A basic goal of complexity theory is to prove that certain complexity classes (e.g., P and NP) are not the same. To do so, we need to exhibit a machine in one class that differs from every machine in the other class in the sense that their answers are different on at least one input. This chapter describes diagonalization–essentially the only general technique known for constructing such a machine.

We already encountered diagonalization in Section 1.5, where it was used to show the existence of uncomputable functions. Here it will be used in more clever ways. We first use diagonalization in Sections 3.1 and 3.2 to prove hierarchy theorems, which show that giving Turing machines more computational resources allows them to solve a strictly larger number of problems. We then use diagonalization in Section 3.3 to show a fascinating theorem of Ladner: If P ≠ NP, then there exist problems that are neither NP-complete nor in P.

Though diagonalization led to some of these early successes of complexity theory, researchers concluded in the 1970s that diagonalization alone may not resolve P versus NP and other interesting questions; Section 3.4 describes their reasoning.