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.
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.
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.
A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth. - Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only. - Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity. - Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon).
We give a deterministic algorithm for approximately counting satisfying assignments of a degree-d polynomial threshold function (PTF). Given a degree-d input polynomial p(x) over Rn and a parameter ε > 0, our algorithm approximates Pr [EQUATION] to within an additive ±ε in time Od,ε(1) · poly(nd). (Since it is NP-hard to determine whether the above probability is nonzero, any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms.) Note that the running time of our algorithm (as a function of nd, the number of coefficients of a degree-d PTF) is a fixed polynomial. The fastest previous algorithm for this problem [Kan12b], based on constructions of unconditional pseudorandom generators for degree-d PTFs, runs in time [EQUATION] for all c > 0.
Let g : {-1, 1}k→ {-1, 1} be any Boolean function and q1, . . . , qkbe any degree-2 polynomials over {-1, 1}n. We give a deterministic algorithm which, given as input explicit descriptions of g, q1, . . ., qkand an accuracy parameter ϵ > 0, approximates Prx~{-1, 1}n[g(sign(q1(x)), . . . , sign(qk(x))) = 1] to within an additive ±ϵ. For any constant ϵ > 0 and k ≥ 1 the running time of our algorithm is a fixed polynomial in n (in fact this is true even for some not-too-small ϵ = 0n(1) and not-too-large k = ωn(1)). This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits. Our algorithm extends a recent result [1] which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function sign(q(x)), corresponding to the k = 1 case of our result. Note that even in the k = 1 case it is NP-hard to determine whether Prx~{-1, 1}n[sign(q(x)) = 1] is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms. Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the k = 1 case in [1]. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for k-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution, i.e., an algorithm for estimating Prx~N(0, 1)n[g(sign(q1(x)), . . . , sign(qk(x))) = 1]. Finally, a third new ingredient is a “regularity lemma” for k-tuples of degree-d polynomial threshold functions. This generalizes both the regularity lemmas of [2], [3] (which apply to a single degree-d polynomial threshold function) and the regularity lemma of Gopalan et al [4] (which applies to a k-tuples of linear threshold functions, i.e., the case d = 1). Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.
The threshold degree of a Boolean function f is the minimum degree of a real polynomial p that represents f in sign: f(x) = sgn p(x). Introduced in the seminal work of Minsky and Papert (1969), this notion is central to some of the strongest algorithmic and complexity-theoretic results for constant-depth circuits. One problem that has remained open for several decades, with applications to computational learning and communication complexity, is to determine the maximum threshold degree of a polynomial-size constant-depth circuit in n variables. The best lower bound prior to our work was Ω(n(d-1)/(2d-1)) for circuits of depth d. We obtain a polynomial improvement for every depth d, with a lower bound of Ω(n3/7) for depth 3 and Ω(√n) for depth d ≥4. The proof contributes a novel approximation-theoretic technique of independent interest, which exploits asymmetry in circuits to prove their hardness for polynomials.