On a Minimization of Variables to Represent Sparse Multi-Valued Input Decision Functions

A multiple-valued input decision function is a mapping $f:P^{\mathrm{n}}\rightarrow\{0,1\}$, where $P=\{0,1,\ \ldots, p-1\}$. This paper considers the learning of such a function. That is, given the TRUE-set $T\subseteq P^{n}$ and the FALSE-set $F\subseteq P^{n}$, obtain a function $f$ such that $f(\vec{a})=1$ for any $\vec{a}\in T$, and $f(\vec{b})=0$ for any $\vec{b}\in F$. We show a method to find a function such that $f$ depends on the least number of variables. Applications of such functions include detection of poisonous mushrooms, hepatitis and breast cancer.