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Space Characterizations of Complexity Measures and Size-Space Trade-Offs in Propositional Proof Systems

Authors Theodoros Papamakarios, Alexander Razborov

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Theodoros Papamakarios
  • Department of Computer Science, University of Chicago, IL, USA
Alexander Razborov
  • University of Chicago, IL, USA
  • Steklov Mathematical Institute, Moscow, Russia

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Theodoros Papamakarios and Alexander Razborov. Space Characterizations of Complexity Measures and Size-Space Trade-Offs in Propositional Proof Systems. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 100:1-100:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.100

Abstract

We identify two new big clusters of proof complexity measures equivalent up to polynomial and log n factors. The first cluster contains, among others, the logarithm of tree-like resolution size, regularized (that is, multiplied by the logarithm of proof length) clause and monomial space, and clause space, both ordinary and regularized, in regular and tree-like resolution. As a consequence, separating clause or monomial space from the (logarithm of) tree-like resolution size is the same as showing a strong trade-off between clause or monomial space and proof length, and is the same as showing a super-critical trade-off between clause space and depth. The second cluster contains width, Σ₂ space (a generalization of clause space to depth 2 Frege systems), both ordinary and regularized, as well as the logarithm of tree-like size in the system R(log). As an application of some of these simulations, we improve a known size-space trade-off for polynomial calculus with resolution. In terms of lower bounds, we show a quadratic lower bound on tree-like resolution size for formulas refutable in clause space 4. We introduce on our way yet another proof complexity measure intermediate between depth and the logarithm of tree-like size that might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof Complexity
  • Resolution
  • Size-Space Trade-offs

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