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.
@InProceedings{bajpai_et_al:LIPIcs.ITCS.2019.8, author = {Bajpai, Swapnam and Krishan, Vaibhav and Kush, Deepanshu and Limaye, Nutan and Srinivasan, Srikanth}, title = {{A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.8}, URN = {urn:nbn:de:0030-drops-101010}, doi = {10.4230/LIPIcs.ITCS.2019.8}, annote = {Keywords: SAT, Polynomial Threshold Functions, Constant-depth Boolean Circuits, Linear Decision Trees, Zero-error randomized algorithms} }
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