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

Abstract

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.

A Uniform One-Dimensional Fragment with Alternation of Quantifiers

“…The first is investigating the decidability, complexity and the status of the finite model property for full AUF 1 without equality, that is to see what happens to our logic if arbitrary blocks of quantifiers, possibly ending with the universal quantifier, are allowed. As already mentioned, in our recent work [10] we answered this question for the three variable restriction, AUF 3 1 , of AUF 1 by showing the exponential model property and NEXPTIME-completeness of its satisfiability problem.…”

“…In the recent paper [10] we note that the fragment with arbitrary blocks of quantifiers AUF 1 and with free use of equality contains infinity axioms (satisfiable formulas without finite models), by constructing the following three-variable formula: ∃xS(x) ∧ ∀x∃y∀z(¬S(y) ∧ R(x, y, z) ∧ (x = z ∨ ¬R(z, y, x))), which has no finite models but is satisfied in the model whose universe is the set of natural numbers, S is true only at 0 and Rxyz is true iff y = x + 1.…”

“…The first step to understand AUF 1 was done in the companion paper [10], where we show the decidability and the finite model property of the three variable restriction of this logic, AUF 3 1 ; in that paper AUF 3 1 is then made a basis for obtaining a rich decidable subclass of the three-variable fragment, FO 3 . Turning now to our current contribution, we first remark that in this paper we still do not answer the question whether the whole AUF 1 has decidable satisfiability.…”