# Discrete Mathematics & Theoretical Computer Science

## Volume 4 n° 2 (2001), pp. 247-254

author: | Vince Grolmusz |
---|---|

title: | A Degree-Decreasing Lemma for (MOD Circuits_{q}-MOD_{p}) |

keywords: | Circuit complexity, modular circuits, composite modulus, Constant Degree Hypothesis |

abstract: | Consider a (MOD
circuit, where the inputs of the bottom
_{q},MOD_{p})MOD gates are
degree-_{p}d polynomials with integer coefficients of the
input variables (p, q are different
primes). Using our main tool --- the Degree Decreasing Lemma --- we
show that this circuit can be converted to a
(MOD
circuit with _{q},MOD_{p})linear polynomials on the input-level with the
price of increasing the size of the circuit. This result has numerous
consequences: for the Constant Degree Hypothesis of Barrington,
Straubing and Thérien, and generalizing the lower bound results of Yan
and Parberry, Krause and Waack, and Krause and Pudlák. Perhaps
the most important application is an exponential lower bound for the
size of
(MOD
circuits computing the _{q},MOD_{p})n fan-in AND, where the input of
each MOD gate at the bottom is
an _{p}arbitrary integer valued function of cn
variables (c<1) plus an arbitrary linear function of
n input variables.
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reference: | Vince Grolmusz (2001),
A Degree-Decreasing Lemma for (MOD Circuits,
_{q}-MOD_{p})Discrete Mathematics and Theoretical Computer Science 4, pp. 247-254 |

bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |

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