The spatial logic of the conflict of convex sets

L. A. Pomortsev

Nečetkie sistemy i mâgkie vyčisleniâ, Tome 18 (2023) no. 1, pp. 63-110

Résumé

The article tests methods of logical branching of propositions of the theory of convex sets. The presented work is based on the concept of the Aristotelian Categorical Syllogism (ACS), which performs multiple entry into the result of logical inference using the conjunction -AND- . Formal inverting of the implications that make up ACS allows interpreting it to perform disjunctive branching of the predicate at the exit from it. As a result, we have a logical decomposition of the predicate into the following statements from it, grouped around the union -OR- . To complete the picture, we add two more types of branching: disjunctive entry and conjunctive exit, which are not difficult for understanding and applying. In mathematics, problems concentrate around ACS branches. Four types of logical branching in combination with Transfer of Control (ToC – same as GoTo) are demonstrated by solving the problem of splitting a vector space into two disjoint Convex Sets. It is shown that the intersection of their closures, hereinafter referred to as the «crack» of the vector space, is a hyperplane.

Détail

Keywords: syllogism, analysis, synthesis, spatial algorithm, logical scheme, block diagram, predicate, implication, quantifier, logical inference, algorithm, branching, simplex, hyperplane, semispace, linear manifold, convex set, local interior, projective independence, symmetry, partition, Voronoi diagram.