A Canonical Partition of the Primes of Logic Functions

TL;DR

This paper introduces algorithms to partition the primes of Boolean functions into essential, unnecessary, and disjoint sets, facilitating minimal-cost sum-of-primes representations through canonical partitions and ancestor set analysis.

Contribution

The paper presents a novel canonical partitioning method for Boolean function primes, including algorithms for identifying essential, unnecessary, and disjoint prime sets based on ancestor sets.

Findings

Defines a canonical partition of primes into essential, unnecessary, and disjoint sets.

Introduces the concept of Ancestor Sets and proves their role in minimal-cost basis selection.

Provides conditions for easily determining minimum-cost covers and partitioning primes for efficient minimization.

Abstract

This paper presents algorithms that relate to the problem of finding a minimum-cost sum-of-primes representation of a Boolean function f when the cost function C is positive and additive. A set of primes whose sum equals f is called a basis for f, so a solution to the problem is a minimum-cost basis. The algorithms construct the following canonical partition of the complete set of primes and identify the members of sets 1, 2, and 3: (1) Essential Primes, which must be part of any basis for f, (2) Unnecessary Primes that cannot be part of a minimum-cost basis for f for any positive additive cost function, (3) Unique disjoint sets of primes, PS1,...,PSN with associated "covering" tables TS1,..., TSN such that any minimum-cost basis consists of the union of the sets Essential Primes, QS1(C), ..., QSN(C) where QSi(C) is contained in PSi and QSi(C) is a minimum-cost "cover" for PSi. Covering is defined by operation Cascade(QS, TS), which has the property that QS covers PS if and only if Cascade(QS, TS) is empty. The key to the results is the study of objects called Ancestor Sets. The Ancestor Theorem proves that if A is an Ancestor Set for f, every minimum-cost basis includes a minimum-cost cover for the set of primes PS in Ancestor Set A and a minimum-cost cover for the set of primes that are not in A (and are not covered by the the union of the Essentials with PS). The PSi in the partition are the sets of primes in canonical disjoint Independent Ancestor Sets Ai, which are easy to generate when the calculation of the primes (and their consensus combinations) is within computational scope. The paper also presents a condition under which QSi(C) can be easily determined, and another condition such that PSi can be broken into disjoint pieces that can be minimized separately.