The Effects of Adding Reachability Predicates in Quantifier-Free Separation Logic

TL;DR

This paper investigates how adding reachability predicates like list segment (ls) to quantifier-free separation logic affects computational complexity, revealing polynomial space decidability in some cases and undecidability in others.

Contribution

It provides the first systematic study of the impact of reachability predicates on the complexity of quantifier-free separation logic, including polynomial and undecidable fragments.

Findings

Restriction without separating implication is polynomial space decidable.

Full extension with implication becomes undecidable.

Various logic fragments are analyzed for computational complexity.

Abstract

The list segment predicate ls used in separation logic for verifying programs with pointers is well-suited to express properties on singly-linked lists. We study the effects of adding ls to the full quantifier-free separation logic with the separating conjunction and implication, which is motivated by the recent design of new fragments in which all these ingredients are used indifferently and verification tools start to handle the magic wand connective. This is a very natural extension that has not been studied so far. We show that the restriction without the separating implication can be solved in polynomial space by using an appropriate abstraction for memory states whereas the full extension is shown undecidable by reduction from first-order separation logic. Many variants of the logic and fragments are also investigated from the computational point of view when ls is added, providing numerous results about adding reachability predicates to quantifier-free separation logic.