On the density and core of the complexity classes

Abstract

We study the density and the complexity core of the polynomial and exponential time classes. We define complexity classes with density function d(n), such as P(d(n)) and NP(d(n)), E(d(n)) and NE(d(n)). Our three basic results indicate the collapse of these classes related to the standard classes:

• 1 -If P=NP then ∀d(n) P(d(n)) = NP(d(n));

• 2 -If P≠ NP then either ∀d(n) P(d(n)) ≠ NP(d(n)) or P(1/¦Σ¦ ⁿ) = NP(1/¦Σ¦ ⁿ);

• 3 -If for any d(n) E(d(n)) ≠ NE(d(n)) then ∀d(n) P(d(n)) ≠ NP(d(n)).

Furthermore, as a lemma to result 2, we show that if P ≠ NP, then for all d(n) computable in polynomial time, either P(d(n)) ≠ NP(d(n)) or P(1−d(n)) ≠ NP(1 − d(n)).

This article was processed using the LAT_EX macro package with LMAMULT style