Restriction on cut rule in cyclic-proof system for symbolic heaps

Symbolic heaps, which are a restricted class of separation logic formulas, with inductive definitions are a suitable language in automated verification systems for memory-manipulating programs. In this context, some related problems, e.g., the entailment problem, have been studied theoretically. In addition, several solvers for the entailment problem based on the proof-search algorithm in cyclic-proof systems, which are proof systems in sequent calculus style with cyclic structure, have been proposed. However, the cut-elimination property generally does not hold for cyclic-proof systems of symbolic heaps with inductive definitions, which means that searching for a cut-free proof is insufficient. In other words, we hope to find a reasonable proof-search algorithm considering the cut rule or we give up on obtaining a complete proof-search procedure. This paper investigates this issue and demonstrates a limit to the challenge of the restrictions on the cut rule in a cyclic-proof system for symbolic heaps. We propose a restricted cut rule, referred to as the presumable cut, which is a relaxed variant of the analytic cut, in which the cut formula must be a subformula of the bottom sequent. This paper demonstrates that the provability of the cyclic-proof system for symbolic heaps becomes strictly weaker by restricting the cut rule to the presumable cut.