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Combinatorics of first order structures and propositional proof systems

By Jan Krajícek


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 \Phi iff \Phi 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 \Phi for which the associated sequence of propositional formulas h\Phi i n, encoding that \Phi holds in Lstructures of size n, have polynomial size P-proofs. That is, Comb(M) contains all \Phi feasibly verifiable in P. Finding M that covers P but does not combinatorially satisfy \Phi thus gives a super polynomial lower bound for the size of P-proofs of h\Phi in. 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 \Lambd

Year: 2009
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