An algebraic approach to problem solution and problem semantics

Abstract

The usual approach to the synthesis of algorithms for the solution of problems in combinatorial mathematics consists of two steps. 1 — Description: the problem is embedded in a general structure which is rich enough to permit a mathematical modelling of the problem. 2 — Solution: the problem is solved by means of techniques "as simple as possible", with respect to some given notion of complexity. We give a formalization of this approach in the framework of category theory, which is general enough to get rid of unessential details. In particular such a framework will be provided by the category of ordered complete Σ-algebras, and we will describe the relation between description and solution by means of a variant of so called "Mezei-Wright like results" [10], relating the concept of least fixed point to that of a suitable natural transformation between functors.