Minor complexities of finite operations

Abstract

In this paper we present a new class of complexity measures, induced by a new data structure for representing $k$-valued functions (operations), called minor decision diagram. The results are presented in terms of Multi-Valued Logic circuits (MVL-circuits), ordered decision diagrams, formulas and minor decomposition trees. When assigning values to some variables in a function $f$ the resulting function is a subfunction of $f$, and when identifying some variables the resulting function is a minor of $f$. A set $M$ of essential variables in $f$ is separable if there is a subfunction of $f$, whose set of essential variables is $M$. The essential arity gap $gap(f)$ of the function $f$ is the minimum number of essential variables in $f$ which become fictive when identifying distinct essential variables in $f$. We prove that, if a function $f$ has non-trivial arity gap ($gap(f)\ge 2$), then all sets of essential variables in $f$ are separable. We define equivalence relations which classify the functions of $k$-valued logic into classes with the same minor complexities. These relations induce transformation groups which are compared with the subgroups of the restricted affine group (RAG) and the groups determined by the equivalence relations with respect to the subfunctions, implementations and separable sets in functions. These methods provide a detailed classification of $n$-ary $k$-valued functions for small values of $n$ and $k$.