On the complexity of hazard-free circuits

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 prov...

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Bibliographic Details
Published in:arXiv.org
Main Authors: Ikenmeyer, Christian, Komarath, Balagopal, Lenzen, Christoph, Lysikov, Vladimir, Mokhov, Andrey, Sreenivasaiah, Karteek
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 04.04.2018
Subjects:
Algorithms
Boolean
Boolean algebra
Boolean functions
Circuit design
Circuits
Complexity
Gates (circuits)
Hazards
Lower bounds
Monotone functions
Multiplication
Upper bounds
ISSN:2331-8422
Online Access:Get full text
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Summary: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.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1711.01904