# Complexity and Expressivity of Uniform One-Dimensional Fragment with Equality

• Emanuel Kieroński
• Antti Kuusisto
Conference paper

DOI: 10.1007/978-3-662-44522-8_31

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8634)
Cite this paper as:
Kieroński E., Kuusisto A. (2014) Complexity and Expressivity of Uniform One-Dimensional Fragment with Equality. In: Csuhaj-Varjú E., Dietzfelbinger M., Ésik Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8634. Springer, Berlin, Heidelberg

## Abstract

Uniform one-dimensional fragment $$UF_1^=$$ 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. $$UF_1^=$$ can be seen as a canonical generalization of two-variable logic, defined in order to be able to deal with relations of arbitrary arities. $$UF_1^=$$ was introduced recently, and it was shown that the satisfiability problem of the equality-free fragment UF1 of $$UF_1^=$$ is decidable. In this article we establish that the satisfiability and finite satisfiability problems of $$UF_1^=$$ are NEXPTIME-complete. We also show that the corresponding problems for the extension of $$UF_1^=$$ with counting quantifiers are undecidable. In addition to decidability questions, we compare the expressivities of $$UF_1^=$$ and two-variable logic with counting quantifiers FOC2. We show that while the logics are incomparable in general, $$UF_1^=$$ is strictly contained in FOC2 when attention is restricted to vocabularies with the arity bound two.

### Keywords

Two-variable logics complexity expressivity