### Abstract

Let C be a threshold logic circuit computing a Boolean function MOD _{m}:{0,1}^{n} → {0,1}, where n ≥1 and m≥2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} ^{n} to C is a multiple of m and, otherwise, C outputs "1." The function MOD_{2} is the so-called PARITY function, and MOD _{n+1} is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output "1" for any input x ∈{ 0, 1} ^{n} . In the paper, we prove that a very simple inequality n/(m-1)≤s ^{e} holds for every circuit C computing MOD _{m} . The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MOD _{m} , and yields a lower bound e=Ω((logn-logm)/loglogn) on e if s=O(polylog(n)). We actually obtain a general result on the so-called generalized mod function, from which the result on the ordinary mod function MOD _{m} immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits.

Original language | English |
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Title of host publication | Mathematical Foundations of Computer Science 2009 - 34th International Symposium, MFCS 2009, Proceedings |

Pages | 724-735 |

Number of pages | 12 |

DOIs | |

Publication status | Published - Sep 28 2009 |

Event | 34th International Symposium on Mathematical Foundations of Computer Science 2009, MFCS 2009 - Novy Smokovec, High Tatras, Slovakia Duration: Aug 24 2009 → Aug 28 2009 |

### 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 | 5734 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 34th International Symposium on Mathematical Foundations of Computer Science 2009, MFCS 2009 |
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Country | Slovakia |

City | Novy Smokovec, High Tatras |

Period | 8/24/09 → 8/28/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Mathematical Foundations of Computer Science 2009 - 34th International Symposium, MFCS 2009, Proceedings*(pp. 724-735). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5734 LNCS). https://doi.org/10.1007/978-3-642-03816-7_61

**Size and energy of threshold circuits computing mod functions.** / Uchizawa, Kei; Nishizeki, Takao; Takimoto, Eiji.

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

*Mathematical Foundations of Computer Science 2009 - 34th International Symposium, MFCS 2009, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5734 LNCS, pp. 724-735, 34th International Symposium on Mathematical Foundations of Computer Science 2009, MFCS 2009, Novy Smokovec, High Tatras, Slovakia, 8/24/09. https://doi.org/10.1007/978-3-642-03816-7_61

}

TY - GEN

T1 - Size and energy of threshold circuits computing mod functions

AU - Uchizawa, Kei

AU - Nishizeki, Takao

AU - Takimoto, Eiji

PY - 2009/9/28

Y1 - 2009/9/28

N2 - Let C be a threshold logic circuit computing a Boolean function MOD m:{0,1}n → {0,1}, where n ≥1 and m≥2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} n to C is a multiple of m and, otherwise, C outputs "1." The function MOD2 is the so-called PARITY function, and MOD n+1 is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output "1" for any input x ∈{ 0, 1} n . In the paper, we prove that a very simple inequality n/(m-1)≤s e holds for every circuit C computing MOD m . The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MOD m , and yields a lower bound e=Ω((logn-logm)/loglogn) on e if s=O(polylog(n)). We actually obtain a general result on the so-called generalized mod function, from which the result on the ordinary mod function MOD m immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits.

AB - Let C be a threshold logic circuit computing a Boolean function MOD m:{0,1}n → {0,1}, where n ≥1 and m≥2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} n to C is a multiple of m and, otherwise, C outputs "1." The function MOD2 is the so-called PARITY function, and MOD n+1 is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output "1" for any input x ∈{ 0, 1} n . In the paper, we prove that a very simple inequality n/(m-1)≤s e holds for every circuit C computing MOD m . The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MOD m , and yields a lower bound e=Ω((logn-logm)/loglogn) on e if s=O(polylog(n)). We actually obtain a general result on the so-called generalized mod function, from which the result on the ordinary mod function MOD m immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits.

UR - http://www.scopus.com/inward/record.url?scp=70349327674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349327674&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-03816-7_61

DO - 10.1007/978-3-642-03816-7_61

M3 - Conference contribution

AN - SCOPUS:70349327674

SN - 3642038158

SN - 9783642038150

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 724

EP - 735

BT - Mathematical Foundations of Computer Science 2009 - 34th International Symposium, MFCS 2009, Proceedings

ER -