Sam Buss and Ramyaa
    "Short Refutations for the Equivalence-Chain Principle for Constant-Depth Formulas"
    Preprint, January 2017

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Abstract: We consider tautologies expressing equivalence-chain properties in the spirit of Thapen and Krajicek, which are candidates for exponentially separating depth k and depth k+1 Frege proof systems. We formulate a special case where the initial member of the equivalence chain is fully specified and the equivalence-chain implications are actually equivalences. This special case is shown to lead to polynomial size resolution refutations. Thus it cannot be used for separating depth k and depth k+1 propositional systems. We state some Hastad switching lemma conditions that restrict the possible propositional proofs in more general situations.

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