Information in Propositional Proofs and Algorithmic Proof Search

We study from the proof complexity perspective the (informal) proof search problem (cf. [17, Sections 1.5 and 21.5]): • 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 iP(τ ) assigning to a tautology a natural number, and we show that: • iP(τ ) characterizes time any P -proof search algorithm has to use on τ , • for a fixed P there is such an information-optimal algorithm (informally: it finds proofs of minimal information content), • a proof system is information-efficiency optimal (its information-efficiency function is minimal up to a multiplicative constant) iff it is p-optimal, • for non-automatizable systems P there are formulas τ with short proofs but having large information measure iP(τ ) . We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for iP(τ ) 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.