Revised: July 10, 2020

Published: October 21, 2020

**Keywords:**random Boolean function, quantifier elimination

**Categories:**complexity theory, lower bounds, average case, Boolean functions, formula complexity, formulas over the reals, quantifier elimination, extension complexity, sign patterns, zero patterns, short

**ACM Classification:**F.1.3, F.2.3

**AMS Classification:**68Q17, 03C10

**Abstract:**
[Plain Text Version]

We say that a first-order formula $A(x_1,\dots,x_n)$ over $\R$ defines a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, if for every $x_1,\dots,x_n\in\{0,1\}$, $A(x_1,\dots,x_n)$ is true iff $f(x_1,\dots,x_n)=1$. We show that:

(i) every $f$ can be defined by a formula of size $O(n)$,

(ii) if $A$ is required to have at most $k\geq 1$ quantifier alternations, there exists an $f$ which requires a formula of size $2^{\Omega(n/k)}$.

The latter result implies several previously known as well as some new lower bounds in computational complexity: a non-constructive version of the lower bound on span programs of Babai, Gál, and Wigderson (Combinatorica 1999); Rothvoß's result (CoRR 2011) that there exist $0/1$ polytopes that require exponential-size linear extended formulations; a similar lower bound by Briët et al. (Math. Program. 2015) on semidefinite extended formulations; and a new result stating that a random Boolean function has exponential linear separation complexity. We note that (i) holds over any field of characteristic zero, and (ii) holds over any real closed or algebraically closed field.