Deciding First-Order Satisfiability when Universal and Existential Variables are Separated

TL;DR

This paper introduces a new decidable fragment of first-order logic characterized by separating universal and existential variables, which generalizes existing fragments and has implications for automated reasoning.

Contribution

The paper presents a novel decidable fragment of first-order logic based on variable separation, extending known fragments without restrictions on quantifier prefixes or predicate arity.

Findings

The new fragment has the finite model property.

Decidability of satisfiability is established with a non-elementary upper bound.

The $orall^* o orall^* o orall^*$ subfragment is NEXPTIME-complete.

Abstract

We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones -- the Bernays-Sch\"onfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property and derive a non-elementary upper bound on the computing time required for deciding satisfiability in the new fragment. For the subfragment of prenex sentences with the quantifier prefix $\exists^* \forall^* \exists^*$ the satisfiability problem is shown to be complete for NEXPTIME. Finally, we discuss how automated reasoning procedures can take advantage of our results.