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A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates

Authors Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, Srikanth Srinivasan



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Author Details

Swapnam Bajpai
  • Indian Institute of Technology, Bombay, Mumbai, India
Vaibhav Krishan
  • Indian Institute of Technology, Bombay, Mumbai, India
Deepanshu Kush
  • Indian Institute of Technology, Bombay, Mumbai, India
Nutan Limaye
  • Indian Institute of Technology, Bombay, Mumbai, India
Srikanth Srinivasan
  • Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India

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Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan. A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 8:1-8:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.8

Abstract

We show that there is a zero-error randomized algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to C in significantly better than brute-force time. Formally, for any constants d,k, there is an epsilon > 0 such that the zero-error randomized algorithm counts the number of satisfying assignments to a given depth-d circuit C made up of k-PTF gates such that C has size at most n^{1+epsilon}. The algorithm runs in time 2^{n-n^{Omega(epsilon)}}. Before our result, no algorithm for beating brute-force search was known for counting the number of satisfying assignments even for a single degree-k PTF (which is a depth-1 circuit of linear size). The main new tool is the use of a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett, Moran and Zhang (FOCS 2017). We show that their ideas fit nicely into a memoization approach that yields the #SAT algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • SAT
  • Polynomial Threshold Functions
  • Constant-depth Boolean Circuits
  • Linear Decision Trees
  • Zero-error randomized algorithms

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