From Proof Complexity to Circuit Complexity via Interactive Protocols

Authors Noel Arteche , Erfan Khaniki , Ján Pich , Rahul Santhanam

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Noel Arteche
  • Lund University, Sweden
  • University of Copenhagen, Denmark
Erfan Khaniki
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
Ján Pich
  • University of Oxford, UK
Rahul Santhanam
  • University of Oxford, UK


Independently, Albert Atserias suggested to us to consider using interactive proof systems to derive this type of connections. We thank Pavel Pudlák for useful comments, and the anonymous reviewers for relevant comments and references. This work was done in part while the first author was visiting the University of Oxford and the Institute of Mathematics of the Czech Academy of Sciences. For the purpose of Open Access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.

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Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam. From Proof Complexity to Circuit Complexity via Interactive Protocols. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Theory of computation → Circuit complexity
  • Theory of computation → Complexity theory and logic
  • proof complexity
  • circuit complexity
  • interactive protocols


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