Developing logic programs from specifications using stepwise refinement

In this paper we demonstrate a refinement calculus for logic programs, which is a framework for developing logic programs from specifications. The paper is written in a tutorial-style, using a running example to illustrate how the refinement calculus is used to develop logic programs. The paper also presents an overview of some of the advanced features of the calculus, including the introduction of higher-order procedures and the refinement of abstract data types.

Stability of nonlinear difference inclusions

The stability of difference inclusions x(k+1) is an element of F(x(k)) is studied, where F(x) = {F(x, gimel) : is an element of Lambda} and the selections F(., gimel) : E -->E assume values in a Banach space E, partially ordered by a cone K. It is assumed that the operators F(.,gimel) are heterotone or pseudoconcave. The main results concern asymptotically stable absorbing sets, and include the case of a single equilibrium point. The results are applied to a number of practical problems.

Stability of linear difference and differential inclusions

Any given n X n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N x N matrix B of spectral radius p(B) arbitrarily close to p(A). A difference inclusion x(k+1) is an element of Ax(k), where A is a compact set of matrices, is asymptotically stable if and only if A can be extended to a set B of nonnegative matrices B with \ \B \ \ (1) < 1 or \ \B \ \ (infinity) < 1. Similar results are derived for differential inclusions.

Towards a definition of "difference" in osteoarthritis

To assess existing information regarding detectable differences in osteoarthritis (OA), a systematic literature search was conducted up to December 1999. Thirty-three articles were considered methodologically relevant to the definition and categorization of detectable differences in OA. It was determined that the musculoskeletal literature contains a wealth of information that relates to observed changes, much of which is derived from the clinical trials literature, but there have been relatively few methodological studies that have systematically evaluated the nature, categorization, and relevance of the change. Furthermore, most of those that have been published take the perspective of an individual or groups of experts other than that of the patient. This summary of the current literature reveals that the diverse sources of information go part way towards developing an understanding of detectable differences and their importance in the area of OA research and clinical practice. Stakeholders' interests as well as factors that modulate perceptions of importance need to be taken under consideration. In particular, the patient's perspective of the importance of change at an individual level requires further evaluation. This area of clinical research is relatively underdeveloped, but there is considerable opportunity for progress.

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.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.