### Abstract

In this paper, we investigate a relationship between energy and size of a threshold circuit processing a simple task, called P_{LR}
^{n}, that was introduced in a context of pattern recognition. Formally, P _{LR}
^{n}: {0, 1}^{n} x {0, 1}^{n} → {0,1} is defined as follows: For every x = (x_{1},x_{2},⋯, x_{n}) ∈ {0, 1}^{n} and y = (y_{1},y _{2},⋯,yn) ∈ {0, 1}^{n}, P_{LR} ^{n}(x, y) = 1 if there exists a pair of indices i and j such that i < j and x_{i} = y_{j} = 1; and P_{LR} ^{n}(x,y) = 0 otherwise. We prove that P_{LR}
^{n} can be computed by a threshold circuit of energy e and size s = O (e · n ^{2}/^{e-1}) for any integer e, 3 ≤ e ≤ 2log_{2} n + 1. Our result implies that one can construct an energy-efficient circuit computing P_{LR}
^{n} if it is allowable to use large size. Moreover, we focus on an extreme case where a threshold circuit has energy e = 1, and show that PLR can be computed by a threshold circuit of energy e = 1 and size s = ⌊n/2⌋, while P_{LR}
^{n} cannot be computed by any threshold circuit of energy e = 1 and size s ≤ ⌊n/2⌋ - 1.

Original language | English |
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Title of host publication | Theory and Applications of Models of Computation - 10th International Conference, TAMC 2013, Proceedings |

Publisher | Springer-Verlag |

Pages | 248-259 |

Number of pages | 12 |

ISBN (Print) | 9783642382352 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

Event | 10th International Conference on Theory and Applications of Models of Computation, TAMC 2013 - Hong Kong, China Duration: 2013 May 20 → 2013 May 22 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7876 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th International Conference on Theory and Applications of Models of Computation, TAMC 2013 |
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Country | China |

City | Hong Kong |

Period | 13/5/20 → 13/5/22 |

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

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## Cite this

*Theory and Applications of Models of Computation - 10th International Conference, TAMC 2013, Proceedings*(pp. 248-259). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7876 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-642-38236-9_23