Computer Science

We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure κ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

The class FORMULA[s]∘G consists of Boolean functions computable by size- s de Morgan formulas whose leaves are any Boolean functions from a class G . We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[ n 1.99 ]∘G , for classes G of functions with low communication complexity. Let R (k) (G) be the maximum k -party NOF randomized communication complexity of G . We show: (1) The Generalized Inner Product function GI P k n cannot be computed in FORMULA[s]∘G on more than 1/2+ε fraction of inputs for s=o ⎛ ⎝ ⎜ ⎜ n 2 (k⋅ 4 k ⋅ R (k) (G)⋅log(n/ε)⋅log(1/ε)) 2 ⎞ ⎠ ⎟ ⎟ . As a corollary, we get an average-case lower bound for GI P k n against FORMULA[ n 1.99 ]∘PT F k−1 . (2) There is a PRG of seed length n/2+O( s √ ⋅ R (2) (G)⋅log(s/ε)⋅log(1/ε)) that ε -fools FORMULA[s]∘G . For FORMULA[s]∘LTF , we get the better seed length O( n 1/2 ⋅ s 1/4 ⋅log(n)⋅log(n/ε)) . This gives the first non-trivial PRG (with seed length o(n) ) for intersections of n half-spaces in the regime where ε≤1/n . (3) There is a randomized 2 n−t -time # SAT algorithm for FORMULA[s]∘G , where t=Ω ( n s √ ⋅ log 2 (s)⋅ R (2) (G) ) 1/2 . In particular, this implies a nontrivial #SAT algorithm for FORMULA[ n 1.99 ]∘LTF . (4) The Minimum Circuit Size Problem is not in FORMULA[ n 1.99 ]∘XOR . On the algorithmic side, we show that FORMULA[ n 1.99 ]∘XOR can be PAC-learned in time 2 O(n/logn) .

We prove that for every decision tree, the absolute values of the Fourier coefficients of given order ℓ≥1 sum to at most c ℓ ( d ℓ )(1+logn ) ℓ−1 − − − − − − − − − − − − − √ , where n is the number of variables, d is the tree depth, and c>0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with ℓ, becoming trivial already at ℓ= d − − √ . As an application, we obtain, for every integer k≥1, a partial Boolean function on n bits that has bounded-error quantum query complexity at most ⌈k/2⌉ and randomized query complexity Ω ~ ( n 1−1/k ). This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015). Prior to our work, the best known separation was polynomially weaker: O(1) versus Ω( n 2/3−ϵ ) for any ϵ>0 (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of O(logn) versus Ω( n 1−ϵ ) for bounded-error quantum versus randomized communication complexity, for any ϵ>0. The best previous separation was polynomially weaker: O(logn) versus Ω( n 2/3−ϵ ) (implicit in Tal, FOCS 2020).

A Boolean function f:{0,1 } n →{0,1} is k -linear if it returns the sum (over the binary field F 2 ) of k coordinates of the input. In this paper, we study property testing of the classes k -Linear, the class of all k -linear functions, and k -Linear ∗ , the class ∪ k j=0 j -Linear. We give a non-adaptive distribution-free two-sided ϵ -tester for k -Linear that makes O(klogk+ 1 ϵ ) queries. This matches the lower bound known from the literature. We then give a non-adaptive distribution-free one-sided ϵ -tester for k -Linear ∗ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided ϵ -tester for k -Linear must make at least Ω ~ (k)logn+Ω(1/ϵ) queries. The latter bound, almost matches the upper bound O(klogn+1/ϵ) known from the literature. We then show that any adaptive uniform-distribution one-sided ϵ -tester for k -Linear must make at least Ω ~ ( k − − √ )logn+Ω(1/ϵ) queries.

The GKS game was formulated by Justin Gilmer, Michal Koucky, and Michael Saks in their research of the sensitivity conjecture. Mario Szegedy invented a protocol for the game with the cost of O( n 0.4732 ) . Then a protocol with the cost of O( n 0.4696 ) was obtained by DeVon Ingram who used a bipartite matching. We propose a slight improvement of Ingram's method and design a protocol with cost of O( n 0.4693 ) .

We show that simple modifications to van der Hoeven's forward and inverse truncated Fourier transforms allow the algorithms to be performed in-place, and with only a linear overhead in complexity.

A Boolean constraint satisfaction problem (CSP), Max-CSP (f) , is a maximization problem specified by a constraint f:{??,1 } k ?�{0,1} . An instance of the problem consists of m constraint applications on n Boolean variables, where each constraint application applies the constraint to k literals chosen from the n variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the n ~variables. In the (γ,β) -approximation version of the problem for parameters γ?�β�?[0,1] , the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work we consider the approximability of Max-CSP (f) in the (dynamic) streaming setting, where constraints are inserted (and may also be deleted in the dynamic setting) one at a time. We completely characterize the approximability of all Boolean CSPs in the dynamic streaming setting. Specifically, given f , γ and β we show that either (1) the (γ,β) -approximation version of Max-CSP (f) has a probabilistic dynamic streaming algorithm using O(logn) space, or (2) for every ε>0 the (γ?��?β+ε) -approximation version of Max-CSP (f) requires Ω( n ??????) space for probabilistic dynamic streaming algorithms. We also extend previously known results in the insertion-only setting to a wide variety of cases, and in particular the case of k=2 where we get a dichotomy and the case when the satisfying assignments of f support a distribution on {??,1 } k with uniform marginals.

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP( C ,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B , to find the number of homomorphisms from A to B . Flum and Grohe showed that #CSP( C ,-) is solvable in polynomial time if C has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT ≠ #W[1], there are no other cases of #CSP( C ,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04]. We show that, assuming FPT ≠ W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP( C ,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP( C ,-). In particular, our condition generalises the case when C is closed under taking minors.

The ϵ -approximate degree de g ϵ (f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to error ϵ . The approximate degree of f is at least k iff there exists 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−ϵ . Our contributions are: We give a simple new construction of a dual polynomial for the AND function, certifying that de g ϵ (f)≥Ω( nlog1/ϵ − − − − − − − √ ) . 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 read-once DNF is Ω( n − − √ ) . 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 error at most K 3/2 ⋅exp(−Ω( k 2 /K)) for all k<K<n/64 . This 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(−Ω(de g 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 analyses draw new connections between approximate degree and concentration phenomena. As a corollary, we show that for any d<n/64 , any degree d polynomial approximating a symmetric function f to error 1/3 must have ℓ 1 -norm at least K −3/2 exp(Ω(de g 1/3 (f ) 2 /d)) , which we also show to be tight for any d>de g 1/3 (f) . These upper and lower bounds were also previously only known in the case f= AND.

The theorem states that: Every Boolean function can be ϵ−approximated by a Disjunctive Normal Form (DNF) of size O ϵ ( 2 n /logn) . This paper will demonstrate this theorem in detail by showing how this theorem is generated and proving its correctness. We will also dive into some specific Boolean functions and explore how these Boolean functions can be approximated by a DNF whose size is within the universal bound O ϵ ( 2 n /logn) . The Boolean functions we interested in are: Parity Function: the parity function can be ϵ−approximated by a DNF of width (1−2ϵ)n and size 2 (1−2ϵ)n . Furthermore, we will explore the lower bounds on the DNF's size and width. Majority Function: for every constant 1/2<ϵ<1 , there is a DNF of size 2 O( n √ ) that can ϵ−approximated the Majority Function on n bits. Monotone Functions: every monotone function f can be ϵ−approximated by a DNF g of size 2 n−Ωϵ(n) satisfying g(x)≤f(x) for all x.