  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. Krajicek and Pudlak (1989) show that this question is equivalent to the
existence of an algorithm that is optimal on all propositional tautologies.
Monroe (2009) recently gave a conjecture implying that such algorithm does not
exist.
  We show that in the presence of errors such optimal algorithms 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
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.
