Karchmer-Wigderson Games for Hazard-Free Computation

Authors Christian Ikenmeyer, Balagopal Komarath, Nitin Saurabh

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Christian Ikenmeyer
  • University of Warwick, Coventry, UK
Balagopal Komarath
  • IIT Gandhinagar, India
Nitin Saurabh
  • IIT Hyderabad, India


We thank Igor Sergeev for his support with the literature. Nitin Saurabh would like to thank Yuval Filmus for helpful discussions. We would also like to thank anonymous reviewers for useful suggestions that greatly improved the presentation.

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Christian Ikenmeyer, Balagopal Komarath, and Nitin Saurabh. Karchmer-Wigderson Games for Hazard-Free Computation. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 74:1-74:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Communication complexity
  • Theory of computation → Concurrency
  • Hardware → Combinational circuits
  • Hazard-free computation
  • monotone computation
  • Karchmer-Wigderson games
  • communication complexity
  • lower bounds


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