32 Search Results for "Pudlák, Pavel"


Document
RANDOM
On the Communication Complexity of Finding a King in a Tournament

Authors: Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)


Abstract
A tournament is a complete directed graph. A source in a tournament is a vertex that has no in-neighbours (every other vertex is reachable from it via a path of length 1), and a king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king. In particular, a maximum out-degree vertex is a king. The tasks of finding a king and a maximum out-degree vertex in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of finding a king, of finding a maximum out-degree vertex, and of finding a source (if it exists) in a tournament, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: - We show that the communication task of finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. As a result, known bounds on the communication complexity of CIS [Yannakakis, JCSS'91, Göös, Pitassi, Watson, SICOMP'18] imply a bound of Θ̃(log² n) for finding a source (if it exists, or outputting that there is no source) in a tournament. - The deterministic and randomized communication complexities of finding a king are Θ(n). The quantum communication complexity of finding a king is Θ̃(√n). - The deterministic, randomized, and quantum communication complexities of finding a maximum out-degree vertex are Θ(n log n), Θ̃(n) and Θ̃(√n), respectively. Our upper bounds above hold for all partitions of edges, and the lower bounds for a specific partition of the edges. One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness. An interesting point to note here is that while the deterministic query complexity of finding a king has been open for over two decades [Shen, Sheng, Wu, SICOMP'03], we are able to essentially resolve the complexity of this problem in a model (communication complexity) that is usually harder to analyze than query complexity.

Cite as

Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Communication Complexity of Finding a King in a Tournament. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 64:1-64:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{mande_et_al:LIPIcs.APPROX/RANDOM.2024.64,
  author =	{Mande, Nikhil S. and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin},
  title =	{{On the Communication Complexity of Finding a King in a Tournament}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{64:1--64:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.64},
  URN =		{urn:nbn:de:0030-drops-210571},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.64},
  annote =	{Keywords: Communication complexity, tournaments, query complexity}
}
Document
Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree

Authors: Olaf Beyersdorff, Tim Hoffmann, Kaspar Kasche, and Luc Nicolas Spachmann

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)


Abstract
We initiate an in-depth proof-complexity analysis of polynomial calculus (𝒬-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of 𝒬-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for 𝒬-PC, similar in spirit, yet different from the classic size-degree formula for propositional PC by Impagliazzo, Pudlák and Sgall (1999). We use the circuit characterisation together with the size-degree relation to obtain various new lower bounds on proof size in 𝒬-PC. This leads to incomparability results for 𝒬-PC systems over different fields.

Cite as

Olaf Beyersdorff, Tim Hoffmann, Kaspar Kasche, and Luc Nicolas Spachmann. Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{beyersdorff_et_al:LIPIcs.MFCS.2024.27,
  author =	{Beyersdorff, Olaf and Hoffmann, Tim and Kasche, Kaspar and Spachmann, Luc Nicolas},
  title =	{{Polynomial Calculus for Quantified Boolean Logic: Lower Bounds Through Circuits and Degree}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.27},
  URN =		{urn:nbn:de:0030-drops-205834},
  doi =		{10.4230/LIPIcs.MFCS.2024.27},
  annote =	{Keywords: proof complexity, QBF, polynomial calculus, circuits, lower bounds}
}
Document
Pebble Games and Algebraic Proof Systems

Authors: Lisa-Marie Jaser and Jacobo Torán

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)


Abstract
Analyzing refutations of the well known pebbling formulas Peb(G) we prove some new strong connections between pebble games and algebraic proof system, showing that there is a parallelism between the reversible, black and black-white pebbling games on one side, and the three algebraic proof systems Nullstellensatz, Monomial Calculus and Polynomial Calculus on the other side. In particular we prove that for any DAG G with a single sink, if there is a Monomial Calculus refutation for Peb(G) having simultaneously degree s and size t then there is a black pebbling strategy on G with space s and time t+s. Also if there is a black pebbling strategy for G with space s and time t it is possible to extract from it a MC refutation for Peb(G) having simultaneously degree s and size ts. These results are analogous to those proven in [Susanna F. de Rezende et al., 2021] for the case of reversible pebbling and Nullstellensatz. Using them we prove degree separations between NS, MC and PC, as well as strong degree-size tradeoffs for MC. We also notice that for any directed acyclic graph G the space needed in a pebbling strategy on G, for the three versions of the game, reversible, black and black-white, exactly matches the variable space complexity of a refutation of the corresponding pebbling formula Peb(G) in each of the algebraic proof systems NS,MC and PC. Using known pebbling bounds on graphs, this connection implies separations between the corresponding variable space measures.

