Clones of Pivotally Decomposable Functions

Abstract

Abstract-This paper is a contribution to the understanding of the relation between pivotal decompositions of operations on a set A and clones on the same set. In this preliminary study we establish sufficient conditions on a pivotal operation Π so that the corresponding class of Π-decomposable operations constitutes a clone, and discuss the normal form representations that such pivotal operations induce. We also outline several open questions, providing directions for further research. I. MOTIVATIONSeveral classes of operations have the remarkable feature that each member f : A n → A is decomposable into simpler operations that are then combined by a single operation, in order to retrieve the values of the original operation f . A noteworthy example is the class of Boolean operations f : {0, 1} n → {0, 1} that can be decomposed into expressions of the formfor x = (x 1 , . . . , x n ) ∈ {0, 1} n and k ∈ [n] and where x c k denotes the n-tuple obtained from x by substituting its kth component by c ∈ {0, 1}. Such decomposition scheme is referred to as Shannon decomposition (or Shannon expansion). Boolean operations are similarly decomposable into expressions in the language of Boolean latticeswhereMore recent examples include the class of polynomial operations over a distributive lattice (essentially, combinations of variables and constants using the lattice operations ∧ and ∨) that were shown in [11] to be decomposable into expressions of the formwhere med is the ternary lattice polynomial given by Note that decomposition schemes (1), (2) and (3) share the same general form, namely,• in (1) we have Π(x, y, z) = xy + (1 − x)z,• in (2) we have Π(x, y, z) = (x ∧ y) ∨ (x ∧ z), and• in (3) we have Π(x, y, z) = med(x, y, z).These facts were observed in [12] where these decomposition schemes, called pivotal decompositions, were investigated. In particular, it was observed that not every ternary operation serves as a pivotal operation.In this paper we are interested in classes of pivotally decomposable operations that constitute clones of operations over a set A. Our motivation is rooted in [2] where a study of normal form representations of Boolean operations was presented and based on compositions of Boolean clones. In particular, it was shown that normal form representations of Boolean operations that have the ternary operator med as the only logical connective, allow shorter representations than the classical DNF, CNF and polynomial representations.As we will see, clones of pivotally decomposable operations provide nice normal form representations having a unique operation as logical connective, namely, the corresponding pivotal operation.The paper is organised as follows. After recalling in Section II basic notions and terminology needed throughout the paper, we define the concept of pivotal operation and that of pivotally decomposable class in Section III and introduce the notion of normal form associated with a pivotal operation and mention some related properties. In Section IV we establish sufficient conditions o...

Pivotal decomposition schemes inducing clones of operations

“…Proposition 2.5. If Π is a Π-decomposable pivotal operation that satisfies (8) and (6), then it satisfies (7). In particular, Π is a symmetric operation.…”

“…These preliminary efforts were then further pursued under the observation that certain classes of pivotal operations fulfill certain closure requirements, notably, closure under functional composition. This led to the study [6] of those classes of pivotally decomposable operations that constitute clones. In particular, we presented conditions on pivotal operations to ensure that the corresponding classes of pivotally decomposable operations constitute clones.…”