Preliminary Report on Input Cover Number as a Metric for Propositional Resolution Proofs

Input Cover Number (denoted by ) is introduced as a metric for diÆculty of propositional resolution derivations. If F = fCig is the input CNF formula, then F(D) is de ned as the minimum number of clauses Ci needed to form a superset of (i.e., cover) clause D. Input Cover Number provides a re nement of the clause-width metric in the sense that it applies to families of formulas whose clause width grows with formula size, such as pigeon-hole formulas PHP(m;n) and GT(n). Although these two families have much di erent general-resolution complexities, it is known that both require (n) clause width (after transforming to 3CNF). It is shown here that is (n) for pigeon-hole formulas and is (1) for GT(n) formulas and variants of GT(n).