An Extension of Khrapchenko's Theorem

Abstract Khrapchenko's theorem is a classical result yielding lower bounds on the formula complexity of Boolean functions over the unate basis. In particular, it can yield a tight n2 lower bound on the formula complexity of the parity function. In this note we consider an extended definition of formula size in which each variable may be assigned a different cost. We then generalize Khrapchenko's theorem to cover this new definition and in particular derive a ‖c‖ 1 2 =(∑c i 1 2 ) 2 lower bou nd on the generalized formula size complexity of the parity function of n variables with cost vector c=(c1,…,cn). This bound is shown to be tight to within a factor of 2 by methods similar to Huffman coding. The extended definition of formula size arises naturally in cases where formulae for compound functions like ƒ(g1,…;,gn) are sought.