Karchmer-Wigderson Games for Hazard-free Computation

Christian Ikenmeyer; Balagopal Komarath; Nitin Saurabh

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arXiv · 2107.05128

Karchmer-Wigderson Games for Hazard-free Computation

Christian Ikenmeyer·Balagopal Komarath·Nitin Saurabh

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Abstract

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 significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.

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BibTeX RIS

Christian Ikenmeyer, Balagopal Komarath, Nitin Saurabh. 2021-07-11. Karchmer-Wigderson Games for Hazard-free Computation. https://arxiv.org/abs/2107.05128

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