Lattices of Logical Fragments over Words

TL;DR

This paper develops a purely syntactic framework for logical fragments over words, characterizing their lattice structures and closure properties, and relates definability to syntactic morphisms, extending prior results without using model-theoretic methods.

Contribution

It introduces a syntactic notion of logical fragments over words, characterizes their lattice structures, and connects definability to syntactic morphisms, extending Straubing's work without model-theoretic tools.

Findings

Logical fragments form lattices with specific closure properties.

Definability often characterized by syntactic morphisms.

Acyclic Sigma_2 fragment matches FO^2 expressive power.

Abstract

This paper introduces an abstract notion of fragments of monadic second-order logic. This concept is based on purely syntactic closure properties. We show that over finite words, every logical fragment defines a lattice of languages with certain closure properties. Among these closure properties are residuals and inverse C-morphisms. Here, depending on certain closure properties of the fragment, C is the family of arbitrary, non-erasing, length-preserving, length-multiplying, or length-reducing morphisms. In particular, definability in a certain fragment can often be characterized in terms of the syntactic morphism. This work extends a result of Straubing in which he investigated certain restrictions of first-order logic formulae. In contrast to Straubing's model-theoretic approach, our notion of a logical fragment is purely syntactic and it does not rely on Ehrenfeucht-Fraisse games. As motivating examples, we present (1) a fragment which captures the stutter-invariant part of piecewise-testable languages and (2) an acyclic fragment of Sigma_2. As it turns out, the latter has the same expressive power as two-variable first-order logic FO^2.