Abstract

Topical but classical results concerning the incidence relationship between prime clauses and implicants of a monotone Boolean function are derived by applying a general theory of computational equivalence and replaceability to distributive lattices. A non-standard combinatorial model for the free distributive lattice FDL(n) is described, and a correspondence between monotone Boolean functions and partitions of a standard Cayley diagram for the symmetric group is derived.

Preliminary research on-classifying and characterising the simple paths and circuits that are the blocks of this partition is summarised. It is shown in particular that each path and circuit corresponds to a characteristic configuration of implicants and clauses. The motivation for the research and expected future directions are briefly outlined.

Introduction

Models of Boolean formulae expressed in terms of the incidence relationship between the prime implicants and clauses of a function were first discovered several years ago, but they have recently been independently rediscovered by several authors, and have attracted renewed interest. They have been used in proving lower bounds by Karchmer and Wigderson and subsequently by Razborov. More general investigations aimed at relating the complexity of functions to the model have also been carried out by Newman [20].

This paper demonstrates the close connection between these classical models for monotone Boolean formulae and circuits and a general theory of computational equivalence as it applies to FDL(n): the (finite) distributive lattice freely generated by n elements. It also describes how the incidence relationships between prime implicants and clauses associated with monotone Boolean functions can be viewed as built up from a characteristic class of incidence patterns between relatively small subsets of implicants and clauses.