On the complexity of hazard-free circuits

TL;DR

This paper establishes exponential lower bounds for hazard-free circuit complexity, showing that some functions require exponentially larger hazard-free circuits than their general counterparts, and provides efficient conversion methods.

Contribution

It proves the first unconditional exponential lower bounds for hazard-free circuits and introduces an efficient conversion method from general to hazard-free circuits.

Findings

Hazard-free complexity generalizes monotone complexity.

Certain functions require exponential hazard-free circuits.

Efficient conversion from Boolean to hazard-free circuits with polynomial overhead.

Abstract

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.