A Rogues’ Gallery of Complexity Classes

Abstract

To the computer scientist, structure is meaning. Seeking to understand nature’s diverse problems with man’s humble resources, we simplify our task by grouping similarly structured problems. The resulting complexity classes, such as P, NP, and PSPACE, are simply families of problems that can be solved with a certain underlying computational power. The range of interesting computational powers is broad—deterministic, nondeterministic, probabilistic, unique, table lookup, etc.—and a suitably rich palette has been developed to reflect these powers—P, NP, PP, UP, P/poly, etc. These classes can themselves be studied in terms of their internal structure and behavior. This chapter briefly reviews the definitions, meanings, and histories of the central complexity classes covered in this book.

The form is the meaning, and indeed the classic Greek mind, with an integrity of perception lost by later cultures which separated the two, firmly identified them.

— Vincent Scully, The Earth, the Temple, and the Gods [Scu62]