Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes

In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class C? For concrete languages L (such as TAUT or SAT) and concrete promise classes C (such as NP∩coNP, UP, BPP, disjoint NP-pairs etc.), these questions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new characterizations for Q1 and Q2 that apply to almost all promise classes C and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount on advice is available in the underlying machine model. This continues a recent line of research on proof systems with advice started by Cook and Kraj´ıˇcek [6].

• Type: Conference
• Year: 2009
• Where: CSR
• Authors: Olaf Beyersdorff, Zenon Sadowski
• Keywords: CSR 2009, Promise Class C, Promise Classes, Proof Systems, Theoretical Computer Science