11 research outputs found
Functional equations, constraints, definability of function classes, and functions of Boolean variables
The paper deals with classes of functions of several variables defined on an arbitrary set A and taking values in a possibly different set B. Definability of function classes by functional equations is shown to be equivalent to definability by relational constraints, generalizing a fact established by Pippenger in the case A = B = {0,1}. Conditions for a class of functions to be definable by constraints of a particular type are given in terms of stability under certain functional compositions. This leads to a correspondence between functional equations with particular algebraic syntax and relational constraints with certain invariance properties with respect to clones of operations on a given set. When A = {0,1} and B is a commutative ring, such B-valued functions of n variables are represented by multilinear polynomials in n indeterminates in B[X1,..., Xn], Functional equations are given to describe classes of field-valued functions of a specified bounded degree. Classes of Boolean and pseudo-Boolean functions are covered as particular cases
Generalizations of Swierczkowski's lemma and the arity gap of finite functions
Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an
at least quaternary operation on a finite set A and every operation obtained
from f by identifying a pair of variables is a projection, then f is a
semiprojection. We generalize this lemma in various ways. First, it is extended
to B-valued functions on A instead of operations on A and to essentially at
most unary functions instead of projections. Then we characterize the arity gap
of functions of small arities in terms of quasi-arity, which in turn provides a
further generalization of Swierczkowski's Lemma. Moreover, we explicitly
classify all pseudo-Boolean functions according to their arity gap. Finally, we
present a general characterization of the arity gaps of B-valued functions on
arbitrary finite sets A.Comment: 11 pages, proofs simplified, contents reorganize
Explicit Descriptions of Bisymmetric Sugeno Integrals
We provide sufficient conditions for a Sugeno integral to be bisymmetric. We explicitly describe bisymmetric Sugeno integrals over chains
Linearly definable classes of Boolean functions
International audienceIn this paper we address the question "How many properties of Boolean functions can be defined by means of linear equations?" It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors and stable classes, a topic that has been widely investigated in recent years by several authors including Maurice Pouzet
Join-irreducible Boolean functions
This paper is a contribution to the study of a quasi-order on the set
of Boolean functions, the \emph{simple minor} quasi-order. We look at
the join-irreducible members of the resulting poset . Using a
two-way correspondence between Boolean functions and hypergraphs,
join-irreducibility translates into a combinatorial property of hypergraphs. We
observe that among Steiner systems, those which yield join-irreducible members
of are the -2-monomorphic Steiner systems. We also describe
the graphs which correspond to join-irreducible members of .Comment: The current manuscript constitutes an extension to the paper
"Irreducible Boolean Functions" (arXiv:0801.2939v1
Stability of Boolean function classes with respect to clones of linear functions
We consider classes of Boolean functions stable under compositions both from
the right and from the left with clones. Motivated by the question how many
properties of Boolean functions can be defined by means of linear equations, we
focus on stability under compositions with the clone of linear idempotent
functions. It follows from a result by Sparks that there are countably many
such linearly definable classes of Boolean functions. In this paper, we refine
this result by completely describing these classes. This work is tightly
related with the theory of function minors, stable classes, clonoids, and
hereditary classes, topics that have been widely investigated in recent years
by several authors including Maurice Pouzet and his coauthors.Comment: 44 page
Stability of Boolean function classes with respect to clones of linear functions
We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors, stable classes, clonoids, and hereditary classes, topics that have been widely investigated in recent years by several authors including Maurice Pouzet and his coauthors