# Input Distance and Lower Bounds for Propositional Resolution Proof Length

• Allen Van Gelder
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3569)

## Abstract

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 min i |DC i |. 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).

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