arXiv subjects

Igor C. Oliveira

Publications and source records attributed to Igor C. Oliveira.

At least 19 recordsLinked to original sources

arXiv2026

A Theory for Probabilistic Polynomial-Time Reasoning

In this work, we propose a new bounded arithmetic theory, denoted $APX_1$, designed to formalize a broad class of probabilistic arguments commonly used in theoretical computer science. Under plausible assumptions, $APX_1$ is strictly weaker than previously proposed frameworks, such as the theory $APC_1$ introduced in the seminal work of Jerabek (2007). From a computational standpoint, $APX_1$ is closely tied to approximate counting and to the central question in derandomization, the prBPP versus prP problem, whereas $APC_1$ is linked to the dual weak pigeonhole principle and to the existence of Boolean functions with exponential circuit complexity. A key motivation for introducing $APX_1$ is that its weaker axioms expose finer proof-theoretic structure, making it a natural setting for several lines of research, including unprovability of complexity conjectures and reverse mathematics of randomized lower bounds. In particular, the framework we develop for $APX_1$ enables the formulation of precise questions concerning the provability of prBPP=prP in deterministic feasible mathematics. Since the (un)provability of P versus NP in bounded arithmetic has long served as a central theme in the field, we expect this line of investigation to be of particular interest. Our technical contributions include developing a comprehensive foundation for probabilistic reasoning from weaker axioms, formalizing non-trivial results from theoretical computer science in $APX_1$, and establishing a tailored witnessing theorem for its provably total TFNP problems. As a byproduct of our analysis of the minimal proof-theoretic strength required to formalize statements arising in theoretical computer science, we resolve an open problem regarding the provability of $AC^0$ lower bounds in $PV_1$, which was considered in earlier works by Razborov (1995), Krajicek (1995), and Muller and Pich (2020).

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arXiv2025

Polynomial-Time Pseudodeterministic Construction of Primes

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. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an *unconditional* polynomial-time randomized algorithm $B$ such that, for infinitely many values of $n$, $B(1^n)$ outputs a canonical $n$-bit prime $p_n$ with high probability. More generally, we prove that for every dense property $Q$ of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying $Q$. This improves upon a subexponential-time construction of Oliveira and Santhanam. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell, using a variant of the Shaltiel--Umans generator.

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arXiv2025

On the Unprovability of Circuit Size Bounds in Intuitionistic $\mathsf{S}^1_2$

We show that there is a constant $k$ such that Buss's intuitionistic theory $\mathsf{IS}^1_2$ 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. We complement this result by showing that the upper bound $\mathsf{NP} \subseteq \mathsf{coNSIZE}[n^k]$ is unprovable in $\mathsf{IS}^1_2$. In order to establish our main result, we obtain new unconditional lower bounds against refuters that might be of independent interest. In particular, we show that there is no efficient refuter for the lower bound $\mathsf{NP} \nsubseteq \mathsf{i.o.}\text{-}\mathsf{coNP}/\mathsf{poly}$, addressing in part a question raised by Atserias (2006).

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arXiv2025

Meta-Mathematics of Computational Complexity Theory

We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity bounds within theories of bounded arithmetic. This includes the techniques employed and related open problems, such as the (non)existence of a feasible proof that P = NP.

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arXiv2025

Boolean Circuit Complexity and Two-Dimensional Cover Problems

We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the graph complexity framework of Pudlák, Rödl, and Savický (1988). For convenience, we formalize these ideas in the more general setting of "discrete complexity", i.e., the natural set-theoretic formulation of circuit complexity, variants of communication complexity, graph complexity, and other measures. We show that random graphs have linear graph cover complexity, and that explicit super-logarithmic graph cover complexity lower bounds would have significant consequences in circuit complexity. We then use discrete complexity, the fusion method, and a result of Karchmer and Wigderson (1993) to introduce nondeterministic graph complexity. This allows us to establish a connection between graph complexity and nondeterministic circuit complexity. Finally, complementing these results, we describe an exact characterization of the power of the fusion method in discrete complexity. This is obtained via an adaptation of a result of Nakayama and Maruoka (1995) that connects the fusion method to the complexity of "cyclic" Boolean circuits, which generalize the computation of a circuit by allowing cycles in its specification.

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arXiv2023

Constant-depth circuits vs. monotone circuits

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 ${\sf AC^0}$ that requires monotone circuits of depth $Ω(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai and Gurevich (1987). In addition, our separation holds for a monotone graph property, which was unknown even in the context of ${\sf AC^0}$ versus ${\sf mAC^0}$. - For every $k \geq 1$, there is a monotone function in ${\sf AC^0}[\oplus]$ that requires monotone circuits of size $\exp(Ω(\log^k n))$. This makes progress towards a question posed by Grigni and Sipser (1992). These results show that constant-depth circuits can be more efficient than monotone circuits when computing monotone functions. In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an ${\sf AC^0}$ circuit of size $s$ and depth $d$ can be computed by a monotone circuit of size $2^{n - n/O(\log s)^{d-1}}$. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in ${\sf AC^0}$ admits a polynomial size monotone circuit, then ${\sf NC^2}$ is not contained in ${\sf NC^1}$ . Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems established by Göös et al. (2019) via lifting techniques. Adapting results of Schaefer (1978) and Allender et al. (2009), we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.

