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1. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/s00224-022-10106-8  

   Vyas, Nikhil ; Williams, R. Ryan ( November 2022 , Theory of Computing Systems)

   We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$$C$, by showing that$\mathcal { C}$$C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$$C$. Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$${\sum }_i{x}_i$. We show that for every sparsef, and for all “typical”$\mathcal { C}$$C$, faster#SAT algorithms for$\mathcal { C}$$C$circuits imply lower bounds against the circuit class$f \circ \mathcal { C}$$f\circ C$, which may bestrongerthan$\mathcal { C}$$C$itself. In particular:

   #SAT algorithms forn^k-size$\mathcal { C}$$C$-circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   #SAT algorithms for$2^{n^{{\varepsilon }}}$$2^{n^{\epsilon }}$-size$\mathcal { C}$$C$-circuits running in$2^{n-n^{{\varepsilon }}}$$2^{n-n^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

    

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2. Proof Complexity Lower Bounds from Algebraic Circuit Complexity

   https://doi.org/10.4086/toc.2021.v017a010  

   Forbes, Michael A. ; Shpilka, Amir ; Tzameret, Iddo ; Wigderson, Avi ( January 2021 , Theory of Computing)

   Raz, Ran (Ed.)

   We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are circuits from restricted boolean circuit classes. Essentially all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). Our main contributions are two general methods of converting certain algebraic lower bounds into proof complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean circuit lower bounds are turned into related proof-complexity ones, especially the "feasible interpolation" technique. We establish algebraic lower bounds of these forms for several explicit polynomials, against a variety of classes, and infer the relevant proof complexity bounds. These yield separations between IPS subsystems, which we complement by simulations to create a partial structure theory for IPS systems. Our first method is a functional lower bound, a notion of Grigoriev and Razborov, which is a function f' from n-bit strings to a field, such that any polynomial f agreeing with f' on the boolean cube requires large algebraic circuit complexity. We develop functional lower bounds for a variety of circuit classes (sparse polynomials, depth-3 powering formulas, read-once algebraic branching programs and multilinear formulas) where f'(x) equals 1/p(x) for a constant-degree polynomial p depending on the relevant circuit class. We believe these lower bounds are of independent interest in algebraic complexity, and show that they also imply lower bounds for the size of the corresponding IPS refutations for proving that the relevant polynomial p is non-zero over the boolean cube. In particular, we show super-polynomial lower bounds for refuting variants of the subset-sum axioms in these IPS subsystems. Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (non-zero) multiples require large algebraic circuit complexity. By extending known techniques, we give lower bounds for multiples for various restricted circuit classes such sparse polynomials, sums of powers of low-degree polynomials, and roABPs. These results are of independent interest, as we argue that lower bounds for multiples is the correct notion for instantiating the algebraic hardness versus randomness paradigm of Kabanets and Impagliazzo. Further, we show how such lower bounds for multiples extend to lower bounds for refutations in the corresponding IPS subsystem. 

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3. Approximating the noise sensitivity of a monotone Boolean function

   Rubinfeld, R. ; Vasilyan, A. ( January 2019 , Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019)

   The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure. The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O'Donnell, 2014]). Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm's query complexity is close to optimal in terms of its dependence on n. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias. 

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4. Approximate Degree, Secret Sharing, and Concentration Phenomena

   https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.71  

   Bogdanov, Andrej ; Mande, Nikhil S. ; Thaler, Justin ; Williamson, Christopher ( September 2019 , Leibniz international proceedings in informatics)

   The epsilon-approximate degree, deg_epsilon(f), of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are: - We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution. - We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1-norm at least K^{-3/2} * exp ({Omega (deg_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND. 

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5. Approximating the Distance to Monotonicity of Boolean Functions

   https://doi.org/10.1137/1.9781611975994.123  

   Pallavoor, Ramesh Krishnan ; Raskhodnikova, Sofya ; Waingarten, Erik ( January 2020 , Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020)

   We design a nonadaptive algorithm that, given a Boolean function f: {0, 1}^n → {0, 1} which is α-far from monotone, makes poly(n, 1/α) queries and returns an estimate that, with high probability, is an O-tilde(\sqrt{n})-approximation to the distance of f to monotonicity. Furthermore, we show that for any constant k > 0, approximating the distance to monotonicity up to n^(1/2−k)-factor requires 2^{n^k} nonadaptive queries, thereby ruling out a poly(n, 1/α)-query nonadaptive algorithm for such approximations. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. Approximating the distance to a property is closely related to tolerantly testing that property. Our lower bound stands in contrast to standard (non-tolerant) testing of monotonicity that can be done nonadaptively with O-tilde(n/ε^2) queries. We obtain our lower bound by proving an analogous bound for erasure-resilient testers. An α-erasure-resilient tester for a desired property gets oracle access to a function that has at most an α fraction of values erased. The tester has to accept (with probability at least 2/3) if the erasures can be filled in to ensure that the resulting function has the property and to reject (with probability at least 2/3) if every completion of erasures results in a function that is ε-far from having the property. Our method yields the same lower bounds for unateness and being a k-junta. These lower bounds improve exponentially on the existing lower bounds for these properties. 

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