We introduce two new complexity measures for Boolean functions, which we name sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [ŠS04] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory.

As a surprising application we show that sumPI²(f) is lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [Hâs98], are in fact special cases of our method. The second quantity we introduce, maxPI(f), is always at least as large as sumPI(f), and is derived from sumPI in such a way that maxPI²(f) remains a lower bound on formula size.

Our main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares of the spectral norms of its submatrices. The generality of this lemma gives that our methods can also be used to lower bound the communication complexity of relations, and a related combinatorial quantity, the rectangle partition number.

To exhibit the strengths and weaknesses of our methods, we look at the sumPI and maxPI complexity of a few examples, including the recursive majority of three function, a function defined by Ambainis [Amb03], and the collision problem.