One-Dimensional Logic over Words

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

One-Dimensional Guarded Fragments

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

Decidability Issues for Two-Variable Logics with Several Linear Orders

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

Finite Satisfiability of Unary Negation Fragment with Transitivity

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

Two-Variable Universal Logic with Transitive Closure

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 2-ExpTime. 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

One-Dimensional Logic over Trees

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

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.Peer reviewe

Complexity and Expressivity of Uniform One-Dimensional Fragment with Equality

Uniform one-dimensional fragment UF is a formalism obtained from first-order logic by limiting quantification to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulas with two or more variables. The fragment can be seen as a canonical generalization of two-variable logic, defined in order to be able to deal with relations of arbitrary arities. The fragment was introduced recently, and it was shown that the satisfiability problem of the equality-free fragment of UF is decidable. In this article we establish that the satisfiability and finite satisfiability problems of UF are NEXPTIME-complete. We also show that the corresponding problems for the extension of UF with counting quantifiers are undecidable. In addition to decidability questions, we compare the expressivities of UF and two-variable logic with counting quantifiers FOC^2. We show that while the logics are incomparable in general, UF is strictly contained in FOC^2 when attention is restricted to vocabularies with the arity bound two

Modal Logics Definable by Universal Three-Variable Formulas

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