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

Abstract

Abstract. 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 Krajíček [6].

Different Approaches to Proof Systems

“…We remark that optimal proof systems are known to imply complete sets for various promise classes [KMT03], and this relation also holds in the presence of advice [BS09]. A related line of research has shown strong time and space hierarchy theorems for randomised and other semantic classes which use advice [FS04,FST05,vMP07,KvM08].…”

“…A prominent open question posed in [KP89] is whether there exists a strongest proof system, called a (p-)optimal proof system, which (p-)simulates all proof systems for . This question has interesting consequences such as existence of complete languages for promise classes [KMT03,BS09]. Despite a considerable research effort the existence of optimal proof systems is still open (cf.…”

Optimal Acceptors and Optimal Proof Systems

“…This is indeed the case; however, in order to keep the closedness under reductions (with advice) the advice given must have length O(1) and not a specific constant number of bits. For exactly one bit of advice one gets only an NP pair that is hard for disjoint NP without advice under many-one reductions without advice [BS09].…”

On p-Optimal Proof Systems and Logics for PTIME

“…A proof system is p-optimal if it simulates any other proof system in polynomial time. 1 In their fundamental paper [13] Krajícek and Pudlák derive a series of statements equivalent to the existence of a p-optimal proof system for TAUT and state the conjecture: Conjecture 1. There is no p-optimal proof system for TAUT.…”

“…There are artificial logics capturing polynomial time, but they do not fulfill a natural requirement to logics in this context: (1) There is an algorithm that decides whether A is a model of ϕ for all structures A and sentences ϕ of the logic and that does this for fixed ϕ in time polynomial in the size A of A.…”