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:

    “The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
    —Franz Grillparzer (1791–1872)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)