On Optimal Heuristic Randomized Semidecision Procedures, with Application to Proof Complexity
The existence of a ($p$-)optimal propositional proof system is a major open question in (proof) complexity; many people conjecture that such systems do not exist. Kraj\'{\i}\v{c}ek and Pudl\'{a}k \cite{KP} show that this question is equivalent to the existence of an algorithm that is optimal\footnote{Recent papers \cite{Monroe}
call such algorithms \emph{$p$-optimal} while traditionally Levin's algorithm was called \emph{optimal}. We follow the older tradition. Also there is some mess in terminology here, thus please see formal definitions in Sect.~\ref{sec:prelim} below.} on all propositional tautologies. Monroe \cite{Monroe} recently gave a conjecture implying that such algorithm does not exist.
We show that in the presence of errors such optimal algorithms \emph{do} exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow the algorithm to claim a small number of false ``theorems'' (according to any polynomial-time samplable distribution on non-tautologies) and err with bounded probability on other inputs.
Our result can also be viewed as the existence of an optimal proof system in a class of proof systems obtained by generalizing automatizable proof systems.
Propositional proof complexity
optimal algorithm
453-464
Regular Paper
Edward A.
Hirsch
Edward A. Hirsch
Dmitry
Itsykson
Dmitry Itsykson
10.4230/LIPIcs.STACS.2010.2475
Creative Commons Attribution-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nd/3.0/legalcode