We show that there is a 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 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 even for a single degree-2 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.

This is a joint work with Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, and Srikanth Srinivasan.