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DOI: 10.4230/LIPIcs.MFCS.2018.78
URN: urn:nbn:de:0030-drops-96607
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9660/
Rajgopal, Ninad ; Santhanam, Rahul ; Srinivasan, Srikanth
Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds
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Abstract
We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p.
BibTeX - Entry
@InProceedings{rajgopal_et_al:LIPIcs:2018:9660,
author = {Ninad Rajgopal and Rahul Santhanam and Srikanth Srinivasan},
title = {{Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds}},
booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
pages = {78:1--78:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-086-6},
ISSN = {1868-8969},
year = {2018},
volume = {117},
editor = {Igor Potapov and Paul Spirakis and James Worrell},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9660},
URN = {urn:nbn:de:0030-drops-96607},
doi = {10.4230/LIPIcs.MFCS.2018.78},
annote = {Keywords: circuit satisfiability, circuit lower bounds, polynomial method, derandomization}
}
| Keywords: | circuit satisfiability, circuit lower bounds, polynomial method, derandomization | |
| Seminar: | 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) | |
| Issue date: | 2018 | |
| Date of publication: | 27.08.2018 |
