Nondeterminisic Sublinear Time Has Measure 0 in P

TL;DR

This paper proves that the class of languages decidable in nondeterministic sublinear time has measure zero in P, strengthening previous results and using DNF width across all major measure notions.

Contribution

It extends prior work by Cai, Sivakumar, and Strauss, showing that all nondeterministic sublinear time languages have measure zero in P, using DNF width techniques.

Findings

Nondeterministic sublinear time class has measure 0 in P.

Result applies to all four major measure notions on P.

Improves previous measure results for NTIME classes.

Abstract

The measure hypothesis is a quantitative strengthening of the P != NP conjecture which asserts that NP is a nonnegligible subset of EXP. Cai, Sivakumar, and Strauss (1997) showed that the analogue of this hypothesis in P is false. In particular, they showed that NTIME[n^{1/11}] has measure 0 in P. We improve on their result to show that the class of all languages decidable in nondeterministic sublinear time has measure 0 in P. Our result is based on DNF width and holds for all four major notions of measure on P.