Cite as

Lisa-Marie Jaser and Jacobo Torán. Pebble Games and Algebraic Proof Systems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{jaser_et_al:LIPIcs.MFCS.2024.64,
  author =	{Jaser, Lisa-Marie and Tor\'{a}n, Jacobo},
  title =	{{Pebble Games and Algebraic Proof Systems}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{64:1--64:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.64},
  URN =		{urn:nbn:de:0030-drops-206200},
  doi =		{10.4230/LIPIcs.MFCS.2024.64},
  annote =	{Keywords: Proof Complexity, Algebraic Proof Systems, Pebble Games}
}
Document
Consistent Ultrafinitist Logic

Authors: Michał J. Gajda

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)


Abstract
Ultrafinitism postulates that we can only compute on relatively short objects, and numbers beyond a certain value are not available. This approach would also forbid many forms of infinitary reasoning and allow removing certain paradoxes stemming from enumeration theorems. For a computational application of ultrafinitist logic, we need more than a proof system, but a logical framework to express both proofs, programs, and theorems in a single framework. We present its inference rules, reduction relation, and self-encoding to allow direct proving of the properties of ultrafinitist logic within itself. We also provide a justification why it can express all bounded Turing programs, and thus serve as a "logic of computability".

Cite as

Michał J. Gajda. Consistent Ultrafinitist Logic. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{gajda:LIPIcs.TYPES.2023.5,
  author =	{Gajda, Micha{\l} J.},
  title =	{{Consistent Ultrafinitist Logic}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.5},
  URN =		{urn:nbn:de:0030-drops-204833},
  doi =		{10.4230/LIPIcs.TYPES.2023.5},
  annote =	{Keywords: ultrafinitism, bounded Turing completeness, logic of computability, decidable logic, explicit complexity, strict finitism}
}
Document
New Lower Bounds for Polynomial Calculus over Non-Boolean Bases

Authors: Yogesh Dahiya, Meena Mahajan, and Sasank Mouli

Published in: LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)


Abstract
In this paper, we obtain new size lower bounds for proofs in the Polynomial Calculus (PC) proof system, in two different settings. - When the Boolean variables are encoded using ±1 (as opposed to 0,1): We establish a lifting theorem using an asymmetric gadget G, showing that for an unsatisfiable formula F, the lifted formula F∘G requires PC size 2^{Ω(d)}, where d is the degree required to refute F. Our lower bound does not depend on the number of variables n, and holds over every field. The only previously known size lower bounds in this setting were established quite recently in [Sokolov, STOC 2020] using lifting with another (symmetric) gadget. The size lower bound there is 2^{Ω((d-d₀)²/n)} (where d₀ is the degree of the initial equations arising from the formula), and is shown to hold only over the reals. - When the PC refutation proceeds over a finite field 𝔽_p and is allowed to use extension variables: We show that there is an unsatisfiable AC⁰[p] formula with N variables for which any PC refutation using N^{1+ε(1-δ)} extension variables, each of arity at most N^{1-ε} and size at most N^c, must have size exp(Ω(N^{εδ}/polylog N)). Our proof achieves these bounds by an XOR-ification of the generalised PHP^{m,r}_n formulas from [Razborov, CC 1998]. The only previously known lower bounds for PC in this setting are those obtained in [Impagliazzo-Mouli-Pitassi, CCC 2023]; in those bounds the number of extension variables is required to be sub-quadratic, and their arity is restricted to logarithmic in the number of original variables. Our result generalises these, and demonstrates a tradeoff between the number and the arity of extension variables. Since our tautology is represented by a small AC⁰[p] formula, our results imply lower bounds for a reasonably strong fragment of AC⁰[p]-Frege.

Cite as

Yogesh Dahiya, Meena Mahajan, and Sasank Mouli. New Lower Bounds for Polynomial Calculus over Non-Boolean Bases. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{dahiya_et_al:LIPIcs.SAT.2024.10,
  author =	{Dahiya, Yogesh and Mahajan, Meena and Mouli, Sasank},
  title =	{{New Lower Bounds for Polynomial Calculus over Non-Boolean Bases}},
  booktitle =	{27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-334-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{305},
  editor =	{Chakraborty, Supratik and Jiang, Jie-Hong Roland},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.10},
  URN =		{urn:nbn:de:0030-drops-205327},
  doi =		{10.4230/LIPIcs.SAT.2024.10},
  annote =	{Keywords: Proof Complexity, Polynomial Calculus, degree, Fourier basis, extension variables}
}
Document
Strategy Extraction by Interpolation

Authors: Friedrich Slivovsky

Published in: LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)


Abstract
In applications, QBF solvers are often required to generate strategies. This typically involves a process known as strategy extraction, where a Boolean circuit encoding a strategy is computed from a proof. It has previously been observed that Craig interpolation in propositional logic can be seen as a special case of QBF strategy extraction. In this paper we explore this connection further and show that, conversely, any strategy for a false QBF corresponds to a sequence of interpolants in its complete (Herbrand) expansion. Inspired by this correspondence, we present a new strategy extraction algorithm for the expansion-based proof system Exp+Res. Its asymptotic running time matches the best known bound of O(mn) for a proof with m lines and n universally quantified variables. We report on experiments comparing this algorithm with a strategy extraction algorithm based on combining partial strategies, as well as with round-based strategy extraction.

