Irreducible Topological Components of an Arbitrary Boolean Truth Function and Generation of Their Minimal Coverings

Abstract

An extension of the work initiated by Quine in reducing an arbitrary Boolean truth function to its minimal form is presented. Apart from the unique parts of the form, the entire class of nonunique forms is discussed. The portion of the truth table that is left uncovered by the unique parts of the solution is partitioned into topologically invariant components of which it is the direct sum. Each component may be covered independently of the others. The generation of the set of coverings of a component is developed around a central theorem: A union of cells, all basic to a particular vertex, contains no further cells basic to that vertex. A proof of the theorem is given. The components are components in the topological sense and are preserved under changes of representation. The discussion focuses on the general, unrefinable structure of a Boolean function, as opposed to practical means for calculating its minimal coverings.