Cyclic-proof Systems for Symbolic Heaps Require Cut Formulas Outside Initial Signatures

Abstract

Separation logic is an extension of Hoare logic for verifying memory-manipulating programs. Formulas for pre- and post-conditions in separation logic are often restricted to symbolic heaps with inductive predicates for automated verification. The entailment checking problem between symbolic heaps has been actively investigated in this context. One potential solution is to construct entailment provers based on cyclic-proof systems. Cyclic-proof systems are a reasonable way to reason entailments with inductive predicates. However, several cyclic-proof systems, including for symbolic-heap separation logic, do not satisfy the cut-elimination property. Hence, a cut-free proof search is insufficient, and a heuristic search for cut formulas is required to apply the cut rule. This paper investigates the search space for cut formulas in the cyclic-proof system for symbolic heap. We prove that the proof system does not satisfy the cut-restriction property with the initial signature cuts, of which cut formulas contain only the inductive predicates in the signature of the conclusion entailments. In other words, the provability is properly weakened by restricting cut formulas to those in the initial signature. From this, it follows that it may be necessary to introduce new inductive predicates while finding cut formulas for proof search.