An extension of Khrapchenko's theorem

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 {norm of matrix}c{norm of matrix} _{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,...,c_n). 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 f{hook}(g₁,...;,g_n) are sought.