All reports by Author Igor Carboni Oliveira:

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TR24-145
| 2nd October 2024
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Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor Carboni Oliveira, Dimitrios Tsintsilidas#### Provability of the Circuit Size Hierarchy and Its Consequences

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TR24-083
| 18th April 2024
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Lijie Chen, Jiatu Li, Igor Carboni Oliveira#### On the Unprovability of Circuit Size Bounds in Intuitionistic $S^1_2$

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TR24-060
| 4th April 2024
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Lijie Chen, Jiatu Li, Igor Carboni Oliveira#### Reverse Mathematics of Complexity Lower Bounds

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TR23-076
| 24th May 2023
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Lijie Chen, Zhenjian Lu, Igor Carboni Oliveira, Hanlin Ren, Rahul Santhanam#### Polynomial-Time Pseudodeterministic Construction of Primes

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TR23-069
| 11th May 2023
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Bruno Pasqualotto Cavalar, Igor Carboni Oliveira#### Constant-depth circuits vs. monotone circuits

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TR23-035
| 22nd March 2023
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Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira#### A Duality Between One-Way Functions and Average-Case Symmetry of Information

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TR23-022
| 11th March 2023
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Jiatu Li, Igor Carboni Oliveira#### Unprovability of strong complexity lower bounds in bounded arithmetic

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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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TR21-041
| 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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TR20-021
| 21st February 2020
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Rahul Ilango, Bruno Loff, Igor Carboni Oliveira#### NP-Hardness of Circuit Minimization for Multi-Output Functions

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TR19-168
| 20th November 2019
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Igor Carboni Oliveira, Lijie Chen, Shuichi Hirahara, Ján Pich, Ninad Rajgopal, Rahul Santhanam#### Beyond Natural Proofs: Hardness Magnification and Locality

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TR19-077
| 30th May 2019
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Jan Bydzovsky, Igor Carboni Oliveira, Jan Krajicek#### Consistency of circuit lower bounds with bounded theories

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TR19-073
| 17th May 2019
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Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan#### Parity helps to compute Majority

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TR19-064
| 23rd April 2019
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Igor Carboni Oliveira#### Randomness and Intractability in Kolmogorov Complexity

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TR18-159
| 11th September 2018
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Igor Carboni Oliveira, Rahul Santhanam, Roei Tell#### Expander-Based Cryptography Meets Natural Proofs

Revisions: 2

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TR18-158
| 11th September 2018
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Igor Carboni Oliveira, Ján Pich, Rahul Santhanam#### Hardness magnification near state-of-the-art lower bounds

Revisions: 1

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TR18-139
| 10th August 2018
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Igor Carboni Oliveira, Rahul Santhanam#### Hardness Magnification for Natural Problems

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TR18-122
| 3rd July 2018
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Igor Carboni Oliveira, Rahul Santhanam#### Pseudo-derandomizing learning and approximation

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TR18-030
| 9th February 2018
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Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam#### NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits

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TR17-173
| 6th November 2017
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Igor Carboni Oliveira, Ruiwen Chen, Rahul Santhanam#### An Average-Case Lower Bound against ACC^0

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TR16-197
| 7th December 2016
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Igor Carboni Oliveira, Rahul Santhanam#### Conspiracies between Learning Algorithms, Circuit Lower Bounds and Pseudorandomness

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TR16-196
| 5th December 2016
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Igor Carboni Oliveira, Rahul Santhanam#### Pseudodeterministic Constructions in Subexponential Time

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TR16-071
| 1st May 2016
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Jan Krajicek, Igor Carboni Oliveira#### Unprovability of circuit upper bounds in Cook's theory PV

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TR15-123
| 31st July 2015
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Xi Chen, Igor Carboni Oliveira, Rocco Servedio#### Addition is exponentially harder than counting for shallow monotone circuits

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TR15-008
| 14th January 2015
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Igor Carboni Oliveira, Siyao Guo, Tal Malkin, Alon Rosen#### The Power of Negations in Cryptography

Revisions: 1

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TR14-173
| 13th December 2014
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Igor Carboni Oliveira, Rahul Santhanam#### Majority is incompressible by AC$^0[p]$ circuits

Revisions: 1

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TR14-144
| 30th October 2014
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Eric Blais, Clement Canonne, Igor Carboni Oliveira, Rocco Servedio, Li-Yang Tan#### Learning circuits with few negations

Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor Carboni Oliveira, Dimitrios Tsintsilidas

The Circuit Size Hierarchy CSH$^a_b$ states that if $a > b \geq 1$ then the set of functions on $n$ variables computed by Boolean circuits of size $n^a$ is strictly larger than the set of functions computed by circuits of size $n^b$. This result, which is a cornerstone of circuit ... more >>>

Lijie Chen, Jiatu Li, Igor Carboni Oliveira

We show that there is a constant $k$ such that Buss's intuitionistic theory $\mathbf{IS}^1_2$ does not prove that SAT requires co-nondeterministic circuits of size at least $n^k$. To our knowledge, this is the first unconditional unprovability result in bounded arithmetic in the context of worst-case fixed-polynomial size circuit lower bounds. ... more >>>

Lijie Chen, Jiatu Li, Igor Carboni Oliveira

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are necessary to prove a given theorem. In this work, we systematically explore the reverse mathematics of complexity lower bounds. We explore reversals in the setting of bounded arithmetic, with Cook's theory $\mathbf{PV}_1$ as the base ... more >>>

Lijie Chen, Zhenjian Lu, Igor Carboni Oliveira, Hanlin Ren, Rahul Santhanam

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time.

