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Abstract

In this paper, we prove two general lower bounds for algebraic decision trees which test membership in a set $S\subseteq\Re^n$ which is defined by linear inequalities. Let $rank(S)$ be the maximal dimension of a linear subspace contained in the closure of $S$. % \endgraf First we prove that any decision tree which uses multilinear functions (i.e.~arbitrary products of linear functions) must have depth at least $n-rank(S)$. This solves an open question raised by A.C.~Yao and can be used to show that multilinear functions are not really more powerful than simple comparisons between the input variables when computing the largest $k$ elements of $n$ given numbers. Yao could only prove this result in the special case when products of at most two linear functions are used. Our proof is based on a dimension argument. It seems to be the first time that such an approach yields good lower bounds for nonlinear decision trees. % \endgraf Surprisingly, we can use the same methods to give an alternative proof for Rabin's fundamental Theorem, namely that the depth of any decision tree using arbitrary analytic functions is at least $n-rank(S)$. Since we show that Rabin's original proof is incorrect, our proof of Rabin's Theorem is not only the first correct one but also generalizes the Theorem to a wider class of functions.