A refinement calculus for logic programs

Abstract

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.

Developing Logic Programs from Specifications Using Stepwise Refinement

“…To satisfy this condition, the recursive parameter must be bound. In this paper we have introduced recursion somewhat informally, but a more formal approach based the use of a refinement law for introducing recursion may be found in [12].…”

“…The refinement calculus approach as described in [12] has been extended in several directions. One of these is data refinement [8,6], where the type of a program variable is refined to some other type, usually for implementation purposes, as illustrated in Sect.…”

“…The specification X = 1 may be understood as binding X to 1 in states where X is unbound; it succeeds in states where X is bound to 1, and fails if X is bound to something other that 1. In our semantics we model the meaning of specification command, and any command in general, as a function from sets of bindings to sets of bindings, where a binding maps variables to values [12]. A binding represents an answer, providing a (single) value for every free variable.…”

“…A least fixed point always exists for our recursive programs [9], though for non-terminating recursions the fixed point is the worst possible program, abort. The refinement rule for introducing recursion prevents us from deriving non-terminating recursions in our refinements [12].…”

“…The specification constructs include a specification command that allows the effect of a program to be specified in terms of a general predicate, and an assumption command that defines the range of values for which a program is expected to work. A semantics for the refinement calculus has been given which models commands (both specifications and code) as partial functions from sets of bindings of program variables to subsets of those bindings [12]. A tool has been developed to support the refinement calculus [15], based on the Isabelle/HOL theorem prover.…”