Composition of Post classes and normal forms of Boolean functions

Abstract. We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone.

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“…Note that decomposition schemes (1), (2) and (3) share the same general form, namely, f (x) = Π(x k , f (x 1 k ), f (x 0 k )). Indeed,…”

Section: Introductionmentioning

confidence: 99%

Pivotal decomposition schemes inducing clones of operations

Couceiro

Teheux

2017

Beitr Algebra Geom

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“…Remark 17 A comparative study of normal form representations of Boolean functions was presented in Couceiro et al [7] where it was shown that the so-called median normal form representation, in which Boolean functions are expressed as repeated applications of the median function to variables, negated variables, and constants, provides a more efficient representation than the classical conjunctive normal form, disjunctive normal form and polynomial representations (the latter are also called Zhegalkin polynomial representations due to [32] or Reed-Muller polynomial representations due to [25,28]). Even though algorithms for converting the classical CNF, DNF, and polynomial representations into this median normal form were provided, no hint was given on how to produce median representations, e.g., from truth tables.…”

Section: Theorem 16 ([24 Theorem 17])mentioning

confidence: 99%

“…Indeed, by setting an ordering of variables, say, the canonical ordering of variables, we can repeatedly apply Theorem 16 to the variables of any given function in order to derive a nested formula made of medians applied to variables and constants. By making use of tools in [7], namely the decomposition of any Boolean function as a nondecreasing function composed with variables and negated variables, this procedure can be easily extended to any Boolean function. Unfortunately, this approach seems to produce median expressions which are not optimal in the sense of [7].…”

Section: Theorem 16 ([24 Theorem 17])mentioning

confidence: 99%

“…By making use of tools in [7], namely the decomposition of any Boolean function as a nondecreasing function composed with variables and negated variables, this procedure can be easily extended to any Boolean function. Unfortunately, this approach seems to produce median expressions which are not optimal in the sense of [7]. To this extent one needs to find rules to simplify the median expressions produced by the above algorithm.…”

Section: Theorem 16 ([24 Theorem 17])mentioning

confidence: 99%

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Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

Couceiro

Marichal

2010

Fuzzy Sets and Systems

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We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions.

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“…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.…”

Section: Motivationmentioning

confidence: 99%

Clones of Pivotally Decomposable Functions

Couceiro

Teheux

2015

2015 IEEE International Symposium on Multiple-Valued Logic

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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...

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Composition of Post classes and normal forms of Boolean functions

Cited by 17 publications

References 14 publications

Pivotal decomposition schemes inducing clones of operations

Pivotal decomposition schemes inducing clones of operations

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

Clones of Pivotally Decomposable Functions

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