A Refinement Calculus for Logic Programs

Existing refinement calculi provide frameworks for the stepwise development of imperative programs from specifications. This paper presents a refinement calculus for deriving logic programs. The calculus contains a wide-spectrum logic programming language, including executable constructs such as sequential conjunction, disjunction, and existential quantification, as well as specification constructs such as general predicates, assumptions and universal quantification. A declarative semantics is defined for this wide-spectrum language based on executions. Executions are partial functions from states to states, where a state is represented as a set of bindings. The semantics is used to define the meaning of programs and specifications, including parameters and recursion. To complete the calculus, a notion of correctness-preserving refinement over programs in the wide-spectrum language is defined and refinement laws for developing programs are introduced. The refinement calculus is illustrated using example derivations and prototype tool support is discussed.Comment: 36 pages, 3 figures. To be published in Theory and Practice of Logic Programming (TPLP

A Declarative Semantics for Logic Program Refinement

The refinement calculus provides a framework for the stepwise development of imperative programs from specifications. This paper presents a semantics for a refinement calculus for deriving logic programs. The calculus contains a wide-spectrum logic programming language, including executable constructs such as sequential conjunction, disjunction, and existential quantification, as well as specifications constructs (general predicates and assumptions) and universal quantification. A semantics is defined for this wide-spectrum language based on {\em executions}, which are partial functions from states to states, where a state is represented as a set of bindings. This execution semantics is used to define the meaning of programs and specifications, including parameters and recursion. To complete the calculus, a notion of correctness-preserving refinement over programs in the wide-spectrum language is defined and a refinement law for introducing recursive procedures is presented

Refinement of higher-order logic programs

A refinement calculus provides a method for transforming specifications to executable code, maintaining the correctness of the code with respect to its specification. In this paper we extend the refinement calculus for logic programs to include higher-order programming capabilities in specifications and programs, such as procedures as terms and lambda abstraction. We use a higher-order type and term system to describe programs, and provide a semantics for the higher-order language and refinement. The calculus is illustrated by refinement examples

Refinement Calculus of Reactive Systems

Refinement calculus is a powerful and expressive tool for reasoning about sequential programs in a compositional manner. In this paper we present an extension of refinement calculus for reactive systems. Refinement calculus is based on monotonic predicate transformers, which transform sets of post-states into sets of pre-states. To model reactive systems, we introduce monotonic property transformers, which transform sets of output traces into sets of input traces. We show how to model in this semantics refinement, sequential composition, demonic choice, and other semantic operations on reactive systems. We use primarily higher order logic to express our results, but we also show how property transformers can be defined using other formalisms more amenable to automation, such as linear temporal logic (suitable for specifications) and symbolic transition systems (suitable for implementations). Finally, we show how this framework generalizes previous work on relational interfaces so as to be able to express systems with infinite behaviors and liveness properties

A Relational Logic for Higher-Order Programs

Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called Relational Higher Order Logic (RHOL), for proving relational properties of a simply typed $\lambda$-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavour of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.Comment: Submitted to ICFP 201

Predicate Transformers and Linear Logic, yet another denotational model

International audienceIn the refinement calculus, monotonic predicate transformers are used to model specifications for (imperative) programs. Together with a natural notion of simulation, they form a category enjoying many algebraic properties. We build on this structure to make predicate transformers into a de notational model of full linear logic: all the logical constructions have a natural interpretation in terms of predicate transformers (i.e. in terms of specifications). We then interpret proofs of a formula by a safety property for the corresponding specification

rCOS: A refinement calculus for object systems

This article presents a mathematical characterization of object-oriented concepts by defining an observation-oriented semantics for a relational objectoriented language with a rich variety of features including subtypes, visibility, inheritance, type casting, dynamic binding and polymorphism. The language is expressive enough for the specification of object-oriented designs and programs. We also propose a calculus based on this model to support both structural and behavioral refinement of object-oriented designs. We take the approach of the development of the design calculus based on the standard predicate logic in Hoare and He’s Unifying Theories of Programming (UTP). We also consider object reference in terms of object identity as values and mutually dependent methods