### Abstract

A complexity measure for threshold circuits, called the energy complexity, has been proposed to measure an amount of energy consumed during computation in the brain. Biological neurons need more energy to transmit a "spike" than not to transmit one, and hence the energy complexity of a threshold circuit is defined as the number of gates in the circuit that output "1" during computation. Since the firing activity of neurons in the brain is quite sparse, the following question arises: what Boolean functions can or cannot be computed by threshold circuits with small energy complexity. In the paper, we partially answer the question, that is, we show that there exists a tradeoff among three complexity measures of threshold circuits: the energy complexity, size, and depth. The tradeoff implies an exponential lower bound on the size of constant-depth threshold circuits with small energy complexity for a large class of Boolean functions.

Original language | English |
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Title of host publication | Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007 |

Pages | 169-178 |

Number of pages | 10 |

DOIs | |

Publication status | Published - Oct 2 2007 |

Externally published | Yes |

Event | 22nd Annual IEEE Conference on Computational Complexity, CCC 2007 - San Diego, CA, United States Duration: Jun 13 2007 → Jun 16 2007 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
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ISSN (Print) | 1093-0159 |

### Other

Other | 22nd Annual IEEE Conference on Computational Complexity, CCC 2007 |
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Country | United States |

City | San Diego, CA |

Period | 6/13/07 → 6/16/07 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science
- Computational Mathematics

### Cite this

*Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007*(pp. 169-178). [4262761] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2007.4

**An exponential lower bound on the size of constant-depth threshold circuits with small energy complexity.** / Uchizawa, Kei; Takimoto, Eiji.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007.*, 4262761, Proceedings of the Annual IEEE Conference on Computational Complexity, pp. 169-178, 22nd Annual IEEE Conference on Computational Complexity, CCC 2007, San Diego, CA, United States, 6/13/07. https://doi.org/10.1109/CCC.2007.4

}

TY - GEN

T1 - An exponential lower bound on the size of constant-depth threshold circuits with small energy complexity

AU - Uchizawa, Kei

AU - Takimoto, Eiji

PY - 2007/10/2

Y1 - 2007/10/2

N2 - A complexity measure for threshold circuits, called the energy complexity, has been proposed to measure an amount of energy consumed during computation in the brain. Biological neurons need more energy to transmit a "spike" than not to transmit one, and hence the energy complexity of a threshold circuit is defined as the number of gates in the circuit that output "1" during computation. Since the firing activity of neurons in the brain is quite sparse, the following question arises: what Boolean functions can or cannot be computed by threshold circuits with small energy complexity. In the paper, we partially answer the question, that is, we show that there exists a tradeoff among three complexity measures of threshold circuits: the energy complexity, size, and depth. The tradeoff implies an exponential lower bound on the size of constant-depth threshold circuits with small energy complexity for a large class of Boolean functions.

AB - A complexity measure for threshold circuits, called the energy complexity, has been proposed to measure an amount of energy consumed during computation in the brain. Biological neurons need more energy to transmit a "spike" than not to transmit one, and hence the energy complexity of a threshold circuit is defined as the number of gates in the circuit that output "1" during computation. Since the firing activity of neurons in the brain is quite sparse, the following question arises: what Boolean functions can or cannot be computed by threshold circuits with small energy complexity. In the paper, we partially answer the question, that is, we show that there exists a tradeoff among three complexity measures of threshold circuits: the energy complexity, size, and depth. The tradeoff implies an exponential lower bound on the size of constant-depth threshold circuits with small energy complexity for a large class of Boolean functions.

UR - http://www.scopus.com/inward/record.url?scp=34748903048&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34748903048&partnerID=8YFLogxK

U2 - 10.1109/CCC.2007.4

DO - 10.1109/CCC.2007.4

M3 - Conference contribution

SN - 0769527809

SN - 9780769527802

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 169

EP - 178

BT - Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007

ER -