Replaceability and computational equivalence for monotone boolean functions

Abstract

Replacement rules have played an important role in the study of monotone boolean function complexity. In this paper, notions of replaceability and computational equivalence are formulated in an abstract algebraic setting, and examined in detail for finite distributive lattices — the appropriate algebraic context for monotone boolean functions. It is shown that when computing an element f of a finite distributive lattice D, the elements of D partition into classes of computationally equivalent elements, and define a quotient of D in which all intervals of the form [t ∧ f, t ∨ f] are boolean. This quotient is an abstract simplicial complex with respect to ordering by replaceability. Other results include generalisations and extensions of known theorems concerning replacement rules for monotone boolean networks. Possible applications of computational equivalence in developing upper and lower bounds on monotone boolean function complexity are indicated, and new directions of research both abstract mathematical and computational, are suggested.