Automating OBDD proofs is NP-hard

Authors Dmitry Itsykson , Artur Riazanov

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Dmitry Itsykson
  • Steklov Institute of Mathematics at St. Petersburg, Russia
Artur Riazanov
  • Steklov Institute of Mathematics at St. Petersburg, Russia


The authors thank Anastasia Sofronova and Michal Garl´ık for fruitful discussions, Anastasia Sofronova for her comments on the draft. They also thank anonymous referees for their feedback. Artur Riazanov additionally thanks Dmitry Sokolov for his patient explanations of the lifting technique in [Garg et al., 2018].

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Dmitry Itsykson and Artur Riazanov. Automating OBDD proofs is NP-hard. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We prove that the proof system OBDD(∧, weakening) is not automatable unless P = NP. The proof is based upon the celebrated result of [Albert Atserias and Moritz Müller, 2019] about the hardness of automatability for resolution. The heart of the proof is lifting with multi-output indexing gadget from resolution block-width to dag-like multiparty number-in-hand communication protocol size with o(n) parties, where n is the number of variables in the non-lifted formula. A similar lifting theorem for protocols with n+1 participants was proved by [Göös et al., 2020] to establish the hardness of automatability result for Cutting Planes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • proof complexity
  • OBDD
  • automatability
  • lifting
  • dag-like communication


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