P Versus NP Problem - Formal Definition For NP-completeness

Formal Definition For NP-completeness

There are many equivalent ways of describing NP-completeness.

Let be a language over a finite alphabet .

is NP-complete if, and only if, the following two conditions are satisfied:

  1. ; and
  2. any is polynomial-time-reducible to (written as ), where if, and only if, the following two conditions are satisfied:
    1. There exists such that \forall w\in\Sigma^{*}(w\in L^{'}\Leftrightarrow f(w)\in L); and
    2. there exists a polynomial-time Turing machine that halts with on its tape on any input .

Read more about this topic:  P Versus NP Problem

Famous quotes containing the words formal and/or definition:

    “True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
    —Marcel Proust (1871–1922)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)