### Abstract

In this paper we show that there is a close relationship between the energy complexity and the depth of threshold circuits computing any Boolean function although they have completely different physical meanings. Suppose that a Boolean function f can be computed by a threshold circuit C of energy complexity e and hence at most e threshold gates in C output "1" for any input to C. We prove that the function f can also be computed by a threshold circuit C′ of the depth 2e+1 and hence the parallel computation time of C′ is 2e+1. If the size of C is s, that is, there are s threshold gates in C, then the size s′ of C′ is s′=2es+1. Thus, if the size s of C is polynomial in the number n of input variables, then the size s′ of C′ is polynomial in n, too.

Original language | English |
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Pages (from-to) | 3938-3946 |

Number of pages | 9 |

Journal | Theoretical Computer Science |

Volume | 411 |

Issue number | 44-46 |

DOIs | |

Publication status | Published - Oct 25 2010 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*411*(44-46), 3938-3946. https://doi.org/10.1016/j.tcs.2010.08.006

**Energy and depth of threshold circuits.** / Uchizawa, Kei; Nishizeki, Takao; Takimoto, Eiji.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 411, no. 44-46, pp. 3938-3946. https://doi.org/10.1016/j.tcs.2010.08.006

}

TY - JOUR

T1 - Energy and depth of threshold circuits

AU - Uchizawa, Kei

AU - Nishizeki, Takao

AU - Takimoto, Eiji

PY - 2010/10/25

Y1 - 2010/10/25

N2 - In this paper we show that there is a close relationship between the energy complexity and the depth of threshold circuits computing any Boolean function although they have completely different physical meanings. Suppose that a Boolean function f can be computed by a threshold circuit C of energy complexity e and hence at most e threshold gates in C output "1" for any input to C. We prove that the function f can also be computed by a threshold circuit C′ of the depth 2e+1 and hence the parallel computation time of C′ is 2e+1. If the size of C is s, that is, there are s threshold gates in C, then the size s′ of C′ is s′=2es+1. Thus, if the size s of C is polynomial in the number n of input variables, then the size s′ of C′ is polynomial in n, too.

AB - In this paper we show that there is a close relationship between the energy complexity and the depth of threshold circuits computing any Boolean function although they have completely different physical meanings. Suppose that a Boolean function f can be computed by a threshold circuit C of energy complexity e and hence at most e threshold gates in C output "1" for any input to C. We prove that the function f can also be computed by a threshold circuit C′ of the depth 2e+1 and hence the parallel computation time of C′ is 2e+1. If the size of C is s, that is, there are s threshold gates in C, then the size s′ of C′ is s′=2es+1. Thus, if the size s of C is polynomial in the number n of input variables, then the size s′ of C′ is polynomial in n, too.

UR - http://www.scopus.com/inward/record.url?scp=77957669961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957669961&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2010.08.006

DO - 10.1016/j.tcs.2010.08.006

M3 - Article

AN - SCOPUS:77957669961

VL - 411

SP - 3938

EP - 3946

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 44-46

ER -