A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits
Consider a (MOD q, MOD p) circuit, where the inputs of the bottom MOD p gates are degree- 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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2001-12-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Online Access: | http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/145 |
| _version_ | 1811277841530093568 |
|---|---|
| author | Vince Grolmusz |
| author_facet | Vince Grolmusz |
| author_sort | Vince Grolmusz |
| collection | DOAJ |
| description | Consider a (MOD q, MOD p) circuit, where the inputs of the bottom MOD p gates are degree- 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 q, MOD p) circuit with 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 q, MOD p) circuits computing the n fan-in AND, where the input of each MOD p gate at the bottom is an arbitrary integer valued function of cn variables (c<1) plus an arbitrary linear function of n input variables. |
| first_indexed | 2024-04-13T00:25:07Z |
| format | Article |
| id | doaj.art-e9c5fe90550048088a7d6c1d6ad79b73 |
| institution | Directory Open Access Journal |
| issn | 1462-7264 1365-8050 |
| language | English |
| last_indexed | 2024-04-13T00:25:07Z |
| publishDate | 2001-12-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-e9c5fe90550048088a7d6c1d6ad79b732022-12-22T03:10:39ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1462-72641365-80502001-12-0142A Degree-Decreasing Lemma for (M O D q- M O D p) CircuitsVince GrolmuszConsider a (MOD q, MOD p) circuit, where the inputs of the bottom MOD p gates are degree- 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 q, MOD p) circuit with 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 q, MOD p) circuits computing the n fan-in AND, where the input of each MOD p gate at the bottom is an arbitrary integer valued function of cn variables (c<1) plus an arbitrary linear function of n input variables.http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/145 |
| spellingShingle | Vince Grolmusz A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits Discrete Mathematics & Theoretical Computer Science |
| title | A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits |
| title_full | A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits |
| title_fullStr | A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits |
| title_full_unstemmed | A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits |
| title_short | A Degree-Decreasing Lemma for (M O D q- M O D p) Circuits |
| title_sort | degree decreasing lemma for m o d q m o d p circuits |
| url | http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/145 |
| work_keys_str_mv | AT vincegrolmusz adegreedecreasinglemmaformodqmodpcircuits AT vincegrolmusz degreedecreasinglemmaformodqmodpcircuits |