Cite as

Friedrich Slivovsky. Strategy Extraction by Interpolation. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{slivovsky:LIPIcs.SAT.2024.28,
  author =	{Slivovsky, Friedrich},
  title =	{{Strategy Extraction by Interpolation}},
  booktitle =	{27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-334-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{305},
  editor =	{Chakraborty, Supratik and Jiang, Jie-Hong Roland},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.28},
  URN =		{urn:nbn:de:0030-drops-205509},
  doi =		{10.4230/LIPIcs.SAT.2024.28},
  annote =	{Keywords: QBF, Expansion, Strategy Extraction, Interpolation}
}
Document
A Technique for Hardness Amplification Against AC⁰

Authors: William M. Hoza

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


Abstract
We study hardness amplification in the context of two well-known "moderate" average-case hardness results for AC⁰ circuits. First, we investigate the extent to which AC⁰ circuits of depth d can approximate AC⁰ circuits of some larger depth d + k. The case k = 1 is resolved by Håstad, Rossman, Servedio, and Tan’s celebrated average-case depth hierarchy theorem (JACM 2017). Our contribution is a significantly stronger correlation bound when k ≥ 3. Specifically, we show that there exists a linear-size AC⁰_{d + k} circuit h : {0, 1}ⁿ → {0, 1} such that for every AC⁰_d circuit g, either g has size exp(n^{Ω(1/d)}), or else g agrees with h on at most a (1/2 + ε)-fraction of inputs where ε = exp(-(1/d) ⋅ Ω(log n)^{k-1}). For comparison, Håstad, Rossman, Servedio, and Tan’s result has ε = n^{-Θ(1/d)}. Second, we consider the majority function. It is well known that the majority function is moderately hard for AC⁰ circuits (and stronger classes). Our contribution is a stronger correlation bound for the XOR of t copies of the n-bit majority function, denoted MAJ_n^{⊕ t}. We show that if g is an AC⁰_d circuit of size S, then g agrees with MAJ_n^{⊕ t} on at most a (1/2 + ε)-fraction of inputs, where ε = (O(log S)^{d - 1} / √n)^t. To prove these results, we develop a hardness amplification technique that is tailored to a specific type of circuit lower bound proof. In particular, one way to show that a function h is moderately hard for AC⁰ circuits is to (a) design some distribution over random restrictions or random projections, (b) show that AC⁰ circuits simplify to shallow decision trees under these restrictions/projections, and finally (c) show that after applying the restriction/projection, h is moderately hard for shallow decision trees with respect to an appropriate distribution. We show that (roughly speaking) if h can be proven to be moderately hard by a proof with that structure, then XORing multiple copies of h amplifies its hardness. Our analysis involves a new kind of XOR lemma for decision trees, which might be of independent interest.

Cite as

William M. Hoza. A Technique for Hardness Amplification Against AC⁰. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{hoza:LIPIcs.CCC.2024.1,
  author =	{Hoza, William M.},
  title =	{{A Technique for Hardness Amplification Against AC⁰}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{1:1--1:20},
  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.2024.1},
  URN =		{urn:nbn:de:0030-drops-203977},
  doi =		{10.4230/LIPIcs.CCC.2024.1},
  annote =	{Keywords: Bounded-depth circuits, average-case lower bounds, hardness amplification, XOR lemmas}
}
Document
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
Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

Authors: Xin Li and Yan Zhong

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


Abstract
Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation. This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include: - An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works. - An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).

Cite as

Xin Li and Yan Zhong. Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{li_et_al:LIPIcs.CCC.2024.10,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{10:1--10:14},
  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.2024.10},
  URN =		{urn:nbn:de:0030-drops-204060},
  doi =		{10.4230/LIPIcs.CCC.2024.10},
  annote =	{Keywords: Randomness Extractors, Affine, Read-once Linear Branching Programs, Low-degree polynomials, AC⁰ circuits}
}
Document
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)


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@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
Local Enumeration and Majority Lower Bounds

Authors: Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard

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


Abstract
Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics.

