Short refutations for an equivalence-chain principle for constant-depth formulas

We consider tautologies expressing equivalence-chain properties in the spirit of Thapen and Kraj́ıček, 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 H̊astad switching lemma conditions that restrict the possible propositional proofs in more general situations. Supported in part by NSF grant CCR-1213151. Buss also thanks the Simons Institute for a supporting a visit to the Special Program on Logical Structures in Computation in August and September 2016 where a main portion of this work was carried out. Ramyaa gratefully acknowledges the support from Simons Institute in Theoretical Computer Science and its Fall 2016 Program on Logical Structures in Computation for hosting her during the period of this work.