Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs

Authors Nicola Galesi, Dmitry Itsykson, Artur Riazanov, Anastasia Sofronova

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Nicola Galesi
  • Dipartimento di Informatica, Sapienza Università di Roma, Via Salaria 113, Rome, Italy
Dmitry Itsykson
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics of the , Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
Artur Riazanov
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics of the , Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
Anastasia Sofronova
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics of the , Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
  • St. Petersburg State University, 7-9 Universitetskaya Emb., St. Petersburg, Russia


The authors thank Navid Talebanfard for discussions on the lower bound. Nicola also thanks Paul Wollan for introducing him to the tree-cutwidth. Dmitry is a Young Russian Mathematics award winner and would like to thank sponsors and jury of the contest.

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Nicola Galesi, Dmitry Itsykson, Artur Riazanov, and Anastasia Sofronova. Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2^{tw(G)^{Omega(1/d)}} in depth-d Frege systems for d<(K log n)/(log log n), where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2^{tw(G)^{O(1/d)}} poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Tseitin formula
  • treewidth
  • AC0-Frege


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