Space complexity in polynomial calculus

Research output: Chapter in Book/Report/Conference proceedingPaper in conference proceeding


During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovich et. al.'02], where the lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could refute any k-CNF formula in constant space. We prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich'02]. - For PCR, we prove an Omega(n) space lower bound for a bit wise encoding of the functional pigeonhole principle with m pigeons and n holes. These formulas have width O(log(n)), and hence this is an exponential improvement over [Alekhnovich'02] measured in the width of the formulas. - We then present another encoding of the pigeonhole principle that has constant width, and prove an Omega(n) space lower bound in PCR for these formulas as well. - We prove an Omega(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHP(m, n) with m pigeons and n holes, and show that this is tight. - We prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not known to be the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way.


External organisations
  • University of Toronto
  • Sapienza University of Rome
  • KTH Royal Institute of Technology
  • Academy of Sciences of the Czech Republic
  • Technion - Israel Institute of Technology
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Computer Science


  • k-CNF, pcr, polynomial calculus, proof complexity, resolution, space
Original languageEnglish
Title of host publicationProceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012
PublisherIEEE - Institute of Electrical and Electronics Engineers Inc.
Number of pages11
ISBN (Print)9780769547084
Publication statusPublished - 2012
Publication categoryResearch
Externally publishedYes
EventIEEE Computer Society Technical Committee on Mathematical Foundations of Computing - Porto, Portugal
Duration: 2012 Jun 262012 Jun 29

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159


ConferenceIEEE Computer Society Technical Committee on Mathematical Foundations of Computing