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Combinatorics of first order structures and propositional proof systems

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Abstract.

We define the notion of a combinatorics of a first order structure, and a relation of covering between first order structures and propositional proof systems. Namely, a first order structure M combinatorially satisfies an L-sentence Φ iff Φ holds in all L-structures definable in M. The combinatorics Comb(M) of M is the set of all sentences combinatorially satisfied in M. Structure M covers a propositional proof system P iff M combinatorially satisfies all Φ for which the associated sequence of propositional formulas 〈Φ〉 n , encoding that Φ holds in L-structures of size n, have polynomial size P-proofs. That is, Comb(M) contains all Φ feasibly verifiable in P. Finding M that covers P but does not combinatorially satisfy Φ thus gives a super polynomial lower bound for the size of P-proofs of 〈Φ〉 n . We show that any proof system admits a class of structures covering it; these structures are expansions of models of bounded arithmetic. We also give, using structures covering proof systems R *(log) and PC, new lower bounds for these systems that are not apparently amenable to other known methods. We define new type of propositional proof systems based on a combinatorics of (a class of) structures.

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  1. Mathematical Institute, Academy of Sciences, Žitná, 25, Prague 1, 115 67, The Czech Republic

    Jan Krajíček

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  1. Jan Krajíček
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Correspondence to Jan Krajíček.

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Partially supported by grant # A 101 99 01 of the Academy of Sciences of the Czech Republic and by project LN00A056 of The Ministry of Education of the Czech Republic.

Also member of the Institute for Theoretical Computer Science of the Charles University. A part of this work was done while visiting the Mathematical Institute, Oxford.

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Krajíček, J. Combinatorics of first order structures and propositional proof systems. Arch. Math. Logic 43, 427–441 (2004). https://doi.org/10.1007/s00153-003-0186-y

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  • DOI: https://doi.org/10.1007/s00153-003-0186-y

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