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Documents authored by Cook, Joshua

Document
DOI: 10.4230/LIPIcs.CCC.2024.5
Explicit Time and Space Efficient Encoders Exist Only with Random Access

Authors: Joshua Cook and Dana Moshkovitz

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

Abstract
We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n^{1 + o(1)} and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [Divsalar et al., 1998] and the codes described in [Gál et al., 2013]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length. In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access.
Cite as

Joshua Cook and Dana Moshkovitz. Explicit Time and Space Efficient Encoders Exist Only with Random Access. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 5:1-5:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cook_et_al:LIPIcs.CCC.2024.5,
  author =	{Cook, Joshua and Moshkovitz, Dana},
  title =	{{Explicit Time and Space Efficient Encoders Exist Only with Random Access}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{5:1--5:54},
  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.2024.5},
  URN =		{urn:nbn:de:0030-drops-204015},
  doi =		{10.4230/LIPIcs.CCC.2024.5},
  annote =	{Keywords: Time-Space Trade Offs, Error Correcting Codes, Encoders, Explicit Constructions, Streaming Lower Bounds, Sequential Access, Time-Space Lower Bounds, Lossless Condensers, Invertible Condensers, Condensers}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.47
Efficient Interactive Proofs for Non-Deterministic Bounded Space

Authors: Joshua Cook and Ron D. Rothblum

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time Õ(n+S²). This improves on the best previous bound of Õ(n+S³) and matches the result for deterministic space bounded algorithms, up to polylog(S) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T, space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time Õ(n + S log(T) + S d) and the prover runs in time 2^O(S). For d = O(log(T)), this matches the best known interactive proofs for deterministic algorithms, up to polylog(S) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log(T). We also improve the best prior verifier time for unbounded alternations by a factor of S. Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in.
Cite as

Joshua Cook and Ron D. Rothblum. Efficient Interactive Proofs for Non-Deterministic Bounded Space. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 47:1-47:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cook_et_al:LIPIcs.APPROX/RANDOM.2023.47,
  author =	{Cook, Joshua and Rothblum, Ron D.},
  title =	{{Efficient Interactive Proofs for Non-Deterministic Bounded Space}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{47:1--47:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.47},
  URN =		{urn:nbn:de:0030-drops-188727},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.47},
  annote =	{Keywords: Interactive Proofs, Alternating Algorithms, AC0\lbrack2\rbrack, Doubly Efficient Proofs}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.55
Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE

Authors: Joshua Cook and Dana Moshkovitz

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
We prove that for some constant a > 1, for all k ≤ a, MATIME[n^{k+o(1)}]/1 ⊄ SIZE[O(n^k)], for some specific o(1) function. This is a super linear polynomial circuit lower bound. Previously, Santhanam [Santhanam, 2007] showed that there exists a constant c > 1 such that for all k > 1: MATIME[n^{ck}]/1 ⊄ SIZE[O(n^k)]. Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c. Using ideas from Murray and Williams [Murray and Williams, 2018], one can show for all k > 1: MATIME [n^{10 k²}]/1 ⊄ SIZE[O(n^k)]. To prove this result, we construct the first PCP for SPACE[n] with quasi-linear verifier time: our PCP has a Õ(n) time verifier, Õ(n) space prover, O(log(n)) queries, and polynomial alphabet size. Prior to this work, PCPs for SPACE[O(n)] had verifiers that run in Ω(n²) time. This PCP also proves that NE has MIP verifiers which run in time Õ(n).
Cite as

Joshua Cook and Dana Moshkovitz. Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cook_et_al:LIPIcs.APPROX/RANDOM.2023.55,
  author =	{Cook, Joshua and Moshkovitz, Dana},
  title =	{{Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{55:1--55:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.55},
  URN =		{urn:nbn:de:0030-drops-188805},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.55},
  annote =	{Keywords: MA, PCP, Circuit Complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.14
More Verifier Efficient Interactive Protocols for Bounded Space

