Information in propositional proofs and algorithmic proof search
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We study from the proof complexity perspective the (informal) proof search problem: Is there an optimal way to search for propositional proofs? We note that 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(τ) assigning to a tautology a natural number, and we show that: - i_P(τ) characterizes time any P-proof search algorithm has to use on τ and that 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 τ with short proofs but having large information measure i_P(τ). We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for i_P(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.
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