Polynomial-time Random Oracles, Nondeterministic Sublinear Time, and Boolean Function Complexity - Sekoni, Adewale (ProQuest Dissertations & Theses, 2018)
We use resource-bounded measure to study nondeterministic sublinear time, random oracles that separate complexity classes, and complexity measures of Boolean functions. 1. Nondeterministic Sublinear Time. 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[n1/11] has measure 0 in P. We improve on their result to show that the class NTIME[o (n)] 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. 2. Random Oracles that Separate Complexity Classes. Bennett and Gill (1981) showed that PA ≠ NP A ≠ coNPA for a random oracle A, with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We show that PA ≠ NPA for every oracle A that is p-betting-game random. Ideally, we would extend this result to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation, either P ≠ PSPACE or BPP ≠ EXP. Rossman, Servedio, and Tan (2015) showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend their result to show that PH A is infinite relative to oracles A that are p-betting-game random. We show that if PHA separates at even its first level, then the unrelativized separation NP ≠ EXP holds. 3. Complexity Measures of Boolean Functions. The study of the nonuniform circuit-size complexity of uniform complexity classes using resource-bounded measure was initiated by Lutz (1992). He showed almost all languages in ESPACE have maximum circuit-size complexity. This was subsequently improved to a resource-bounded dimension result in the third level of the exponential time hierarchy by Hitchcock and Vinodchandran (2006). We use both resource-bounded measure and dimension to study other complexity measures of Boolean functions. The main complexity measures we consider are DNF size, DNF width, DNF-of-parities size, and influence.