Lower Bounds on Formula Size of BooleanFunctions using

KK orner 7] deened the notion of graph-entropy. He used it in 8] to simplify the proof of the Fredman-Komlos lower bound for the family size of perfect hash functions. We use this information theoretic notion to obtain a general method for formula size lower bounds. This method can be applied to low-complexity functions for which the other known general methods ((11, 12, 3] and see also 17]) do not apply. Speciically the results are: 1. A new general lower bound on the formula size of quadratic Boolean functions. 2. As a corollary we get an (n 2 logn) lower bound for the function that decides whether a graph of n vertices has a cycle of length four, and to the function that decides whether a graph has a vertex of degree at least two. 3. A simple proof of a result of Krichevskii, 10] , stating that the formula size for the threshold-2 Boolean function with n variables is at least n log n. 4. A simple proof of a lower bound rst proved by Snir, 16], stating that a W V W formula for n-variable threshold-k function, where all ^ gates have fan in k, has the size of (n log n ? log(k ? 1) log k ? log(k ? 1)) = (nk log n k) Notation: 1. Let X be a nite set, interpreted as Boolean variables. A formula is a rooted tree whose leaves are labeled with members of X or their negations, and whose internal Partially supported by American-Israeli binational science foundation, grant no. 87-00082.