All reports by Author Zhenjian Lu:

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TR22-056
| 18th April 2022
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Zhenjian Lu, Igor Carboni Oliveira, Marius Zimand#### Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

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TR21-171
| 2nd December 2021
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Bruno Pasqualotto Cavalar, Zhenjian Lu#### Algorithms and Lower Bounds for Comparator Circuits from Shrinkage

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TR21-041
| 15th March 2021
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Zhenjian Lu, Igor Carboni Oliveira#### An Efficient Coding Theorem via Probabilistic Representations and its Applications

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TR21-040
| 15th March 2021
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Lijie Chen, Zhenjian Lu, Xin Lyu, Igor Carboni Oliveira#### Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC1

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TR21-039
| 15th March 2021
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Zhenjian Lu, Igor Carboni Oliveira, Rahul Santhanam#### Pseudodeterministic Algorithms and the Structure of Probabilistic Time

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TR20-018
| 18th February 2020
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Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor Oliveira#### Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

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TR19-022
| 23rd February 2019
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Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis#### Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

Revisions: 1

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TR18-115
| 11th June 2018
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Valentine Kabanets, Zhenjian Lu#### Satisfiability and Derandomization for Small Polynomial Threshold Circuits

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TR18-012
| 20th January 2018
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Valentine Kabanets, Zhenjian Lu#### Nisan-Wigderson Pseudorandom Generators for Circuits with Polynomial Threshold Gates

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TR17-026
| 17th February 2017
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Valentine Kabanets, Daniel Kane, Zhenjian Lu#### A Polynomial Restriction Lemma with Applications

Zhenjian Lu, Igor Carboni Oliveira, Marius Zimand

The classical coding theorem in Kolmogorov complexity states that if an $n$-bit string $x$ is sampled with probability $\delta$ by an algorithm with prefix-free domain then K$(x) \leq \log(1/\delta) + O(1)$. In a recent work, Lu and Oliveira [LO21] established an unconditional time-bounded version of this result, by showing that ... more >>>

Bruno Pasqualotto Cavalar, Zhenjian Lu

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first superlinear lower bound against comparator circuits was proved only recently by Gál and Robere (ITCS 2020), who established ... more >>>

Zhenjian Lu, Igor Carboni Oliveira

A probabilistic representation of a string $x \in \{0,1\}^n$ is given by the code of a randomized algorithm that outputs $x$ with high probability [Oliveira, ICALP 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ... more >>>

Lijie Chen, Zhenjian Lu, Xin Lyu, Igor Carboni Oliveira

We develop a general framework that characterizes strong average-case lower bounds against circuit classes $\mathcal{C}$ contained in $\mathrm{NC}^1$, such as $\mathrm{AC}^0[\oplus]$ and $\mathrm{ACC}^0$. We apply this framework to show:

- Generic seed reduction: Pseudorandom generators (PRGs) against $\mathcal{C}$ of seed length $\leq n -1$ and error $\varepsilon(n) = n^{-\omega(1)}$ can ... more >>>

Zhenjian Lu, Igor Carboni Oliveira, Rahul Santhanam

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.

1. A new pseudorandom generator and its consequences: We build on techniques developed to ... more >>>

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor Oliveira

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be ... more >>>

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function $f$ can be computed by a Boolean circuit of size at most $\theta$, for a given parameter $\theta$. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a ... more >>>

Valentine Kabanets, Zhenjian Lu

A polynomial threshold function (PTF) is defined as the sign of a polynomial $p\colon\bool^n\to\mathbb{R}$. 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 ... more >>>

Valentine Kabanets, Zhenjian Lu

We show how the classical Nisan-Wigderson (NW) generator [Nisan & Wigderson, 1994] yields a nontrivial pseudorandom generator (PRG) for circuits with sublinearly many polynomial threshold function (PTF) gates. For the special case of a single PTF of degree $d$ on $n$ inputs, our PRG for error $\epsilon$ has the seed ... more >>>

Valentine Kabanets, Daniel Kane, Zhenjian Lu

A polynomial threshold function (PTF) of degree $d$ is a boolean function of the form $f=\mathrm{sgn}(p)$, where $p$ is a degree-$d$ polynomial, and $\mathrm{sgn}$ is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is ... more >>>