The methodology of reducing from communication complexity was introduced by Blais, Brody, and Matulef [54], and our description of it is based on [136].

Introduction

Our perspective in this book is mainly algorithmic. Hence, we view complexity lower bounds mainly as justifications for the failure to provide better algorithms (i.e., algorithms of lower complexity). The lower bounds that we shall be discussing are lower bounds on the query complexity of testers. These lower bounds are of an information theoretic nature, and so they cannot (and do not) rely on computational assumptions.

We start with two brief preliminary discussions. The first discussion is very abstract and vague: it concerns the difficulty of establishing lower bounds. The second discussion is very concrete: it highlights the fact that computational complexity considerations play no role in this chapter, a fact that is most evident in the avoidance of the uniformity condition.

What Makes Lower Bounds Hard to Prove? Proving lower bounds is often more challenging than proving upper bounds, since one has to defeat all possible methods (or algorithms) rather than show that one of them works. Indeed, it seems harder to cope with a universal quantifier than with an existential one, but one should bear in mind that a second quantifier of opposite nature follows the first one.