# Mathematical Logic Quarterly

## Short refutations for an equivalence‐chain principle for constant‐depth formulas

### Abstract

We consider tautologies expressing equivalence‐chain properties in the spirit of Thapen and Krajíček, 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 Håstad switching lemma conditions that restrict the possible propositional proofs in more general situations.

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