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Emanuel Kieronski

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15 papers

2 author rows

LPAR Conference 2023 Conference Paper

An excursion to the border of decidability: between two- and three-variable logic

Oskar Fiuk
Emanuel Kieronski

With respect to the number of variables the border of decidability lies between 2 and 3: the two-variable fragment of first-order logic, FO2, has an exponential model property and hence NExpTime-complete satisfiability problem, while for the three-variable fragment, FO3, satisfiability is undecidable. In this paper we propose a rich subfragment of FO3, containing full FO2 (without equality), and show that it retains the finite model property and NExpTime complexity. Our fragment is obtained as an extension of the uniform one-dimensional variant of FO3.

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Highlights Conference 2021 Conference Abstract

Finite Model Theory of the Triguarded Fragment and Related Logics

Emanuel Kieronski

The Triguarded Fragment (TGF) is among the most expressive decidable fragments of first-order logic, subsuming both its two-variable and guarded fragments without equality. We show that the TGF has the finite model property (providing a tight doubly exponential bound on the model size) and hence finite satisfiability coincides with satisfiability known to be N2ExpTime-complete. Using similar constructions, we also establish 2ExpTime-completeness for finite satisfiability of the constant-free (tri)guarded fragment with transitive guards.

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LPAR Conference 2020 Conference Paper

The Triguarded Fragment with Transitivity

Emanuel Kieronski
Adam Malinowski

The triguarded fragment of first-order logic is an extension of the guarded fragment in which quantification for subformulas with at most two free variables need not be guarded. Thus, it unifies two prominent decidable logics: the guarded fragment and the two-variable fragment. Its satisfiability problem is known to be undecidable in the presence of equality, but becomes decidable when equality is forbidden. We consider an extension of the tri- guarded fragment without equality by transitive relations, allowing them to be used only as guards. We show that the satisfiability problem for the obtained formalism is decidable and 2-ExpTime-complete, that is, it is of the same complexity as for the analogous exten- sion of the classical guarded fragment. In fact, in our satisfiability test we use a decision procedure for the latter as a subroutine. We also show how our approach, consisting in exploiting some existing results on guarded logics, can be used to reprove some known facts, as well as to derive some other new results on triguarded logics.

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MFCS Conference 2019 Conference Paper

Finite Satisfiability of Unary Negation Fragment with Transitivity

Daniel Danielski
Emanuel Kieronski

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.

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JELIA Conference 2019 Conference Paper

On the Complexity of Graded Modal Logics with Converse

Bartosz Bednarczyk
Emanuel Kieronski
Piotr Witkowski 0001

Abstract A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By “traditional” classes of frames we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.

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MFCS Conference 2019 Conference Paper

One-Dimensional Guarded Fragments

Emanuel Kieronski

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.

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CSL Conference 2017 Conference Paper

Extending Two-Variable Logic on Trees

Bartosz Bednarczyk
Witold Charatonik
Emanuel Kieronski

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.

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MFCS Conference 2017 Conference Paper

One-Dimensional Logic over Trees

Emanuel Kieronski
Antti Kuusisto

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.

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CSL Conference 2016 Conference Paper

One-Dimensional Logic over Words

Emanuel Kieronski

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.

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CSL Conference 2015 Conference Paper

Uniform One-Dimensional Fragments with One Equivalence Relation

Emanuel Kieronski
Antti Kuusisto

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.

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Highlights Conference 2013 Conference Abstract

Elementary modal logics -- decidability and complexity

Jakub Michaliszyn
Jan Otop
Emanuel Kieronski

Modal logic is called "elementary" if it is considered over class of frames definable by a first-order formula. Different formulae lead to different logics, and many of them find their applications in computer science. The aim of the Highlights conference talk is to present a wide picture of the latest research on the satisfiability problem of elementary modal logics. The talk will provide all the necessarily definitions, discuss the motivation, summarise the recent research papers on this topic, and discuss some open problems.

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CSL Conference 2012 Conference Paper

Two-Variable Universal Logic with Transitive Closure

Emanuel Kieronski
Jakub Michaliszyn

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.

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CSL Conference 2011 Conference Paper

Decidability Issues for Two-Variable Logics with Several Linear Orders

Emanuel Kieronski

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.

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LPAR Conference 2007 Conference Paper

On Finite Satisfiability of the Guarded Fragment with Equivalence or Transitive Guards

Emanuel Kieronski
Lidia Tendera

Abstract The guarded fragment of first-order logic, GF, enjoys the finite model property, so the satisfiability and the finite satisfiability problems coincide. We are concerned with two extensions of the two-variable guarded fragment that do not possess the finite model property, namely, GF 2 with equivalence and GF 2 with transitive guards. We prove that in both cases every finitely satisfiable formula has a model of at most double exponential size w. r. t. its length. To obtain the result we invent a strategy of building finite models that are formed from a number of multidimensional grids placed over a cylindrical surface. The construction yields a 2Nexptime -upper bound on the complexity of the finite satisfiability problem for these fragments. For the case with equivalence guards we improve the bound to 2Exptime.

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CSL Conference 2005 Conference Paper

Results on the Guarded Fragment with Equivalence or Transitive Relations

Emanuel Kieronski

Abstract We study the problem of the satisfiability of guarded formulas in models in which some distinguished binary symbols are interpreted as equivalence relations or as transitive relations. We sharpen the undecidability result for the two-variable guarded fragment with transitive relations by reducing the number of transitive relations to two. We prove that the satisfiability problem for the two-variable guarded fragment with two equivalence relations is 2EXPTIME-complete. We consider the guarded fragment with equivalence relations in guards and show that this variant is easily reducible to the variant with transitive relations in guards. However, in the case of two variables, the version with equivalence relations is easier: NEXPTIME-complete. Finally we show that the decidability results for the guarded fragment with either equivalence relations or transitive relations in guards cannot be generalized to the loosely guarded fragment.

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