Figure 1.

Computational complexity. (a) The feasibility thesis asserts that there is a fundamental qualitative difference between algorithms that run in polynomial time (P time) (e.g. schoolbook multiplication), and algorithms that run in exponential time (EXP time) (e.g. position evaluation in a generalized game) [2,11–18]. As problem size increases P time algorithms can still feasibly (efficiently) be executed on a physical computer, whereas EXP time algorithms cannot. The feasibility thesis also asserts that NP algorithms cannot feasibly be executed, but this is less clear as this assumes P ≠ NP. (b) Complexity classes are related through the subset relationship: log time ⊆ P time ⊆ NP ⊆ PSPACE ⊆ EXP time [2,11–18]. Little is known of the exact details of these relationships, e.g. does P = NP? (Online version in colour.)