Automated lemma synthesis in symbolic-heap separation logic

Abstract

The symbolic-heap fragment of separation logic has been actively developed and advocated for verifying the memory-safety property of computer programs. At present, one of its biggest challenges is to effectively prove entailments containing inductive heap predicates. These entailments are usually proof obligations generated when verifying programs that manipulate complex data structures like linked lists, trees, or graphs.To assist in proving such entailments, this paper introduces a lemma synthesis framework, which automatically discovers lemmas to serve as eureka steps in the proofs. Mathematical induction and template-based constraint solving are two pillars of our framework. To derive the supporting lemmas for a given entailment, the framework firstly identifies possible lemma templates from the entailment's heap structure. It then sets up unknown relations among each template's variables and conducts structural induction proof to generate constraints about these relations. Finally, it solves the constraints to find out actual definitions of the unknown relations, thus discovers the lemmas. We have integrated this framework into a prototype prover and have experimented it on various entailment benchmarks. The experimental results show that our lemma-synthesisassisted prover can prove many entailments that could not be handled by existing techniques. This new proposal opens up more opportunities to automatically reason with complex inductive heap predicates.

Cyclic program synthesis

“…The techniques and metatheory of cyclic proofs originates in the logic and proof theory community. Most related to our current work is the application of cyclic proof to reasoning about program correctness [7,35,43], as well as to proving pure entailments of Separation Logic with inductively defined predicates [6,41,42]. In particular, our use of cardinalities with Separation Logic coincides with the approach used by Rowe and Brotherston [35].…”

Automated Repair of Heap-Manipulating Programs Using Deductive Synthesis

“…Finally, u # v indicates that the two lists u and v are disjoint, i.e., w.(w ∈ u ∧ w ∈ v). Most of our proposed rules are inspired by the standard entailment checking rules in separation logic literature [47,48]. However, there are two main differences.…”

SL-COMP: competition of solvers for separation logic

“…It employs mathematical induction to prove entailments involving userdefined predicates. In addition, Songbird is also equipped with powerful proof techniques, which include a mutual induction proof system [35] and a lemma synthesis framework [36].…”