From small space to small width in resolution

Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

Research output: Contribution to journalArticlepeer-review


In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexitymeasure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similarmethods.

Original languageEnglish
Article number28
JournalACM Transactions on Computational Logic
Issue number4
StatePublished - 1 Aug 2015
Externally publishedYes


  • PCR
  • Polynomial calculus
  • Polynomial calculus resolution
  • Proof complexity
  • Resolution
  • Space
  • Width
  • degree

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Theoretical Computer Science
  • General Computer Science
  • Logic


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