On the efficiency of normal form systems for representing Boolean functions

Miguel Couceiro (1) , Erkko Lehtonen (2, 3) , Pierre Mercuriali (4) , Romain Péchoux (4)

1 ORPAILLEUR - Knowledge representation, reasonning
2 CSC - Computer Science and Communications Research Unit [Luxembourg]
3 Institut für Algebra [Dresden]
4 MOCQUA - Designing the Future of Computational Models

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.

Mots clés: Efficient representation; Boolean function; Normal form system; Complexity

Domaines: Informatique [cs]; Complexité [cs.CC]; Mathématique discrète [cs.DM]

Miguel Couceiro, Erkko Lehtonen, Pierre Mercuriali, Romain Péchoux. On the efficiency of normal form systems for representing Boolean functions. Theoretical Computer Science, 2020, 813, pp.341-361. ⟨10.1016/j.tcs.2020.01.009⟩. ⟨hal-02153506v2⟩