Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments

TL;DR

This paper investigates the decidability and complexity of interpolant and explicit definability problems in the guarded and two-variable fragments of first-order logic, showing they are decidable but computationally hard.

Contribution

It establishes the decidability and complexity bounds for interpolant and definability problems in GF and FO2 fragments, which lack the Craig interpolation and Beth definability properties.

Findings

In GF, problems are 3ExpTime-complete generally, 2ExpTime-complete with bounded arity.

In FO2, problems are coN2ExpTime upper bound and 2ExpTime lower bound.

Decidability of these problems holds despite the fragments lacking CIP and PBDP.

Abstract

In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3ExpTime-complete in general, and 2ExpTime-complete if the arity of relation symbols is bounded by a constant c not smaller than 3. In FO2, we prove a coN2ExpTime upper bound and a 2ExpTime lower bound for both problems. Thus, both for GF and FO2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO2 under standard complexity assumptions).