Input distance and lower bounds for propositional resolution proof length

• Authors: Allen Van Gelder
• Venue: SAT'05 Proceedings of the 8th international conference on Theory and Applications of Satisfiability Testing
• Year: 2005

Input Distance (Δ) is introduced as a metric for propositional resolution derivations. If $\mathcal{F} = C_i$ is a formula and D is a clause, then $\Delta(\mathcal{D},\mathcal{F})$ is defined as mini |D – Ci|. The Δ for a derivation is the maximum Δ of any clause in the derivation. Input Distance provides a refinement 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 $(W - width(\mathcal{F}))$, where W is the maximum clause width of a narrowest refutation of $\mathcal{F}$. It is shown here that (1) both bounds apply with $(W - width(\mathcal{F}))$ replaced by Δ; (2) for pigeon-hole formulas PHP(m, n), the minimum Δ for any refutation is Ω(n). A similar result is conjectured for the GT(n) family analyzed by Bonet and Galesi (FOCS 1999).