Freefinement is an algorithm that constructs a sound refinement calculus from a verification system under certain conditions. In this paper, a verification system is any formal system for establishing whether an inductively defined term, typically a program, satisfies a specification. Examples of verification systems include Hoare logics and type systems. Freefinement first extends the term language to include specification terms, and builds a verification system for the extended language that is a sound and conservative extension of the original system. The extended system is then transformed into a sound refinement calculus. The resulting refinement calculus can interoperate closely with the verification system - it is even possible to reuse and translate proofs between them. Freefinement gives a semantics to refinement at an abstract level: it associates each term of the extended language with a set of terms from the original language, and refinement simply reduces this set. The paper applies freefinement to a simple type system for the lambda calculus and also to a Hoare logic.

Abstract We survey the well-known algebraic laws of sequential programming, and extend them with some less familiar laws for concurrent programming. We give an algebraic definition of the Hoare triple, and algebraic proofs of all the relevant laws for concurrent separation logic. We give the provable concurrency laws for Milner transitions, for the Back/Morgan refinement calculus, and for Dijkstra’s weakest preconditions. We end with a section in praise of algebra, of which Carroll Morgan is such a master.

Encapsulated abstractions are fundamental in object-oriented programming. A single class may employ multiple abstractions to achieve its purpose. Such abstractions are often related and combined in disciplined ways. This paper explores ways to express, verify and rely on logical relationships between abstractions. It introduces two general specification mechanisms: export clauses for relating abstractions in individual classes, and axiom clauses for relating abstractions in a class and all its descendants. MultiStar, an automatic verification tool based on separation logic and abstract predicate families, implements these mechanisms in a multiple inheritance setting. Several verified examples illustrate MultiStar's underlying logic. To demonstrate the flexibility of our approach, we also used MultiStar to verify the core iterator hierarchy of a popular data structure library.