Complexity and Expressivity of Uniform One-Dimensional Fragment with Equality

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. 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 UF 1 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 FOC 2 . We show that while the logics are incomparable in general, UF = 1 is strictly contained in FOC 2 when attention is restricted to vocabularies with the arity bound two.

Ordered fragments of first-order logic

“…Let us conclude this section by mentioning briefly two complexity results that follow immediately from the literature and which complement the picture emerging from the results listed in Table 1. First, it is easy to translate the algebra GRA(p, s, E, ¬, C, ∩, ∃ 1 , ∃ 0 ) into the one-dimensional uniform fragment UF 1 , which was proved to be NexpTime-complete in [8]. It is also not hard to see that this algebra is NexpTime-hard, and hence the satisfiability problem for this algebra is NexpTime-complete.…”

“…As usual, we will prove this by showing that the logic has the bounded model property. The proof was heavily influenced by a similar model constructions performed in [8] and [7], which were originally influenced by the classical model construction used in [2].…”

Uniform Guarded Fragments

“…A standard technique in proving that the complexity of the satisfiability problem of a given fragment of F O is in NExpTime is to show that each satisfiable sentence of this fragment has a finite model of size at most exponential with respect to the length of the sentence [8,11,14,15]. However, in the case of UGF it seems to be easier to show that we can associate to each of its sentences ϕ a different type of certificate, which is still at most exponential with respect to the length of the sentence, and which can be used to construct a (potentially infinite) model for ϕ.…”

“…Formulas satisfying the first restriction are called one-dimensional, while formulas satisfying the second restriction are called uniform. In [15] it was proved that UF 1 has the finite model property and the complexity of its satisfiability problem is NExpTime-complete, which is the same as for F O 2 [8]. The research around UF 1 and its variants has been quite active, see for instance [11,13,14,16,17].…”

A fine-grained hierarchy of hard problems in the separated fragment

“…The BSR fragment and MFO are among the classical ones (see [3] for references). More recently defined fragments include the two-variable fragment (FO 2 ) [12], [8], the fluted fragment (FL) [15], [16], [14], the guarded fragment (GF) [1], [7], the guarded negation fragment (GNF) [2], and the uniform one-dimensional fragment (UF 1 ) [11]. While GNF and UF 1 are incomparable, GNF extends GF, and UF 1 can be considered as a generalization of FO 2 .…”