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Functional equations, constraints, definability of function classes, and functions of Boolean variables
Abstract
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- Article accepté pour publication ou publié
- pseudo-Boolean functions
- Boolean functions
- field-valued functions of Boolean variables
- linear equations
- multilinear polynomial representations
- ring-valued functions
- function class definability
- relational constraints
- functional equa- tion
- stability
- class composition
- Function classes
- 512
- Algèbre