11 Search Results for "Mertz, Ian"

Document
DOI: 10.4230/LIPIcs.CCC.2024.4
Derandomizing Logspace with a Small Shared Hard Drive

Authors: Edward Pyne

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S²] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S] ⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α ∈ [1,1.5], BPSPACE[S] ⊆ CSPACE[S^α,S^(3-α)]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S], due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S²]. As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
Cite as

Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{pyne:LIPIcs.CCC.2024.4,
  author =	{Pyne, Edward},
  title =	{{Derandomizing Logspace with a Small Shared Hard Drive}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.4},
  URN =		{urn:nbn:de:0030-drops-204006},
  doi =		{10.4230/LIPIcs.CCC.2024.4},
  annote =	{Keywords: Catalytic computation, space-bounded computation, derandomization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.9
Lifting Dichotomies

Authors: Yaroslav Alekseev, Yuval Filmus, and Alexander Smal

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f. In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].
Cite as

Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{alekseev_et_al:LIPIcs.CCC.2024.9,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Smal, Alexander},
  title =	{{Lifting Dichotomies}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.9},
  URN =		{urn:nbn:de:0030-drops-204051},
  doi =		{10.4230/LIPIcs.CCC.2024.9},
  annote =	{Keywords: decision trees, log-rank conjecture, lifting, parity decision trees}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.15
Quantum Automating TC⁰-Frege Is LWE-Hard

Authors: Noel Arteche, Gaia Carenini, and Matthew Gray

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate TC⁰-Frege. This extends the line of results of Krajíček and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and Raz (FOCS, 1997), and Bonet, Domingo, Gavaldà, Maciel, and Pitassi (Computational Complexity, 2004), who showed that Extended Frege, TC⁰-Frege and AC⁰-Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.
Cite as

Noel Arteche, Gaia Carenini, and Matthew Gray. Quantum Automating TC⁰-Frege Is LWE-Hard. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{arteche_et_al:LIPIcs.CCC.2024.15,
  author =	{Arteche, Noel and Carenini, Gaia and Gray, Matthew},
  title =	{{Quantum Automating TC⁰-Frege Is LWE-Hard}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{15:1--15:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.15},
  URN =		{urn:nbn:de:0030-drops-204117},
  doi =		{10.4230/LIPIcs.CCC.2024.15},
  annote =	{Keywords: automatability, post-quantum cryptography, feasible interpolation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.22
Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP

Authors: Theodoros Papamakarios

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We show that for any integer d > 0, depth-d Frege systems are NP-hard to automate. Namely, given a set S of depth-d formulas, it is NP-hard to find a depth-d Frege refutation of S in time polynomial in the size of the shortest such refutation. This extends the result of Atserias and Müller [JACM, 2020] for the non-automatability of resolution - a depth-1 Frege system - to Frege systems of any depth d > 0.
Cite as

Theodoros Papamakarios. Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{papamakarios:LIPIcs.CCC.2024.22,
  author =	{Papamakarios, Theodoros},
  title =	{{Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.22},
  URN =		{urn:nbn:de:0030-drops-204187},
  doi =		{10.4230/LIPIcs.CCC.2024.22},
  annote =	{Keywords: Proof complexity, Automatability, Bounded-depth Frege}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.12
From Proof Complexity to Circuit Complexity via Interactive Protocols

Authors: Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.
Cite as

Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam. From Proof Complexity to Circuit Complexity via Interactive Protocols. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{arteche_et_al:LIPIcs.ICALP.2024.12,
  author =	{Arteche, Noel and Khaniki, Erfan and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{From Proof Complexity to Circuit Complexity via Interactive Protocols}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.12},
  URN =		{urn:nbn:de:0030-drops-201557},
  doi =		{10.4230/LIPIcs.ICALP.2024.12},
  annote =	{Keywords: proof complexity, circuit complexity, interactive protocols}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.6
Bounded Relativization

Authors: Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ℭ, we say that a statement is ℭ-relativizing if the statement holds relative to every oracle 𝒪 ∈ ℭ. It is easy to see that every result that relativizes also ℭ-relativizes for every complexity class ℭ. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0, BPE^{MCSP}/2^{εn} ⊈ SIZE[2ⁿ/n]. We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021). Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ≠ L. For example: - Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ≠ BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ≠ L. - Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ≠ L. - Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ≠ L. In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ≠ EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible.
Cite as

Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren. Bounded Relativization. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 6:1-6:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{hirahara_et_al:LIPIcs.CCC.2023.6,
  author =	{Hirahara, Shuichi and Lu, Zhenjian and Ren, Hanlin},
  title =	{{Bounded Relativization}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{6:1--6:45},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.6},
  URN =		{urn:nbn:de:0030-drops-182764},
  doi =		{10.4230/LIPIcs.CCC.2023.6},
  annote =	{Keywords: relativization, circuit lower bound, derandomization, explicit construction, pseudodeterministic algorithms, interactive proofs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.11
Derandomization with Minimal Memory Footprint

Authors: Dean Doron and Roei Tell

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
Existing proofs that deduce BPL = 𝐋 from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ⊆ DSPACE[c ⋅ S] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC⁰, that were not known before.
Cite as

Dean Doron and Roei Tell. Derandomization with Minimal Memory Footprint. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{doron_et_al:LIPIcs.CCC.2023.11,
  author =	{Doron, Dean and Tell, Roei},
  title =	{{Derandomization with Minimal Memory Footprint}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.11},
  URN =		{urn:nbn:de:0030-drops-182816},
  doi =		{10.4230/LIPIcs.CCC.2023.11},
  annote =	{Keywords: derandomization, space-bounded computation, catalytic space}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.8
Trading Time and Space in Catalytic Branching Programs

Authors: James Cook and Ian Mertz

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
An m-catalytic branching program (Girard, Koucký, McKenzie 2015) is a set of m distinct branching programs for f which are permitted to share internal (i.e. non-source non-sink) nodes. While originally introduced as a non-uniform analogue to catalytic space, this also gives a natural notion of amortized non-uniform space complexity for f, namely the smallest value |G|/m for an m-catalytic branching program G for f (Potechin 2017). Potechin (2017) showed that every function f has amortized size O(n), witnessed by an m-catalytic branching program where m = 2^(2ⁿ-1). We recreate this result by defining a catalytic algorithm for evaluating polynomials using a large amount of space but O(n) time. This allows us to balance this with previously known algorithms which are efficient with respect to space at the cost of time (Cook, Mertz 2020, 2021). We show that for any ε ≥ 2n^(-1), every function f has an m-catalytic branching program of size O_ε(mn), where m = 2^(2^(ε n)). We similarly recreate an improved result due to Robere and Zuiddam (2021), and show that for d ≤ n and ε ≥ 2d^(-1), the same result holds for m = 2^binom(n, ≤ ε d) as long as f is a degree-d polynomial over 𝔽₂. We also show that for certain classes of functions, m can be reduced to 2^(poly n) while still maintaining linear or quasi-linear amortized size. In the other direction, we bound the necessary length, and by extension the amortized size, of any permutation branching program for an arbitrary function between 3n and 4n-4.
Cite as

James Cook and Ian Mertz. Trading Time and Space in Catalytic Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{cook_et_al:LIPIcs.CCC.2022.8,
  author =	{Cook, James and Mertz, Ian},
  title =	{{Trading Time and Space in Catalytic Branching Programs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{8:1--8:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.8},
  URN =		{urn:nbn:de:0030-drops-165708},
  doi =		{10.4230/LIPIcs.CCC.2022.8},
  annote =	{Keywords: complexity theory, branching programs, amortized, space complexity, catalytic computation}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.104
Lifting with Sunflowers

Authors: Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Our proof uses elementary counting together with a novel connection to the sunflower lemma. In addition to a simplified proof, our approach opens up a new avenue of attack towards proving lifting theorems with improved gadget size - one of the main challenges in the area. Focusing on one of the most widely used gadgets - the index gadget - existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to polylogarithmic, similar to the known bounds for the corresponding robust sunflower lemmas.
Cite as

Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with Sunflowers. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 104:1-104:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{lovett_et_al:LIPIcs.ITCS.2022.104,
  author =	{Lovett, Shachar and Meka, Raghu and Mertz, Ian and Pitassi, Toniann and Zhang, Jiapeng},
  title =	{{Lifting with Sunflowers}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{104:1--104:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.104},
  URN =		{urn:nbn:de:0030-drops-157004},
  doi =		{10.4230/LIPIcs.ITCS.2022.104},
  annote =	{Keywords: Lifting theorems, communication complexity, combinatorics, sunflowers}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.32
Bounds on the QAC^0 Complexity of Approximating Parity

