On the complexity of hazard-free circuits

Research output: Contribution to journalJournal articleResearchpeer-review

  • Christian Ikenmeyer
  • Balagopal Komarath
  • Christoph Lenzen
  • Lysikov, Vladimir
  • Andrey Mokhov
  • Karteek Sreenivasaiah

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.

Original languageEnglish
Article number25
JournalJournal of the ACM
Volume66
Issue number4
ISSN0004-5411
DOIs
Publication statusPublished - Aug 2019
Externally publishedYes

    Research areas

  • Boolean circuits, Computational complexity, Hazards, Monotone circuits

ID: 232711266