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)


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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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Allen Van Gelder
    • 1
  1. 1.University of CaliforniaSanta CruzUSA

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