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arXiv2022

Theory and Applications of Probabilistic Kolmogorov Complexity

Diverse applications of Kolmogorov complexity to learning [CIKK16], circuit complexity [OPS19], cryptography [LP20], average-case complexity [Hir21], and proof search [Kra22] have been discovered in recent years. Since the running time of algorithms is a key resource in these fields, it is crucial in the corresponding arguments to consider time-bounded variants of Kolmogorov complexity. While fruitful interactions between time-bounded Kolmogorov complexity and different areas of theoretical computer science have been known for quite a while (e.g., [Sip83, Ko91, ABK+06, AF09], to name a few), the aforementioned results have led to a renewed interest in this topic. The theory of Kolmogorov complexity is well understood, but many useful results and properties of Kolmogorov complexity are not known to hold in time-bounded settings. This creates technical difficulties or leads to conditional results when applying methods from time-bounded Kolmogorov complexity to algorithms and complexity theory. Perhaps even more importantly, in many cases it is necessary to consider randomised algorithms. Since random strings have high complexity, the classical theory of time-bounded Kolmogorov complexity might be inappropriate in such contexts. To mitigate these issues and develop a theory of time-bounded Kolmogorov complexity that survives in the setting of randomised computations, some recent papers [Oli19, LO21, LOS21, GKLO22, LOZ22] have explored probabilistic notions of time-bounded Kolmogorov complexity, such as $\mathsf{rKt}$ complexity [Oli19], $\mathsf{rK}^t$ complexity [LOS21], and $\mathsf{pK}^t$ complexity [GKLO22]. These measures consider different ways of encoding an object via a probabilistic representation. In this survey, we provide an introduction to probabilistic time-bounded Kolmogorov complexity and its applications, highlighting many open problems and research directions.

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arXiv2022

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

The classical coding theorem in Kolmogorov complexity states that if an $n$-bit string $x$ is sampled with probability $δ$ by an algorithm with prefix-free domain then K$(x) \leq \log(1/δ) + O(1)$. In a recent work, Lu and Oliveira [LO21] established an unconditional time-bounded version of this result, by showing that if $x$ can be efficiently sampled with probability $δ$ then rKt$(x) = O(\log(1/δ)) + O(\log n)$, where rKt denotes the randomized analogue of Levin's Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a $O(\log(1/δ))$ bound instead of the information-theoretic optimal $\log(1/δ)$. We show a coding theorem for rKt with a factor of $2$. As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given $x$, the code of the sampler, and $δ$, it outputs, with probability $\ge 0.99$, a probabilistic representation of $x$ that certifies this rKt complexity bound. Assuming the security of cryptographic pseudorandom generators, we show that no efficient coding theorem can achieve a bound of the form rKt$(x) \leq (2 - o(1)) \cdot \log(1/δ) +$ poly$(\log n)$. Under a weaker assumption, we exhibit a gap between efficient coding theorems and existential coding theorems with near-optimal parameters. We consider pK$^t$ complexity [GKLO22], a variant of rKt where the randomness is public and the time bound is fixed. We observe the existence of an optimal coding theorem for pK$^t$, and employ this result to establish an unconditional version of a theorem of Antunes and Fortnow [AF09] which characterizes the worst-case running times of languages that are in average polynomial-time over all P-samplable distributions.

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arXiv2021

Quantum learning algorithms imply circuit lower bounds

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - γ$ by a time $T$ quantum algorithm. We prove that if $γ^2 \cdot T \ll 2^n/n$, then $\mathsf{BQE} \nsubseteq \mathfrak{C}$, where $\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$ is an exponential-time analogue of $\mathsf{BQP}$. This result is optimal in both $γ$ and $T$, since it is not hard to learn any class $\mathfrak{C}$ of functions in (classical) time $T = 2^n$ (with no error), or in quantum time $T = \mathsf{poly}(n)$ with error at most $1/2 - Ω(2^{-n/2})$ via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.

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arXiv2021

Pseudodeterministic Algorithms and the Structure of Probabilistic Time

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $\mathsf{BPTIME}$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows. 1. We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam, FOCS 2004) to construct the first unconditional pseudorandom generator of polynomial stretch computable in pseudodeterministic polynomial time (with one bit of advice) that is secure infinitely often against polynomial-time computations. As an application of this construction, we obtain new results about the complexity of generating and representing prime numbers. 2. Oliveira and Santhanam (STOC 2017) established unconditionally that there is a pseudodeterministic algorithm for the Circuit Acceptance Probability Problem ($\mathsf{CAPP}$) that runs in sub-exponential time and is correct with high probability over any samplable distribution on circuits on infinitely many input lengths. We show that improving this running time or obtaining a result that holds for every large input length would imply new time hierarchy theorems for probabilistic time. In addition, we prove that a worst-case polynomial-time pseudodeterministic algorithm for $\mathsf{CAPP}$ would imply that $\mathsf{BPP}$ has complete problems. 3. We establish an equivalence between pseudodeterministic construction of strings of large $\mathsf{rKt}$ complexity (Oliveira, ICALP 2019) and the existence of strong hierarchy theorems for probabilistic time. More generally, these results suggest new approaches for designing pseudodeterministic algorithms for search problems and for unveiling the structure of probabilistic time.

