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Brian J. d'Auriol, Steve Saladin and Shane Humes, "Linguistic and Non-Linguistic Semantics in the Polytope Model," Technical Report No. 99-01, Department of Mathematics and Computer Science, The University of Akron, Akron, Ohio, 44325-4002, January 1999.
A new method based on geometrical representation for program specification is proposed. The focus of this method is the ordering of groups of computations such that these groups can be abstracted by enclosing geometric objects called polytopes. Additional properties may be added to the geometric objects. Abstraction hierarchies may be formed by considering collections of polytopes. This paper presents the two types of geometrical semantics: linguistic and non-linguistic carried. Linguistic carried semantics is based on translating an initial linguistic specification of a program segment to a corresponding geometric specification whereas non-linguistic carried semantics is based on the initial specification of groups of program components in the geometric domain. Three important properties of geometric semantics are given: (a) the mapping of computations to index points in affine space, the polytope containment of such a group of points, and the enhancements of the polytopes with additional relationships, (b) subspace polytope existence and meaning, and (c) the combination of static and dynamic bindings of domain values to index points in various subspaces. The latter property serves as a defining distinction between linguistic and non-linguistic carried semantics. A formal review of the basic Polytope Model is given; such a succinct but formal presentation of the Polytope Model has not been identified in the literature. Several illustrative examples of geometrical specification are given.
Full Paper (pdf: 292KB)