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Published online by Cambridge University Press: 05 June 2012
Cast a cold eye On life, on death. Horseman, pass by!
W. B. Yeats, “Under Ben Bulben”In this chapter we consider variations on the complexity classes P and NP. We refer specifically to the non-uniform version of P, and to the Polynomial-time Hierarchy (which extends NP). These variations are motivated by relatively technical considerations; still, the resulting classes are referred to quite frequently in the literature.
Summary: Non-uniform polynomial-time (P/poly) captures efficient computations that are carried out by devices that can each handle only inputs of a specific length. The basic formalism ignores the complexity of constructing such devices (i.e., a uniformity condition). A finer formalism that allows for quantifying the amount of non-uniformity refers to so-called “machines that take advice.”
The Polynomial-time Hierarchy (PH) generalizes NP by considering statements expressed by quantified Boolean formulae with a fixed number of alternations of existential and universal quantifiers. It is widely believed that each quantifier alternation adds expressive power to the class of such formulae.
An interesting result that refers to both classes asserts that if NP is contained in P/poly then the Polynomial-time Hierarchy collapses to its second level. This result is commonly interpreted as supporting the common belief that non-uniformity is irrelevant to the P-vs-NP Question; that is, although P/poly extends beyond the class P, it is believed that P/poly does not contain NP.
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