Relationship between density and deterministic complexity of MP-complete languages

Abstract

Let Σ be an arbitrary alphabet.

denotes ɛ∪Σ∪...∪Σⁿ. We say that a function t, t :

is f-sparse iff card

for every natural n. The main theorem of this paper establishes that if CLIQUE has some f-sparse translation into another set, which is calculable by a deterministic Turing machine in time bounded by f, then all the sets belonging to NP are calculable in time bounded by a function polynomially related to f. The proof is constructive and shows the way of constructing a proper algorithm. The simplest and most significant corollary says that if there is an NP-complete language over a single letter alphabet, then P=NP.