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On the Complexity of Hazard-free Circuits


Ikenmeyer,  Christian
Algorithms and Complexity, MPI for Informatics, Max Planck Society;


Lenzen,  Christoph
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Ikenmeyer, C., Komarath, B., Lenzen, C., Lysikov, V., Mokhov, A., & Sreenivasaiah, K. (2017). On the Complexity of Hazard-free Circuits. Retrieved from http://arxiv.org/abs/1711.01904.

Cite as: http://hdl.handle.net/21.11116/0000-0000-3F22-4
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result we establish the NP-completeness of several hazard detection problems.