eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-27
78:1
78:15
10.4230/LIPIcs.MFCS.2018.78
article
Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds
Rajgopal, Ninad
1
Santhanam, Rahul
1
Srinivasan, Srikanth
2
Department of Computer Science, University of Oxford, Oxford, United Kingdom
Department of Mathematics, IIT Bombay, Mumbai, India
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol117-mfcs2018/LIPIcs.MFCS.2018.78/LIPIcs.MFCS.2018.78.pdf
circuit satisfiability
circuit lower bounds
polynomial method
derandomization