Cite as

Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard. Local Enumeration and Majority Lower Bounds. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{gurumukhani_et_al:LIPIcs.CCC.2024.17,
  author =	{Gurumukhani, Mohit and Paturi, Ramamohan and Pudl\'{a}k, Pavel and Saks, Michael and Talebanfard, Navid},
  title =	{{Local Enumeration and Majority Lower Bounds}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{17:1--17:25},
  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.2024.17},
  URN =		{urn:nbn:de:0030-drops-204136},
  doi =		{10.4230/LIPIcs.CCC.2024.17},
  annote =	{Keywords: Depth 3 circuits, k-CNF satisfiability, Circuit lower bounds, Majority function}
}
Document
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)


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@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
Exponential Separation Between Powers of Regular and General Resolution over Parities

Authors: Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák

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


Abstract
Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems.

Cite as

Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 23:1-23:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2024.23,
  author =	{Bhattacharya, Sreejata Kishor and Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
  title =	{{Exponential Separation Between Powers of Regular and General Resolution over Parities}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{23:1--23:32},
  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.2024.23},
  URN =		{urn:nbn:de:0030-drops-204191},
  doi =		{10.4230/LIPIcs.CCC.2024.23},
  annote =	{Keywords: Proof Complexity, Regular Reslin, Branching Programs, Lifting}
}
Document
Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It

Authors: Michal Garlík

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


Abstract
We show that for every integer k ≥ 2, the Res(k) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a result of Atserias and Müller [Atserias and Müller, 2019] to Res(k). We show that if NP is not included in P (resp. QP, SUBEXP) then for every integer k ≥ 1, Res(k) is not automatable in polynomial (resp. quasi-polynomial, subexponential) time.

Cite as

Michal Garlík. Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 33:1-33:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{garlik:LIPIcs.CCC.2024.33,
  author =	{Garl{\'\i}k, Michal},
  title =	{{Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{33:1--33:23},
  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.2024.33},
  URN =		{urn:nbn:de:0030-drops-204295},
  doi =		{10.4230/LIPIcs.CCC.2024.33},
  annote =	{Keywords: reflection principle, feasible disjunction property, k-DNF Resolution}
}
Document
Track A: Algorithms, Complexity and Games
Two-Source and Affine Non-Malleable Extractors for Small Entropy

Authors: Xin Li and Yan Zhong

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


Abstract
Non-malleable extractors are generalizations and strengthening of standard randomness extractors, that are resilient to adversarial tampering. Such extractors have wide applications in cryptography and have become important cornerstones in recent breakthroughs of explicit constructions of two-source extractors and affine extractors for small entropy. However, explicit constructions of non-malleable extractors appear to be much harder than standard extractors. Indeed, in the well-studied models of two-source and affine non-malleable extractors, the previous best constructions only work for entropy rate > 2/3 and 1-γ for some small constant γ > 0 respectively by Li (FOCS' 23). In this paper, we present explicit constructions of two-source and affine non-malleable extractors that match the state-of-the-art constructions of standard ones for small entropy. Our main results include: - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k ≥ log^C n and polynomially small error, matching the parameters of standard extractors by Chattopadhyay and Zuckerman (STOC' 16, Annals of Mathematics' 19) and Li (FOCS' 16). - Two-source and affine non-malleable extractors (over 𝖥₂) for sources on n bits with min-entropy k = O(log n) and constant error, matching the parameters of standard extractors by Li (FOCS' 23). Our constructions significantly improve previous results, and the parameters (entropy requirement and error) are the best possible without first improving the constructions of standard extractors. In addition, our improved affine non-malleable extractors give strong lower bounds for a certain kind of read-once linear branching programs, recently introduced by Gryaznov, Pudlák, and Talebanfard (CCC' 22) as a generalization of several well studied computational models. These bounds match the previously best-known average-case hardness results given by Chattopadhyay and Liao (CCC' 23) and Li (FOCS' 23), where the branching program size lower bounds are close to optimal, but the explicit functions we use here are different. Our results also suggest a possible deeper connection between non-malleable extractors and standard ones.

Cite as

Xin Li and Yan Zhong. Two-Source and Affine Non-Malleable Extractors for Small Entropy. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 108:1-108:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{li_et_al:LIPIcs.ICALP.2024.108,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Two-Source and Affine Non-Malleable Extractors for Small Entropy}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{108:1--108:15},
  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.2024.108},
  URN =		{urn:nbn:de:0030-drops-202512},
  doi =		{10.4230/LIPIcs.ICALP.2024.108},
  annote =	{Keywords: Randomness Extractors, Non-malleable, Two-source, Affine}
}
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