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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

Funding

  • Limaye, Nutan: Funded by Mathematical Research Impact Centric Support (MATRICS), SERB.
  • Srinivasan, Srikanth: Funded by Mathematical Research Impact Centric Support (MATRICS), SERB. Work partially done while visiting the Simons Institute for the Theory of Computing, Berkeley.

Cite AsGet BibTex

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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