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Computer Science, Computational Complexity, cs.CC,
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.