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arXiv2020

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ 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 circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with $\mathsf{PV}$.

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arXiv2019

On monotone circuits with local oracles and clique lower bounds

We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $μ\in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $μ$. 2. For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $μ\leq 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{Θ(\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained by Alon and Boppana (1987).

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arXiv2019

Beyond Natural Proofs: Hardness Magnification and Locality

Hardness magnification reduces major complexity separations (such as $\mathsf{\mathsf{EXP}} \nsubseteq \mathsf{NC}^1$) 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 known for problems that appear to be significantly easier than $Q$, while $Q$ itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [RR97]? Can we adapt known lower bound techniques to establish the desired lower bound for $Q$?

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arXiv2017

Erdős-Ko-Rado for random hypergraphs: asymptotics and stability

We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is asymptotically best possible for $k=Θ(n)$. For this range of $k$ and $p$, we are able to show stability as well. A different behavior occurs when $k = o(n)$. In this case, the lower bound on $p$ is almost optimal. Further, for the small interval $D^{-1}\ll p \leq (n/k)^{1-\varepsilon}D^{-1}$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $Θ(\ln (pD)N D^{-1})$, provided that $k \gg \sqrt{n \ln n}$. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}^k(n,p)$, for essentially all values of $p$ and $k$.

math.CO↗

arXiv2017

Unprovability of circuit upper bounds in Cook's theory PV

We establish unconditionally that for every integer $k \geq 1$ there is a language $L \in \mbox{P}$ 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.

math.LO↗

arXiv2016

Pseudodeterministic Constructions in Subexponential Time

We study pseudodeterministic constructions, i.e., randomized algorithms which output the 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 $1^{|p_n|}$, $A$ outputs $p_n$ with probability $1$. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often. This result follows from a much more general theorem about pseudodeterministic constructions. A property $Q \subseteq \{0,1\}^{*}$ is $γ$-dense if for large enough $n$, $|Q \cap \{0,1\}^n| \geq γ2^n$. We show that for each $c > 0$ at least one of the following holds: (1) There is a pseudodeterministic polynomial time construction of a family $\{H_n\}$ of sets, $H_n \subseteq \{0,1\}^n$, such that for each $(1/n^c)$-dense property $Q \in \mathsf{DTIME}(n^c)$ and every large enough $n$, $H_n \cap Q \neq \emptyset$; or (2) There is a deterministic sub-exponential time construction of a family $\{H'_n\}$ of sets, $H'_n \subseteq \{0,1\}^n$, such that for each $(1/n^c)$-dense property $Q \in \mathsf{DTIME}(n^c)$ and for infinitely many values of $n$, $H'_n \cap Q \neq \emptyset$. We provide further algorithmic applications that might be of independent interest. Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.

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arXiv2016

Conspiracies between Learning Algorithms, Circuit Lower Bounds and Pseudorandomness

We prove several results giving new and stronger connections between learning, circuit lower bounds and pseudorandomness. Among other results, we show a generic learning speedup lemma, equivalences between various learning models in the exponential time and subexponential time regimes, a dichotomy between learning and pseudorandomness, consequences of non-trivial learning for circuit lower bounds, Karp-Lipton theorems for probabilistic exponential time, and NC$^1$-hardness for the Minimum Circuit Size Problem.

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arXiv2015

Near-optimal small-depth lower bounds for small distance connectivity

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{Ω(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were $n^{k^{\exp(-O(d))}}$ (due to Beame, Impagliazzo, and Pitassi [BIP98]) and $n^{Ω((\log k)/d)}$ (following from a recent formula size lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal, since a simple construction gives depth-$d$ circuits of size $n^{O(k^{2/d})}$ for this problem (and strengthening our bound even to $n^{k^{Ω(1/d)}}$ would require proving that undirected connectivity is not in $\mathsf{NC^1}.$) Our proof is by reduction to a new lower bound on the size of small-depth circuits computing a skewed variant of the "Sipser functions" that have played an important role in classical circuit lower bounds [Sip83, Yao85, Hås86]. A key ingredient in our proof of the required lower bound for these Sipser-like functions is the use of \emph{random projections}, an extension of random restrictions which were recently employed in [RST15]. Random projections allow us to obtain sharper quantitative bounds while employing simpler arguments, both conceptually and technically, than in the previous works [Ajt89, BPU92, BIP98, Ros14].

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