or counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem. However, combining the two extensions and adding both binary symbols and counting quantifiers leads to an explosion of this complexity. We also compare the expressive power of the two-variable fragment over trees with its extension with counting quantifiers. It turns out that the two logics are equally expressive, although counting quantifiers do add expressive power in the restricted case of unordered trees.

Bartosz Bednarczyk, Witold Charatonik, and Emanuel Kieronski. Extending Two-Variable Logic on Trees. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bednarczyk_et_al:LIPIcs.CSL.2017.11, author = {Bednarczyk, Bartosz and Charatonik, Witold and Kieronski, Emanuel}, title = {{Extending Two-Variable Logic on Trees}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {11:1--11:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.11}, URN = {urn:nbn:de:0030-drops-76794}, doi = {10.4230/LIPIcs.CSL.2017.11}, annote = {Keywords: two-variable logic, trees, satisfiability, expressivity, counting quantifiers} }

Document

**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. We investigate one-dimensional fragment over words and over omega-words. We show that it is expressively equivalent to the two-variable fragment of first-order logic. We also show that its satisfiability problem is NExpTime-complete. Further, we show undecidability of some extensions, whose two-variable counterparts remain decidable.

Emanuel Kieronski. One-Dimensional Logic over Words. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kieronski:LIPIcs.CSL.2016.38, author = {Kieronski, Emanuel}, title = {{One-Dimensional Logic over Words}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {38:1--38:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.38}, URN = {urn:nbn:de:0030-drops-65782}, doi = {10.4230/LIPIcs.CSL.2016.38}, annote = {Keywords: satisfiability, expressivity, words, fragments of first-order logic} }

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**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

The uniform one-dimensional fragment U1 of first-order logic was introduced recently as a natural generalization of the two-variable fragment FO2 to contexts with relation symbols of all arities. It was shown that U1 has the exponential model property and NEXPTIME-complete satisfiability problem. In this paper we investigate two restrictions of U1 that still contain FO2. We call these logics RU1 and SU1, or the restricted and strongly restricted uniform one-dimensional fragments. We introduce Ehrenfeucht-Fraisse games for the logics and prove that while SU1 and RU1 are expressively equivalent, they are strictly contained in U1. Furthermore, we consider extensions of the logics SU1, RU1 and U1 with unrestricted use of a single built-in equivalence relation E. We prove that while all the obtained systems retain the finite model property, their complexities differ. Namely, the satisfiability problem is NEXPTIME-complete for SU1(E) and 2NEXPTIME-complete for both RU1(E) and U1(E). Finally, we show undecidability of some natural extensions of SU1.

Emanuel Kierónski and Antti Kuusisto. Uniform One-Dimensional Fragments with One Equivalence Relation. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 597-615, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kieronski_et_al:LIPIcs.CSL.2015.597, author = {Kier\'{o}nski, Emanuel and Kuusisto, Antti}, title = {{Uniform One-Dimensional Fragments with One Equivalence Relation}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {597--615}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.597}, URN = {urn:nbn:de:0030-drops-54418}, doi = {10.4230/LIPIcs.CSL.2015.597}, annote = {Keywords: two-variable logic, uniform one-dimensional fragments, complexity, expressivity, equivalence relations} }

Document

**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

We prove that the satisfiability problem for the two-variable, universal fragment of first-order logic with constants (or, alternatively phrased, for the Bernays-Schönfinkel class with two universally quantified variables) remains decidable after augmenting the fragment by the transitive closure of a single binary relation. We give a 2-NExpTime-upper bound and a 2-ExpTime-lower bound for the complexity of the problem. We also study the cases in which the number of constants is restricted. It appears that with two constants the considered fragment has the finite model property and NExpTime-complete satisfiability problem. Adding a third constant does not change the complexity but allows to construct infinity axioms. A fourth constant lifts the lower complexity bound to TwoExpTime. Finally, we observe that we are close to the border between decidability and undecidability: adding a third variable or the transitive closure of a second binary relation lead to undecidability.

Emanuel Kieronski and Jakub Michaliszyn. Two-Variable Universal Logic with Transitive Closure. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 396-410, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kieronski_et_al:LIPIcs.CSL.2012.396, author = {Kieronski, Emanuel and Michaliszyn, Jakub}, title = {{Two-Variable Universal Logic with Transitive Closure}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {396--410}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.396}, URN = {urn:nbn:de:0030-drops-36860}, doi = {10.4230/LIPIcs.CSL.2012.396}, annote = {Keywords: two-variable logic, transitive closure, decidability} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We consider the satisfiability problem for modal logic over classes of structures definable by universal first-order formulas with three variables. We exhibit a simple formula for which the problem is undecidable. This improves an earlier result in which nine variables were used. We also show that for classes defined by three-variable, universal Horn formulas the problem is decidable. This subsumes decidability results for many natural modal logics, including T, B, K4, S4, S5.

Emanuel Kieronski, Jakub Michaliszyn, and Jan Otop. Modal Logics Definable by Universal Three-Variable Formulas. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 264-275, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kieronski_et_al:LIPIcs.FSTTCS.2011.264, author = {Kieronski, Emanuel and Michaliszyn, Jakub and Otop, Jan}, title = {{Modal Logics Definable by Universal Three-Variable Formulas}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {264--275}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.264}, URN = {urn:nbn:de:0030-drops-33495}, doi = {10.4230/LIPIcs.FSTTCS.2011.264}, annote = {Keywords: modal logic, decidability} }

Document

**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

We show that the satisfiability and the finite satisfiability problems for two-variable logic, FO2, over the class of structures with three linear orders, are undecidable. This sharpens an earlier result that FO2 with eight linear orders is undecidable. The theorem holds for a restricted case in which linear orders are the only non-unary relations. Recently, a contrasting result has been shown, that the finite satisfiability problem for FO2 with two linear orders and with no additional non-unary relations is decidable. We observe that our proof can be adapted to some interesting fragments of FO2, in particular it works for the two-variable guarded fragment, GF2, even if the order relations are used only as guards. Finally, we show that GF2 with an arbitrary number of linear orders which can be used only as guards becomes decidable if except linear orders only unary relations are allowed.

Emanuel Kieronski. Decidability Issues for Two-Variable Logics with Several Linear Orders. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 337-351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{kieronski:LIPIcs.CSL.2011.337, author = {Kieronski, Emanuel}, title = {{Decidability Issues for Two-Variable Logics with Several Linear Orders}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {337--351}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.337}, URN = {urn:nbn:de:0030-drops-32415}, doi = {10.4230/LIPIcs.CSL.2011.337}, annote = {Keywords: two-variable logic, linear orders, guarded fragment, decidability} }

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