On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates
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Igor' S. Sergeev
Abstract
We obtain estimates for the complexity of the implementation of n-place Boolean functions by circuits and formulas built of unbounded fan-in conjunction and disjunction gates and either negation gates or negations of variables as inputs. Restrictions on the depth of circuits and formulas are imposed. In a number of cases, the estimates obtained in the paper are shown to be asymptotically sharp. In particular, for the complexity of circuits with variables and their negations on inputs, the Shannon function is asymptotically estimated as
Originally published in Diskretnaya Matematika (2018) 30, N°2, 120–137 (in Russian).
Funding: This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 17-01-00485a).
Acknowledgment
The author is grateful to S. Jukna for extensive and fruitful discussions of the problem, to E. Allender, who brought our attention to the problem under consideration, to S. B. Gashkov for his interest in the present work, and to the referees for valuable comments.
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Articles in the same Issue
- Frontmatter
- Centrally essential rings which are not necessarily unital or associative
- On the asymptotics of degree structure of configuration graphs with bounded number of edges
- Limit distributions of the Pearson statistics for nonhomogeneous polynomial scheme
- On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates
- Limit Poisson law for the distribution of the number of components in generalized allocation scheme
- Finite algebras of Bernoulli distributions
Articles in the same Issue
- Frontmatter
- Centrally essential rings which are not necessarily unital or associative
- On the asymptotics of degree structure of configuration graphs with bounded number of edges
- Limit distributions of the Pearson statistics for nonhomogeneous polynomial scheme
- On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates
- Limit Poisson law for the distribution of the number of components in generalized allocation scheme
- Finite algebras of Bernoulli distributions
