Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds

Authors Ninad Rajgopal, Rahul Santhanam, Srikanth Srinivasan

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Ninad Rajgopal
  • Department of Computer Science, University of Oxford, Oxford, United Kingdom
Rahul Santhanam
  • Department of Computer Science, University of Oxford, Oxford, United Kingdom
Srikanth Srinivasan
  • Department of Mathematics, IIT Bombay, Mumbai, India

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Ninad Rajgopal, Rahul Santhanam, and Srikanth Srinivasan. Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • circuit satisfiability
  • circuit lower bounds
  • polynomial method
  • derandomization


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