Karchmer-Wigderson Games for Hazard-free Computation

TL;DR

This paper introduces a new Karchmer-Wigderson game to analyze hazard-free formulas, establishing stronger lower bounds and optimal constructions that bridge Boolean and monotone complexity.

Contribution

It generalizes existing games to study hazard-free formulas, providing new lower bounds and optimal formulas for the multiplexer function.

Findings

Hazard-free formula size and depth lower bounds are stronger than previous results.

Optimal hazard-free formulas for the multiplexer function are constructed.

An improved universal hazard-free formula size upper bound is achieved.

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.