arXiv subjects

Emanuel Kieronski

Publications and source records attributed to Emanuel Kieronski.

8 recordsLinked to original sources

arXiv2025

Two-Variable Logic for Hierarchically Partitioned and Ordered Data

We study Two-Variable First-Order Logic, FO2, under semantic constraints that model hierarchically structured data. Our first logic extends FO2 with a linear order < and a chain of increasingly coarser equivalence relations E_1, E_2, ... . We show that its finite satisfiability problem is NExpTime-complete. We also demonstrate that a weaker variant of this logic without the linear order enjoys the exponential model property. Our second logic extends FO2 with a chain of nested total preorders. We prove that its finite satisfiability problem is also NExpTime-complete.However, we show that the complexity increases to ExpSpace-complete once access to the successor relations of the preorders is allowed. Our last result is the undecidability of FO2 with two independent chains of nested equivalence relations.

cs.LO↗

arXiv2025

Alternating Quantifiers in Uniform One-Dimensional Fragments with an Excursion into Three-Variable Logic

The uniform one-dimensional fragment of first-order logic was introduced a few years ago as a generalization of the two-variable fragment to contexts involving relations of arity greater than two. Quantifiers in this logic are used in blocks, each block consisting only of existential quantifiers or only of universal quantifiers. In this paper we consider the possibility of mixing both types of quantifiers in blocks. We show the finite (exponential) model property and NExpTime-completeness of the satisfiability problem for two restrictions of the resulting formalism: in the first we require that every block of quantifiers is either purely universal or ends with the existential quantifier, in the second we restrict the number of variables to three; in both equality is not allowed. We also extend the second variation to a rich subfragment of the three-variable fragment (without equality) that still has the finite model property and decidable, NExpTime-complete satisfiability.

cs.LO↗

arXiv2024

On the complexity of Maslov's class $\overline{\text{K}}$

Maslov's class $\overline{\text{K}}$ is an expressive fragment of First-Order Logic known to have decidable satisfiability problem, whose exact complexity, however, has not been established so far. We show that $\overline{\text{K}}$ has the exponential-sized model property, and hence its satisfiability problem is NExpTime-complete. Additionally, we get new complexity results on related fragments studied in the literature, and propose a new decidable extension of the uniform one-dimensional fragment (without equality). Our approach involves a use of satisfiability games tailored to $\overline{\text{K}}$ and a novel application of paradoxical tournament graphs.

cs.LO↗

arXiv2021

One-Dimensional Fragment over Words and Trees

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential (universal) quantifiers that leave at most one variable free. We investigate this fragment over words and trees, presenting a complete classification of the complexity of its satisfiability problem for various navigational signatures, and comparing its expressive power with other important formalisms. These include the two-variable fragment with counting and the unary negation fragment.

cs.LO↗

arXiv2019

Finite Satisfiability of Unary Negation Fragment with Transitivity

We show that the finite satisfiability problem for the unary negation fragment with 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.

cs.LO↗

arXiv2019

One-dimensional guarded fragments

We call a first-order formula one-dimensional if its every maximal block of existential (universal) quantifiers 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, FO2. We denote the resulting formalisms, resp., GF1, and TGF1. We show that GF1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF1 we show that it is decidable, has the finite model property, and its satisfiability problem is TwoExpTime-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.

cs.LO↗

arXiv2018

Unary negation fragment with equivalence relations has the finite model property

We consider an extension of the unary negation fragment of first-order logic in which arbitrarily many binary symbols may be required to be interpreted as equivalence relations. We show that this extension has the finite model property. More specifically, we show that every satisfiable formula has a model of at most doubly exponential size. We argue that the satisfiability (= finite satisfiability) problem for this logic is TwoExpTime-complete. We also transfer our results to a restricted variant of the guarded negation fragment with equivalence relations.

cs.LO↗

arXiv2018

Finite Satisfiability of the Two-Variable Guarded Fragment with Transitive Guards and Related Variants

We consider extensions of the two-variable guarded fragment, GF2, where distinguished binary predicates that occur only in guards are required to be interpreted in a special way (as transitive relations, equivalence relations, pre-orders or partial orders). We prove that the only fragment that retains the finite (exponential) model property is GF2 with equivalence guards without equality. For remaining fragments we show that the size of a minimal finite model is at most doubly exponential. 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. We improve the bounds and obtain optimal ones for all the fragments considered, in particular NExpTime for GF2 with equivalence guards, and 2ExpTime for GF2 with transitive guards. To obtain our results we essentially use some results from integer programming.

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