A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as an NFS B if there is a polynomial P with nonnegative integer coefficients such that the complexity of any Boolean function f with respect to A is at most the value of P in the complexity of f with respect to B. In this paper we study monotonic NFSs, i.e., NFSs whose connectives are increasing or decreasing in each argument. We describe the monotonic NFSs that are optimal, i.e., that are minimal with respect to the latter preorder. We show that these minimal monotonic NFSs are all equivalent. Moreover, we address some natural questions, e.g.: does optimality depend on the arity of connectives? Does it depend on the number of connectives used? We show that optimal monotonic NFSs are exactly those that use a single connective or one connective and the negation. Finally, we show that optimality does not depend on the arity of the connectives.@en

In this document, we consider a median-based calculus to represent monotone Boolean functions efficiently. We study an equational specification of median forms and extend it from the domain of monotone Boolean functions to the domain of polynomial functions over distributive lattices. This specification is sound and complete. We illustrate its usefulness when simplifying median formulas algebraically. Furthermore, we propose a definition of median normal forms (MNF), that are thought of as minimal median formulas with respect to a structural ordering of expressions. We investigate related complexity issues and show that the problem of deciding whether a formula is in MNF, that is the problem of minimizing the median form of a monotone Boolean function, is in ∑₂ ^P. Moreover, we show that it still holds for arbitrary Boolean functions, not necessarily monotone.

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In this document, we consider a median-based calculus for efficiently representing polynomial functions over distributive lattices. We extend an equational specification of median forms from the domain of Boolean functions to the domain of lattice polynomials. We show that it is sound and complete, and we illustrate its usefulness when simplifying median formulas algebraically. Furthermore, we propose a definition of median normal forms (MNF), that are thought of as minimal median formulas with respect to a structural ordering of expressions. We also investigate related complexity issues and show that the problem of deciding whether a formula is in MNF is in Σ^P₂. Moreover, we explore polynomial approximations of solutions to this problem through a sound term rewriting system extracted from the proposed equational specification.

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