Abstract
We show that there is a better-than-brute-force 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 \(\varepsilon > 0\) 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 at most \(n^{1+\varepsilon }\) many wires. The algorithm runs in time \(2^{n-n^{\Omega (\varepsilon )}}.\) 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 with linearly many wires).We give two different algorithms for the case of a single PTF. The first uses a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett and Moran (STOC 2018), and the second uses a proof of Hofmeister (COCOON 1996) for converting a degree-1 PTF to a depth-two threshold circuit with small weights. We show that both these ideas fit nicely into a memoization approach that yields the #SAT algorithms.
Highlights
This paper adds to the growing line of work on circuit-analysis algorithms, where we are given as input a Boolean circuit C from a fixed class C computing a function f : {−1, 1}n → {−1, 1},3 and we are required to compute some parameter of the function f
A striking result of this type was proved by Williams [26] who showed that for many circuit classes C, even co-non-deterministic satisfiability algorithms running in better than brute-force time yield lower bounds against C
The method does not extend to k ≥ 3. We extend this result to a powerful model of circuits called k-Polynomial Threshold Function (PTF) circuits, where each gate computes a k-PTF
Summary
This paper adds to the growing line of work on circuit-analysis algorithms, where we are given as input a Boolean circuit C from a fixed class C computing a function f : {−1, 1}n → {−1, 1},3 and we are required to compute some parameter of the function f. An incomparable result was proved by Williams [29] who obtained algorithms for subexponential-sized circuits from the class ACC0 ◦ LTF, which is a subclass of subexponential TC0.9 For the special case of k-PTFs (and generalizations to sparse PTFs over the {0, 1} basis) with small weights, a #SAT algorithm was devised by Sakai et al [22].10. (These are roughly the strongest size lower bounds we know for general TC0 circuits even in the worst case [11].) Using their ideas, they gave the first (zero-error randomized) improvement to brute-force-search for satisfiability algorithms (and even #SAT algorithms) for constant depth TC0 circuits of size at most n1+εd. One of the reasons for this sparsity restriction is that their strategy does not seem to yield a SAT algorithm for a single degree-2 PTF (which is a depth-1 2-PTF circuit of linear size)
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