Input Distance and Lower Bounds for Propositional Resolution Proof Length

Input Distance ( ) is introduced as a metric for propositional resolution derivations. If F = Ci is a formula and D is a clause, then (D;F) is de ned as minijD Cij. The for a derivation is the maximum of any clause in the derivation. Input Distance provides a re nement of the clause-width metric analyzed by Ben-Sasson and Wigderson (JACM 2001) in that it applies to families whose clause width grows, such as pigeon-hole formulas. They showed two upper bounds on (W width(F)), where W is the maximum clause width of a narrowest refutation of F . It is shown here that (1) both bounds apply with (W width(F)) replaced by ; (2) for pigeon-hole formulas PHP(m;n), the minimum for any refutation is (n). A similar result is conjectured for the GT (n) family analyzed by Bonet and Galesi (FOCS 1999).