Expressiveness Results for an Inductive Logic of Separated Relations

TL;DR

This paper investigates the expressiveness of a Separation Logic of Relations (SLR), comparing it to Monadic Second-Order Logic (MSO), revealing their incomparability on unbounded structures and conditions where SLR surpasses MSO.

Contribution

It introduces SLR as a generalization of Symbolic Heap logic with relational atoms and analyzes its expressiveness relative to MSO on various classes of structures.

Findings

SLR and MSO are incomparable on unbounded treewidth structures.

SLR can be embedded in SO logic in general.

MSO is a strict subset of SLR on models with bounded treewidth.

Abstract

In this paper we study a Separation Logic of Relations (SLR) and compare its expressiveness to (Monadic)Second Order Logic (M)SO. SLR is based on the well-known Symbolic Heap fragment of Separation Logic, whose formulae are composed of points-to assertions, inductively defined predicates, with the separating conjunction as the only logical connective. SLR generalizes the Symbolic Heap fragment by supporting general relational atoms, instead of only points-to assertions. In this paper, we restrict ourselves to finite relational structures, and hence only consider Weak (M)SO, where quantification ranges over finite sets. Our main results are that SLR and MSO are incomparable on structures of unbounded treewidth, while SLR can be embedded in SO in general. Furthermore, MSO becomes a strict subset of SLR, when the treewidth of the models is bounded by a parameter and all vertices attached to some hyperedge belong to the interpretation of a fixed unary relation symbol. We also discuss the problem of identifying a fragment of SLR that is equivalent to MSO over models of bounded treewidth.