... more >>>Bruno Pasqualotto Cavalar, Igor Carboni Oliveira

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:

- For every $k \geq 1$, there is a monotone function in ${\rm AC^0}$ (constant-depth poly-size circuits) that requires monotone circuits of depth $\Omega(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai ... more >>>

Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira

Symmetry of Information (SoI) is a fundamental property of Kolmogorov complexity that relates the complexity of a pair of strings and their conditional complexities. Understanding if this property holds in the time-bounded setting is a longstanding open problem. In the nineties, Longpré and Mocas (1993) and Longpré and Watanabe (1995) ... more >>>

Jiatu Li, Igor Carboni Oliveira

While there has been progress in establishing the unprovability of complexity statements in lower fragments of bounded arithmetic, understanding the limits of Jerabek's theory $\textbf{APC}_1$ (2007) and of higher levels of Buss's hierarchy $\textbf{S}^i_2$ (1986) has been a more elusive task. Even in the more restricted setting of Cook's theory ... more >>>

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 >>>

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 >>>

Rahul Ilango, Bruno Loff, Igor Carboni Oliveira

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an ... more >>>

Igor Carboni Oliveira, Lijie Chen, Shuichi Hirahara, Ján Pich, Ninad Rajgopal, Rahul Santhanam

Hardness magnification reduces major complexity separations (such as $EXP \not\subseteq NC^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19, CT19, OPS19, CMMW19, Oli19, CJW19a] have established results of this form. In the most intriguing cases, the required lower bound is ... more >>>

Jan Bydzovsky, Igor Carboni Oliveira, Jan Krajicek

Proving that there are problems in $P^{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere ... more >>>

Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan

We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC$^0[\oplus]$ circuits. Razborov (1987) and Smolensky (1987, 1993) showed that Majority requires depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\Omega(n^{1/2(d-1)})}$. By using a divide-and-conquer approach, it is easy to show that Majority can ... more >>>

Igor Carboni Oliveira

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin's notion of Kolmogorov complexity from 1984. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.

This complexity measure gives rise to a ... more >>>

Igor Carboni Oliveira, Rahul Santhanam, Roei Tell

We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are:

* The security of Goldreich's PRG and OWF ... more >>>

Igor Carboni Oliveira, Ján Pich, Rahul Santhanam

This work continues the development of hardness magnification. The latter proposes a strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful.

We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs ... more >>>

Igor Carboni Oliveira, Rahul Santhanam

We show that for several natural problems of interest, complexity lower bounds that are barely non-trivial imply super-polynomial or even exponential lower bounds in strong computational models. We term this phenomenon "hardness magnification". Our examples of hardness magnification include:

1. Let MCSP$[s]$ be the decision problem whose YES instances are ... more >>>

Igor Carboni Oliveira, Rahul Santhanam

We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser

[GG11]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed

output with high probability. We explore pseudo-determinism in the settings of learning and ap-

proximation. Our goal is to simulate known randomized algorithms in these settings ...
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Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have ... more >>>

Igor Carboni Oliveira, Ruiwen Chen, Rahul Santhanam

In a seminal work, Williams [Wil14] showed that NEXP (non-deterministic exponential time) does not have polynomial-size ACC^0 circuits. Williams' technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known.

We show that there is a language L in NEXP (resp. EXP^NP) ... more >>>

Igor Carboni Oliveira, Rahul Santhanam

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size at most s(n). We show:

Learning Speedups: If C[$n^{O(1)}$] admits a randomized weak learning algorithm under the uniform ... more >>>

Igor Carboni Oliveira, Rahul Santhanam

We study {\it pseudodeterministic constructions}, i.e., randomized algorithms which output the {\it same solution} on most computation paths. We establish unconditionally that there is an infinite sequence $\{p_n\}_{n \in \mathbb{N}}$ of increasing primes and a randomized algorithm $A$ running in expected sub-exponential time such that for each $n$, on input ... more >>>

Jan Krajicek, Igor Carboni Oliveira

We establish unconditionally that for every integer $k \geq 1$ there is a language $L \in P$ such that it is consistent with Cook's theory PV that $L \notin SIZE(n^k)$. Our argument is non-constructive and does not provide an explicit description of this language.

more >>>Xi Chen, Igor Carboni Oliveira, Rocco Servedio

Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes ... more >>>

Igor Carboni Oliveira, Siyao Guo, Tal Malkin, Alon Rosen

The study of monotonicity and negation complexity for Boolean functions has been prevalent in complexity theory as well as in computational learning theory, but little attention has been given to it in the cryptographic context. Recently, Goldreich and Izsak (2012) have initiated a study of whether cryptographic primitives can be ... more >>>

Igor Carboni Oliveira, Rahul Santhanam

We consider $\cal C$-compression games, a hybrid model between computational and communication complexity. A $\cal C$-compression game for a function $f \colon \{0,1\}^n \to \{0,1\}$ is a two-party communication game, where the first party Alice knows the entire input $x$ but is restricted to use strategies computed by $\cal C$-circuits, ... more >>>

Eric Blais, Clement Canonne, Igor Carboni Oliveira, Rocco Servedio, Li-Yang Tan

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and ... more >>>