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1. Log-Seed Pseudorandom Generators via Iterated Restrictions

   Doron, Dean ; Hatami, Pooya ; Hoza, William ( July 2020 , Leibniz international proceedings in informatics)

   Saraf, Shubhangi (Ed.)

   There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The "iterated restrictions" approach, pioneered by Ajtai and Wigderson [Ajtai and Wigderson, 1989], has provided PRGs with seed length polylog n or even Õ(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(-Ω̃(log n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of ∏_{i=1}^m (1 + y_i). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [Gopalan and Yehudayoff, 2014] on elementary symmetric polynomials. 

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2. Tight Correlation Bounds for Circuits Between AC0 and TC0

   https://doi.org/10.4230/LIPIcs.CCC.2023.18  

   Kumar, Vinayak M. ( January 2023 , Computational Complexity Conference)

   Ta-Shma, Amnon (Ed.)

   We initiate the study of generalized AC⁰ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight ≥ k (up to negations of the input bits), which we denote GC⁰(k). The gate set of this class includes biased LTFs like the k-OR (outputs 1 iff ≥ k bits are 1) and k-AND (outputs 0 iff ≥ k bits are 0), and thus can be seen as an interpolation between AC⁰ and TC⁰. We establish a tight multi-switching lemma for GC⁰(k) circuits, which bounds the probability that several depth-2 GC⁰(k) circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-d size-s AC⁰ circuits lifts to depth-d size-s^{.99} GC⁰(.01 log s) circuits with no loss in parameters (other than hidden constants). Our result has the following applications: - Size-2^Ω(n^{1/d}) depth-d GC⁰(Ω(n^{1/d})) circuits do not correlate with parity (extending a result of Håstad (SICOMP, 2014)). - Size-n^Ω(log n) GC⁰(Ω(log² n)) circuits with n^{.249} arbitrary threshold gates or n^{.499} arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). - There is a seed length O((log m)^{d-1}log(m/ε)log log(m)) pseudorandom generator against size-m depth-d GC⁰(log m) circuits, matching the AC⁰ lower bound of Håstad up to a log log m factor (extending a result of Lyu (CCC, 2022)). - Size-m GC⁰(log m) circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)). 

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3. New PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs

   https://doi.org/10.4230/LIPIcs.ICALP.2023.39  

   Chen, Lijie ; Lyu, Xin ; Tal, Avishay ; Wu, Hongxun ( January 2023 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Etessami, Kousha ; Feige, Uriel ; Puppis, Gabriele (Ed.)

   We give the first pseudorandom generators with sub-linear seed length for the following variants of read-once branching programs (roBPs): 1) First, we show there is an explicit PRG of seed length O(log²(n/ε)log(n)) fooling unbounded-width unordered permutation branching programs with a single accept state, where n is the length of the program. Previously, [Lee-Pyne-Vadhan RANDOM 2022] gave a PRG with seed length Ω(n) for this class. For the ordered case, [Hoza-Pyne-Vadhan ITCS 2021] gave a PRG with seed length Õ(log n ⋅ log 1/ε). 2) Second, we show there is an explicit PRG fooling unbounded-width unordered regular branching programs with a single accept state with seed length Õ(√{n ⋅ log 1/ε} log 1/ε). Previously, no non-trivial PRG (with seed length less than n) was known for this class (even in the ordered setting). For the ordered case, [Bogdanov-Hoza-Prakriya-Pyne CCC 2022] gave an HSG with seed length Õ(log n ⋅ log 1/ε). 3) Third, we show there is an explicit PRG fooling width w adaptive branching programs with seed length O(log n ⋅ log² (nw/ε)). Here, the branching program can choose an input bit to read depending on its current state, while it is guaranteed that on any input x ∈ {0,1}ⁿ, the branching program reads each input bit exactly once. Previously, no PRG with a non-trivial seed length is known for this class. We remark that there are some functions computable by constant-width adaptive branching programs but not by sub-exponential-width unordered branching programs. In terms of techniques, we indeed show that the Forbes-Kelley PRG (with the right parameters) from [Forbes-Kelley FOCS 2018] already fools all variants of roBPs above. Our proof adds several new ideas to the original analysis of Forbes-Kelly, and we believe it further demonstrates the versatility of the Forbes-Kelley PRG. 

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4. Pseudorandomness from Shrinkage

   https://doi.org/10.1145/3230630  

   Impagliazzo, Russell ; Meka, Raghu ; Zuckerman, David ( April 2019 , Journal of the ACM)

   One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer from a quantitative loss in parameters, and hence do not give nontrivial implications for models where we don’t know super-polynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs which are essentially best possible without in turn improving the lower bounds. More specifically, say that a circuit family has shrinkage exponent Γ if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p Γ + o (1) . Our PRG uses a seed of length s 1/(Γ + 1) + o (1) to fool circuits in the family of size s . By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths: (1) For de Morgan formulas, seed length s 1/3+ o (1) ; (2) For formulas over an arbitrary basis, seed length s 1/2+ o (1) ; (3) For read-once de Morgan formulas, seed length s .234... ; (4) For branching programs of size s , seed length s 1/2+ o (1) . The previous best PRGs known for these classes used seeds of length bigger than n /2 to output n bits, and worked only for size s = O ( n ) [8]. 

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5. Pseudorandomness from Shrinkage

   Impagliazzo, R ; Meka, R ; Zuckerman, D ( April 2019 , Journal of the ACM)

   One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer from a quantitative loss in parameters, and hence do not give nontrivial implications for models where we don't know super-polynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs that are essentially best possible without in turn improving the lower bounds. More specifically, say that a circuit family has shrinkage exponent Gamma if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p^{Gamma + o(1)}. Our PRG uses a seed of length s^{1/(Gamma + 1) + o(1)} to fool circuits in the family of size s. By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths: 1. For de Morgan formulas, seed length s^{1/3+o(1)}; 2. For formulas over an arbitrary basis, seed length s^{1/2+o(1)}; 3. For read-once de Morgan formulas, seed length s^{.234...}; 4. For branching programs of size s, seed length s^{1/2+o(1)}. The previous best PRGs known for these classes used seeds of length bigger than n/2 to output n bits, and worked only when the size s=O(n). 

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