Authors: Gregory Rosenthal

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
QAC circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. QAC^0 circuits are QAC circuits of constant depth, and are quantum analogues of AC^0 circuits. We prove the following: - For all d ≥ 7 and ε > 0 there is a depth-d QAC circuit of size exp(poly(n^{1/d}) log(n/ε)) that approximates the n-qubit parity function to within error ε on worst-case quantum inputs. Previously it was unknown whether QAC circuits of sublogarithmic depth could approximate parity regardless of size. - We introduce a class of "mostly classical" QAC circuits, including a major component of our circuit from the above upper bound, and prove a tight lower bound on the size of low-depth, mostly classical QAC circuits that approximate this component. - Arbitrary depth-d QAC circuits require at least Ω(n/d) multi-qubit gates to achieve a 1/2 + exp(-o(n/d)) approximation of parity. When d = Θ(log n) this nearly matches an easy O(n) size upper bound for computing parity exactly. - QAC circuits with at most two layers of multi-qubit gates cannot achieve a 1/2 + exp(-o(n)) approximation of parity, even non-cleanly. Previously it was known only that such circuits could not cleanly compute parity exactly for sufficiently large n. The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which we name "nekomata" after an analogous cat yōkai.
Cite as

Gregory Rosenthal. Bounds on the QAC^0 Complexity of Approximating Parity. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{rosenthal:LIPIcs.ITCS.2021.32,
  author =	{Rosenthal, Gregory},
  title =	{{Bounds on the QAC^0 Complexity of Approximating Parity}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{32:1--32:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.32},
  URN =		{urn:nbn:de:0030-drops-135713},
  doi =		{10.4230/LIPIcs.ITCS.2021.32},
  annote =	{Keywords: quantum circuit complexity, QAC^0, fanout, parity, nekomata}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.84
Short Proofs Are Hard to Find

Authors: Ian Mertz, Toniann Pitassi, and Yuanhao Wei

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
We obtain a streamlined proof of an important result by Alekhnovich and Razborov [Michael Alekhnovich and Alexander A. Razborov, 2008], showing that it is hard to automatize both tree-like and general Resolution. Under a different assumption than [Michael Alekhnovich and Alexander A. Razborov, 2008], our simplified proof gives improved bounds: we show under ETH that these proof systems are not automatizable in time n^f(n), whenever f(n) = o(log^{1/7 - epsilon} log n) for any epsilon > 0. Previously non-automatizability was only known for f(n) = O(1). Our proof also extends fairly straightforwardly to prove similar hardness results for PCR and Res(r).
Cite as

Ian Mertz, Toniann Pitassi, and Yuanhao Wei. Short Proofs Are Hard to Find. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{mertz_et_al:LIPIcs.ICALP.2019.84,
  author =	{Mertz, Ian and Pitassi, Toniann and Wei, Yuanhao},
  title =	{{Short Proofs Are Hard to Find}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{84:1--84:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.84},
  URN =		{urn:nbn:de:0030-drops-106605},
  doi =		{10.4230/LIPIcs.ICALP.2019.84},
  annote =	{Keywords: automatizability, Resolution, SAT solvers, proof complexity}
}
  • Refine by Author
  • 3 Mertz, Ian
  • 2 Arteche, Noel
  • 2 Pitassi, Toniann
  • 1 Alekseev, Yaroslav
  • 1 Carenini, Gaia
  • Show More...

  • Refine by Classification
  • 5 Theory of computation → Proof complexity
  • 4 Theory of computation → Circuit complexity
  • 3 Theory of computation → Complexity theory and logic
  • 3 Theory of computation → Pseudorandomness and derandomization
  • 2 Theory of computation → Communication complexity
  • Show More...

  • Refine by Keyword
  • 3 derandomization
  • 2 proof complexity
  • 2 space-bounded computation
  • 1 Automatability
  • 1 Bounded-depth Frege
  • Show More...

  • Refine by Type
  • 11 document

  • Refine by Publication Year
  • 5 2024
  • 2 2022
  • 2 2023
  • 1 2019
  • 1 2021

© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact