Symmetric functions with non-trivial arity gap

Details Symmetric functions with non-trivial arity gap Symmetric functions with non-trivial arity gap

2010-09-24

arxiv.org/abs/1009.4828v2

Computer Science

Discrete Mathematics

14 pages

Scientific paper

Given an $n$-ary $k-$valued function $f$, $gap(f)$ denotes the essential arity gap of $f$ which is the minimal number of essential variables in $f$ which become fictive when identifying any two distinct essential variables in $f$. The class $G_{p,k}^n$ of all $n$-ary $k-$valued functions $f$ with $2\leq gap(f)$ is explicitly determined by the authors and R. Willard in \cite{s2,sl2,ros}. We prove that if $f\in G_{p,k}^n$ is a symmetric function with non-trivial arity gap and $g=f(x_i=c_i)$ is a subfunction of $f$, where $c_i$ is a constant, then $g\in G_{n-1,k}^{n-1}\cup G_{2,k}^{n-1}$ or $g$ is a constant function. A deep description if the class of all symmetric linear $k$-valued functions with non-trivial arity gap is given.

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