Authors: Joshua Cook

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
Let TISP[T, S], BPTISP[T, S], NTISP[T, S] and CoNTISP[T, S] be the set of languages recognized by deterministic, randomized, nondeterministic, and co-nondeterministic algorithms, respectively, running in time T and space S. Let ITIME[T_V, T_P] be the set of languages recognized by an interactive protocol where the verifier runs in time T_V and the prover runs in time T_P. For S = Ω(log(n)) and T constructible in time log(T) S + n, we prove: TISP[T, S] ⊆ ITIME[Õ(log(T) S + n), 2^O(S)] BPTISP[T, S] ⊆ ITIME[Õ(log(T) S + n), 2^O(S)] NTISP[T, S] ⊆ ITIME[Õ(log(T)^2 S + n), 2^O(S)] CoNTISP[T, S] ⊆ ITIME[Õ(log(T)^2 S + n), 2^O(S)] . The best prior verifier time is from Shamir [Shamir, 1992; Lund et al., 1990]: TISP[T, S] ⊆ ITIME[Õ(log(T)(S + n)), 2^O(log(T)(S + n))]. Our prover is faster, and our verifier is faster when S = o(n). The best prior prover time uses ideas from Goldwasser, Kalai, and Rothblum [Goldwasser et al., 2015]: NTISP[T, S] ⊆ ITIME[Õ(log(T) S² + n), 2^O(S)]. Our verifier is faster when log(T) = o(S), and for deterministic algorithms. To our knowledge, no previous interactive protocol for TISP simultaneously has the same verifier time and prover time as ours. In our opinion, our protocol is also simpler than previous protocols.
Cite as

Joshua Cook. More Verifier Efficient Interactive Protocols for Bounded Space. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cook:LIPIcs.FSTTCS.2022.14,
  author =	{Cook, Joshua},
  title =	{{More Verifier Efficient Interactive Protocols for Bounded Space}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.14},
  URN =		{urn:nbn:de:0030-drops-174067},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.14},
  annote =	{Keywords: Interactive Proofs, Verifier Time, Randomized Space, Nondeterministic Space, Fine Grain Complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.19
Size Bounds on Low Depth Circuits for Promise Majority

Authors: Joshua Cook

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We give two results on the size of AC0 circuits computing promise majority. ε-promise majority is majority promised that either at most an ε fraction of the input bits are 1 or at most ε are 0. - First, we show super-quadratic size lower bounds on both monotone and general depth-3 circuits for promise majority. - For any ε ∈ (0, 1/2), monotone depth-3 AC0 circuits for ε-promise majority have size Ω̃(ε³ n^{2 + (ln(1 - ε))/(ln(ε))}). - For any ε ∈ (0, 1/2), general depth-3 AC0 circuits for ε-promise majority have size Ω̃(ε³ n^{2 + (ln(1 - ε²))/(2ln(ε))}). These are the first quadratic size lower bounds for depth-3 ε-promise majority circuits for ε < 0.49. - Second, we give both uniform and non-uniform sub-quadratic size constant-depth circuits for promise majority. - For integer k ≥ 1 and constant ε ∈ (0, 1/2), there exists monotone non uniform AC0 circuits of depth-(2 + 2 k) computing ε-promise majority with size Õ(n^{1/(1 - 2^{-k})}). - For integer k ≥ 1 and constant ε ∈ (0, 1/2), there exists monotone uniform AC0 circuit of depth-(2 + 2 k) computing ε-promise majority with size n^{1/(1 - (2/3) ^k) + o(1)}. These circuits are based on incremental improvements to existing depth-3 circuits for promise majority given by Ajtai [Miklós Ajtai, 1983] and Viola [Emanuele Viola, 2009] combined with a divide and conquer strategy.
Cite as

Joshua Cook. Size Bounds on Low Depth Circuits for Promise Majority. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cook:LIPIcs.FSTTCS.2020.19,
  author =	{Cook, Joshua},
  title =	{{Size Bounds on Low Depth Circuits for Promise Majority}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.19},
  URN =		{urn:nbn:de:0030-drops-132609},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.19},
  annote =	{Keywords: AC0, Approximate Counting, Approximate Majority, Promise Majority, Depth 3 Circuits, Circuit Lower Bound}
}
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