Generically Automating Separation Logic by Functors, Homomorphisms and Modules

TL;DR

This paper introduces a generic algebraic framework for automating separation logic reasoning, enabling scalable verification of complex data structures within formal semantics, and demonstrates its effectiveness through case studies and formalization in Isabelle/HOL.

Contribution

It develops a systematic algebraic theory for separation logic predicates and creates a generic automation algorithm applicable to various data structures, improving scalability and reusability.

Findings

Automated reasoning matches state-of-the-art systems in complex data structures.

The algebraic models can be instantiated automatically for diverse data structures.

The framework is formalized and verified in Isabelle/HOL.

Abstract

Foundational verification considers the functional correctness of programming languages with formalized semantics and uses proof assistants (e.g., Coq, Isabelle) to certify proofs. The need for verifying complex programs compels it to involve expressive Separation Logics (SLs) that exceed the scopes of well-studied automated proof theories, e.g., symbolic heap. Consequently, automation of SL in foundational verification relies heavily on ad-hoc heuristics that lack a systematic meta-theory and face scalability issues. To mitigate the gap, we propose a theory to specify SL predicates using abstract algebras including functors, homomorphisms, and modules over rings. Based on this theory, we develop a generic SL automation algorithm to reason about any data structures that can be characterized by these algebras. In addition, we also present algorithms for automatically instantiating the algebraic models to real data structures. The instantiation reuses the algebraic models of component structures and preserves their data abstractions. Case studies on formalized imperative semantics show our algorithm can instantiate the algebraic models automatically for a variety of complex data structures. Experimental results indicate the automatically instantiated reasoners from our generic theory show similar results to the state-of-the-art systems made of specifically crafted reasoning rules. The presented theories, proofs, and the verification framework are formalized in Isabelle/HOL.