Completeness of Cyclic Proofs for Symbolic Heaps with Inductive Definitions

Abstract

Separation logic is successful for software verification in both theory and practice. Decision procedure for symbolic heaps is one of the key issues. This paper proposes a cyclic proof system for symbolic heaps with general form of inductive definitions, and shows its soundness and completeness. The decision procedure for entailments of symbolic heaps with inductive definitions is also given. Decidability for entailments of symbolic heaps with inductive definitions is an important question. Completeness of cyclic proof systems is also an important question. The results of this paper answer both questions. The decision procedure is feasible since it is nondeterministic double-exponential time complexity.We define F (m) as obtained from a formula F by replacing every inductive predicate P by P (m) .We define ( = (T 1 , T 2 )) as t1∈T1,t2∈T2,t1 ≡t2 t 1 = t 2 . We write x = T for ( = ({x}, T )).We define ( = (T )) as ( = (T ∪ {nil}, T ∪ {nil})). SemanticsThis subsection gives semantics of the language. We define the following structure: Val = N , Locs = {x ∈ N |x > 0}, Heaps = Locs → f in Val n cell , Stores = Vars → Val. Each s ∈ Stores is called a store. Each h ∈ Heaps is called a heap, and Dom(h) is the domain of h, and Range(h) is the range of h. We write h = h 1 + h 2 when Dom(h 1 ) and Dom(h 2 ) are disjoint and the graph of h is the union of those of h 1 and h 2 . A pair (s, h) is called a heap model, which means a memory state. The value s(x) means the value of the variable x in the model (s, h). Each value a ∈ Dom(h) means an address, and the value of h(a) is the content of the memory cell at address a in the heap h. We suppose each memory cell has n cell elements as its content.The interpretation s(t) for any term t is defined as 0 for nil and s(x) for the variable x.For a formula F we define the interpretation s, h |= F as follows.

A Proof Procedure For Separation Logic With Inductive Definitions and Theory Reasoning

“…As we shall see, SC is always sound, but it is invertible 6 only if the heap decomposition corresponding to the left-hand side coincides with that of the formulas on the right-hand side (see Definition 58). It plays a similar rôle to the rule ( * ) defined in [27] 7 . For instance, assume that the conclusion is p(x) * q(y) ⊢ R p 1 (x) * q 1 (y), p 2 (x) * q 2 (y).…”

“…Other approaches have been proposed to check entailments in various fragments, see e.g., [7,12,16]. In particular, a sound and complete proof procedure is given in [27] for inductive rules satisfying conditions that are strictly more restrictive than those in [15]. In [14] a labeled proof systems is presented for separation logic formulas handling arbitrary inductive definitions and all connectives (including negation and separated implication).…”

Beyond Symbolic Heaps: Deciding Separation Logic With Inductive Definitions

“…While satisfiability and related problems for separation logic are undecidable in general, many decidable fragments have been proposed. Most decidability results have been obtained for the symbolic-heap fragment [6,4,16,9,17,32,20,33]. Symbolic heaps are separation-logic formulas in which atomic predicates can only be combined with the separating conjunction; no other separating connectives or Boolean connectives are allowed.…”

“…Consequently, restricted fragments of SLID definitions have been studied. While it is natural to consider restrictions to trees [17,32,33], it is possible to obtain decidability results for more expressive logics. Iosif et al [16] proved the decidability of a particularly expressive fragment of SLID.…”

Counterexample to cut-elimination in cyclic proof system for first-order logic with inductive definitions

“…A cyclic proof system, or a circular proof system, is a proof system whose proof figure is a tree with cycles [4]. Such proof systems have been used to formalize several logics and theories, such as modal µ-calculus [25,24,1], linear time µ-calculus [10,15,17], linear logic with fixed points [14,18], Gödel-Löb provability logic [21], first-order µ-calculus [23], first-order logic with inductive definitions [6,5,2], arithmetic [22,3], bunched logic [7], separation logic [8,16,19,27], and Kleene algebra [13]. Cyclic proofs are also useful for software verification, including verifying properties of concurrent processes [20], termination of pointer programs [8], and decision procedures for symbolic heaps [9,12,26,27].…”