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INFORMATION IN PROPOSITIONAL PROOFS AND ALGORITHMIC PROOF SEARCH

Published online by Cambridge University Press:  27 September 2021

JAN KRAJÍČEK*
Affiliation:
FACULTY OF MATHEMATICS AND PHYSICS CHARLES UNIVERSITY SOKOLOVSKÁ 83, PRAGUE 186 75, THE CZECH REPUBLIC

Abstract

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 $i_P(\tau )$ assigning to a tautology a natural number, and we show that:

  • $i_P(\tau )$ characterizes time any P-proof search algorithm has to use on $\tau $ ,

  • 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 $\tau $ with short proofs but having large information measure $i_P(\tau )$ .

We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P(\tau )$ 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.

Type
Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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