Separateness of Variables -- A Novel Perspective on Decidable First-Order Fragments

TL;DR

This paper introduces the concept of variable separateness to extend decidable first-order logic fragments, significantly broadening their scope while maintaining decidability and enhancing expressiveness.

Contribution

It presents a novel approach using variable separateness to extend many decidable first-order fragments, preserving decidability and increasing expressive succinctness.

Findings

Extended several first-order fragments while preserving decidability

All extensions include the relational monadic fragment without equality

Some extensions show unbounded succinctness improvements

Abstract

The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of separateness of variables and explore its applicability to the classical decision problem. Two disjoint sets of first-order variables are separated in a given formula if variables from the two sets never co-occur in any atom of that formula. This simple notion facilitates extending many well-known decidable first-order fragments significantly and in a way that preserves decidability. We will demonstrate that for several prefix fragments, several guarded fragments, the two-variable fragment, and for the fluted fragment. Altogether, we will investigate nine such extensions more closely. Interestingly, each of them contains the relational monadic first-order fragment without equality. Although the extensions exhibit the same expressive power as the respective originals, certain logical properties can be expressed much more succinctly. In three cases the succinctness gap cannot be bounded using any elementary function.