An algorithm for determining minimal normal forms of an incomplete truth function

Abstract

In certain applications of truth-function logic it is of interest to consider the minimization problem in a more general form.1 So-called ?algebraic? procedures for determining the irredundant forms of a truth function fall into two categories depending on whether the method requires preliminary expansion of a formula into developed normal form or not. The methods of McCluskey2 and Petrick,3 for example, require canonical expansion of a formula; whereas the table of ratio functions by Gazale4 and a method developed by the author5 determine irredundant forms solely from the list of prime implicants themselves. The latter (referred to hereafter as the ?method of iterated consensus?) arrives at minimal forms by an iterative scheme where the rule of consensus is applied repeatedly to the prime implicants. The purpose of this paper is to indicate how algebraic methods of the second variety apply to the more general case where the class of simplest normal truth functions to be determined is that of formulas equivalent to a given formula ? under the hypothesis that certain conjunctions of letters of ? are always false.