Decision Procedure for Separation Logic with Inductive Definitions and Presburger Arithmetic

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This paper considers the satisfiability problem of symbolic heaps in separation logic with Presburger arithmetic and inductive definitions. First the system without any restrictions is proved to be undecidable. Secondly this paper proposes some syntactic restrictions for decidability. These restrictions are identified based on a new decidable subsystem of Presburger arithmetic with inductive definitions. In the subsystem of arithmetic, every inductively defined predicate represents an eventually periodic set and can be eliminated. The proposed system is quite general as it can handle the satisfiability of the arithmetical parts of fairly complex predicates such as sorted lists and AVL trees. Finally, we prove the decidability by presenting a decision procedure for symbolic heaps with the restricted inductive definitions and arithmetic.