Nested Hoare Triples and Frame Rules for Higher-Order Store

Abstract

Abstract. Separation logic is a Hoare-style logic for reasoning about programs with heap-allocated mutable data structures. As a step toward extending separation logic to high-level languages with ML-style general (higher-order) storage, we investigate the compatibility of nested Hoare triples with several variations of higher-order frame rules.The interaction of nested triples and frame rules can be subtle, and the inclusion of certain frame rules is in fact unsound. A particular combination of rules can be shown consistent by means of a Kripke model where worlds live in a recursively defined ultrametric space. The resulting logic allows us to elegantly prove programs involving stored code. In particular, using recursively defined assertions, it leads to natural specifications and proofs of invariants required for dealing with recursion through the store.

Crowfoot: A Verifier for Higher-Order Store Programs

“…3 gives the syntax for the assertion language. Based on [17], the language allows nested triples to appear in assertions, such that we can reason about stored procedures. The assertion x → ∀a.…”

“…{a → } · (a) {a → }, for example, states that the content at address x is a procedure which satisfies the given Hoare triple. 2 Additions to the logic of [17] are the set and element expressions. In the formula P(e V * ; e S * ), the ; separates integer arguments from set arguments.…”

“…The deep frame rule [5,17] allows one to infer {P } C {Q} ⊗ I from {P } C {Q}, where ⊗ is a deep framing operator. Intuitively this operator adds the invariant I not just to the pre-and post-conditions of the triple {P } C {Q}, but also to all triples nested inside P and Q, at all levels.…”

“…Logics with nested triples [17,11], where assertions can contain Hoare triples which describe the behaviour of code stored on the program's heap, have been proposed as a way to reason modularly about higher-order store programs. Recent developments [17,18] have provided solid theoretical foundations for separation logics with nested triples.…”

“…Recent developments [17,18] have provided solid theoretical foundations for separation logics with nested triples. In this paper we present Crowfoot, the first automatic verification system to apply these developments in practice.…”