Although we view specific (natural) computational problems as secondary to (natural) complexity classes, we do use the former for clarification and illustration of the latter. This appendix provides definitions of such computational problems, grouped according to the type of objects to which they refer (i.e., graphs and Boolean formula).

We start by addressing the central issue of the representation of the various objects that are referred to in the aforementioned computational problems. The general principle is that elements of all sets are “compactly” represented as binary strings (without much redundancy). For example, the elements of a finite set S (e.g., the set of vertices in a graph or the set of variables appearing in a Boolean formula) will be represented as binary strings of length log2 |S|.

Graphs

A simple graph G=(V,E) consists of a finite set of vertices V and a finite set of edges E, where each edge is an unordered pair of vertices; that is, E ⊆ {{u, v} : u, v∈V ∧ u≠v}.