Deciding Satisfiability of Boolean Separation Logic with Inductive Predicates via Small Models

Core Concepts

The authors present a novel decision procedure for a fragment of separation logic with arbitrary nesting of separating conjunctions, boolean conjunctions, disjunctions, and guarded negations, along with support for common variants of linked lists. The decision procedure is based on a model-based translation to SMT, with optimizations that bound the size of predicate instantiations within models, leading to an efficient translation.

Abstract

The paper presents a decision procedure for a fragment of separation logic called boolean separation logic (BSL). BSL allows arbitrary nesting of separating conjunctions and boolean connectives of conjunction, disjunction, and guarded negation, along with support for common inductive predicates like singly-linked lists (SLL), doubly-linked lists (DLL), and nested singly-linked lists (NLL).
The key contributions are:

A small-model property that shows satisfiable BSL formulae have models of bounded size.
A translation-based decision procedure that encodes BSL formulae into SMT, with optimizations to avoid expensive quantifier instantiations.

The translation uses a model-based approach, where the heap is decomposed into "chunks" that can be efficiently encoded.
For inductive predicates, the translation expresses reachability using bounded paths, avoiding unbounded quantification.

Experimental evaluation showing the decision procedure is competitive with state-of-the-art approaches on the symbolic heap fragment, and can handle some formulae beyond their capabilities.
A proof that adding guarded negations to the fragment makes the satisfiability problem PSPACE-hard.

The decision procedure supports a richer boolean structure than previous approaches, which is useful in applications like symbolic execution and entailment checking. The authors demonstrate the usefulness of the supported features through several examples.

Stats

The location bound for a formula φ is defined as bound(φ) = 1 + ⌊P
x∈vars(φ)||x||⌋, where ||xS|| = 2 if S ∈ {S, N} and ||xD|| = 1.5.
The translation of a separating conjunction ψ1 ∗ψ2 uses an over-approximation of the footprints of ψ1 and ψ2, denoted as FP#(ψ1) and FP#(ψ2).
The translation of inductive predicates π(x, y) uses a main path formula to express reachability from x to y, and additional invariants to ensure the correct shape of the data structure.

Quotes

"To the best of our knowledge, no existing, practically applicable decision procedure supports a fragment with such a rich boolean structure and at least basic inductive predicates."
"Our approach to deciding BSL formulae is inspired by previous works on translation of SL to SMT. The early works [26] and [27] translate SL to intermediate theories first. Our approach is closer to the more recent approach of [15], which builds on small-model properties and axiomatizes reachability through pointer links directly."
"We further show that adding guarded negations to BSL makes its satisfiability problem PSPACE-hard."

Key Insights Distilled From

Deciding Boolean Separation Logic via Small Models (Technical Report)

by Tomá... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.18999.pdf

Deeper Inquiries

How can the small-model property and the translation-based decision procedure be extended to support more complex inductive predicates beyond the three considered in the paper?

To extend the small-model property and the decision procedure to support more complex inductive predicates, one approach could be to introduce a more flexible mechanism for handling the bounds on path lengths in the translation process. This could involve developing a more sophisticated method for computing precise bounds on paths in the SL-graphs, allowing for a wider range of inductive predicates to be accurately translated. Additionally, the translation process could be enhanced to handle nested inductive predicates or more intricate relationships between different types of inductive structures. By refining the techniques used to determine path lengths and relationships within the SL-graphs, the decision procedure could be adapted to accommodate a broader set of inductive predicates with varying complexities.

Can the techniques used in this work be applied to other logics with a combination of boolean structure and inductive definitions, such as extensions of separation logic or other graph logics?

The techniques employed in this work, such as the translation-based decision procedure and the use of SL-graphs to capture must-relations in the logic, can indeed be applied to other logics with similar characteristics. For extensions of separation logic or other graph logics that involve a combination of boolean structure and inductive definitions, the concept of translating complex predicates into SMT expressions while considering bounded models can be highly beneficial. By adapting the translation process and the small-model property concept to suit the specific features of the target logic, it is possible to develop decision procedures that can effectively handle the rich interplay between boolean connectives and inductive structures in a variety of formalisms beyond the scope of this paper.

What are the potential applications of the rich boolean structure supported by the decision procedure, beyond the examples provided in the paper, and how can they be explored further?

The rich boolean structure supported by the decision procedure opens up a wide range of potential applications beyond the examples highlighted in the paper. One key application could be in program verification and analysis, where the ability to reason about complex boolean combinations of spatial assertions can enhance the precision and efficiency of verification tools. Additionally, the decision procedure could be utilized in automated reasoning systems for software correctness, security analysis, and model checking, enabling the verification of programs with intricate memory structures and data dependencies. Furthermore, the rich boolean structure can be leveraged in symbolic execution and constraint solving for program synthesis and optimization tasks. Exploring these applications further would involve conducting empirical studies and case studies to evaluate the performance and effectiveness of the decision procedure in real-world scenarios across various domains.