### Abstract

Modular gates are known to be immune for the random restriction techniques of Ajtai, Furst, Saxe, Sipser, Yao and Hadstad. We demonstrate here a random clustering technique which overcomes this difficulty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien, Krause and Pudlak, and others, characterizing symmetric functions computable by small (MOD_{p}, AND_{t}, MOD_{m}) circuits. Applying a degree-decreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien, and other related lower bounds for certain (MOD_{p}, MOD_{m}, AND) circuits. Most of the previous lower bounds on circuits with modular gates used special definitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MOD_{m} gates. Our methods are applicable - and our lower bounds are valid - for the most general modular gates as well.

Original language | English |
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Title of host publication | Annual Symposium on Foundations of Computer Science - Proceedings |

Editors | Anon |

Publisher | IEEE Comp Soc |

Pages | 279-288 |

Number of pages | 10 |

Publication status | Published - 1998 |

Event | Proceedings of the 1998 39th Annual Symposium on Foundations of Computer Science - Palo Alto, CA, USA Duration: Nov 8 1998 → Nov 11 1998 |

### Other

Other | Proceedings of the 1998 39th Annual Symposium on Foundations of Computer Science |
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City | Palo Alto, CA, USA |

Period | 11/8/98 → 11/11/98 |

### Fingerprint

### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*(pp. 279-288). IEEE Comp Soc.

**Lower bounds for (MOD p - MOD m) circuits.** / Grolmusz, Vince; Tardos, G.

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

*Annual Symposium on Foundations of Computer Science - Proceedings.*IEEE Comp Soc, pp. 279-288, Proceedings of the 1998 39th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, USA, 11/8/98.

}

TY - GEN

T1 - Lower bounds for (MOD p - MOD m) circuits

AU - Grolmusz, Vince

AU - Tardos, G.

PY - 1998

Y1 - 1998

N2 - Modular gates are known to be immune for the random restriction techniques of Ajtai, Furst, Saxe, Sipser, Yao and Hadstad. We demonstrate here a random clustering technique which overcomes this difficulty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien, Krause and Pudlak, and others, characterizing symmetric functions computable by small (MODp, ANDt, MODm) circuits. Applying a degree-decreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien, and other related lower bounds for certain (MODp, MODm, AND) circuits. Most of the previous lower bounds on circuits with modular gates used special definitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MODm gates. Our methods are applicable - and our lower bounds are valid - for the most general modular gates as well.

AB - Modular gates are known to be immune for the random restriction techniques of Ajtai, Furst, Saxe, Sipser, Yao and Hadstad. We demonstrate here a random clustering technique which overcomes this difficulty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien, Krause and Pudlak, and others, characterizing symmetric functions computable by small (MODp, ANDt, MODm) circuits. Applying a degree-decreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien, and other related lower bounds for certain (MODp, MODm, AND) circuits. Most of the previous lower bounds on circuits with modular gates used special definitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MODm gates. Our methods are applicable - and our lower bounds are valid - for the most general modular gates as well.

UR - http://www.scopus.com/inward/record.url?scp=0032320248&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032320248&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0032320248

SP - 279

EP - 288

BT - Annual Symposium on Foundations of Computer Science - Proceedings

A2 - Anon, null

PB - IEEE Comp Soc

ER -