Abstract

We study from the proof complexity perspective the proof search problem : •Is there an optimal way to search for propositional proofs?We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists.To characterize precisely the time proof search algorithms need for individual formulas we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system P we attach information-efficiency function $i_P$ assigning to a tautology a natural number, and we show that: • $i_P$ characterizes time any P-proof search algorithm has to use on $\tau $,•for a fixed P there is such an information-optimal algorithm,•a proof system is information-efficiency optimal iff it is p-optimal,•for non-automatizable systems P there are formulas $\tau $ with short proofs but having large information measure $i_P$.We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search.