One-dimensional fragment of first-order logic

TL;DR

This paper introduces a new decidable fragment of first-order logic called the one-dimensional fragment, which limits quantification to at most one free variable and is closed under Boolean operations, offering a fresh perspective on modal logic decidability.

Contribution

The paper defines the one-dimensional fragment, proves its decidability via a reduction to monadic logic, and explores its boundaries and expressivity compared to related logics.

Findings

The one-dimensional fragment is decidable.

Minor syntax modifications lead to undecidability.

It is incomparable in expressivity with guarded negation and two-variable logic with counting.

Abstract

We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited 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 formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and non-uniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest.