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Proofs of Space [chapter]

Stefan Dziembowski, Sebastian Faust, Vladimir Kolmogorov, Krzysztof Pietrzak
<span title="">2015</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this work, we put forward an alternative concept for PoWs -so-called proofs of space (PoS), where a service requestor must dedicate a significant amount of disk space as opposed to computation.  ...  Proofs of work (PoW) have been suggested by Dwork and Naor (Crypto'92) as protection to a shared resource.  ...  In this paper we put forward the concept of proofs of space where the resource in question is disk space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-48000-7_29">doi:10.1007/978-3-662-48000-7_29</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6xt3wcmjnbc3vmecxghp5nrofq">fatcat:6xt3wcmjnbc3vmecxghp5nrofq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20161102141628/http://eprint.iacr.org/2013/796.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/fe/7c/fe7c1437c4da50f9fd9cfadc8a1ac6c59ea3ffe5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-48000-7_29"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Proofs of Catalytic Space

Krzysztof Pietrzak, Michael Wagner
<span title="2018-12-21">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cjckxhgqp5dobfucnrxpxnsy4u" style="color: black;">Innovations in Theoretical Computer Science</a> </i> &nbsp;
Proofs of space (PoS) [Dziembowski et al., CRYPTO'15] are proof systems where a prover can convince a verifier that he "wastes" disk space.  ...  We discuss how some of these variants can be used as proofs of catalytic space (PoCS), a notion we put forward in this work, and which basically is a PoS where most of the space required by the prover  ...  Introduction Proofs of Space (PoS) A proof of space (PoS) [16, 30, 3] is an interactive proof system in which a prover P can convince a verifier V that it "wastes" a large amount of disk-space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.itcs.2019.59">doi:10.4230/lipics.itcs.2019.59</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/innovations/Pietrzak19.html">dblp:conf/innovations/Pietrzak19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/voopwoywivda7fdq77ock2vjwm">fatcat:voopwoywivda7fdq77ock2vjwm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220202103412/https://drops.dagstuhl.de/opus/volltexte/2018/10152/pdf/LIPIcs-ITCS-2019-59.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a0/99/a099571500c5c1ac8e76c14bd06ab7abc516b876.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.itcs.2019.59"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Proof Theory of Riesz Spaces and Modal Riesz Spaces [article]

Christophe Lucas, Matteo Mio
<span title="2022-02-16">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We design hypersequent calculus proof systems for the theories of Riesz spaces and modal Riesz spaces and prove the key theorems: soundness, completeness and cut elimination.  ...  These are then used to obtain completely syntactic proofs of some interesting results concerning the two theories.  ...  First Contribution: Proof Theory of Riesz Spaces.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.11185v9">arXiv:2004.11185v9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oiwnu2hf5vfqne6sd5pu52qdpy">fatcat:oiwnu2hf5vfqne6sd5pu52qdpy</a> </span>
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Proofs of Space: When Space Is of the Essence [chapter]

Giuseppe Ateniese, Ilario Bonacina, Antonio Faonio, Nicola Galesi
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper we put forward a new concept dubbed proof of space.  ...  To compute such a proof, the prover must use a specified amount of space, i.e., we are not interested in the number of accesses to the main memory (as in memory-bound proof of work) but rather on the amount  ...  Weak Proof of Space The concept of proof of space can lead to multiple interpretations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-10879-7_31">doi:10.1007/978-3-319-10879-7_31</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tm6osznpdzcd7d5howg2njos2i">fatcat:tm6osznpdzcd7d5howg2njos2i</a> </span>
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Proof of Space from Stacked Expanders [chapter]

Ling Ren, Srinivas Devadas
<span title="">2016</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Recently, proof of space (PoS) has been suggested as a more egalitarian alternative to the traditional hash-based proof of work.  ...  In PoS, a prover proves to a verifier that it has dedicated some specified amount of space.  ...  At the moment, PoTS and PoPS are still far less efficient than PoW in terms of proof size and verifier complexity. A PoW is a single hash, while a PoS consists of hundreds (or more) of Merkle paths.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-53641-4_11">doi:10.1007/978-3-662-53641-4_11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fz5nszj5jngtpmaqtpnvw427j4">fatcat:fz5nszj5jngtpmaqtpnvw427j4</a> </span>
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Proof Theory of Riesz Spaces and Modal Riesz Spaces

Christophe Lucas, Matteo Mio
<span title="2022-02-17">2022</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c67srop7pbe3nciquoxzy3d6pm" style="color: black;">Logical Methods in Computer Science</a> </i> &nbsp;
We design hypersequent calculus proof systems for the theories of Riesz spaces and modal Riesz spaces and prove the key theorems: soundness, completeness and cut elimination.  ...  These are then used to obtain completely syntactic proofs of some interesting results concerning the two theories.  ...  First Contribution: Proof Theory of Riesz Spaces.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.46298/lmcs-18(1:32)2022">doi:10.46298/lmcs-18(1:32)2022</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y5okiphl7zc3fc2iy3pwe76oqe">fatcat:y5okiphl7zc3fc2iy3pwe76oqe</a> </span>
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Topologically Fragmental Space and the Proof of Hadwiger's Conjecture [article]

Cao Zexin
<span title="2005-04-04">2005</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The proof of four color theorem and Hadwiger's conjecture is obtained by researching them and their cross-sections.  ...  And we can see in the context, that those conjectures are not of graph theory, but such topological space.  ...  The proof of theorem 4.1 is illustrated in Fig.6 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0311475v3">arXiv:math/0311475v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tdfztgpsdrgjvdckvshy2cytbu">fatcat:tdfztgpsdrgjvdckvshy2cytbu</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-math0311475/math0311475.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0311475v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

A combinatorial proof of the Degree Theorem in Auter space [article]

Robert McEwen, Matthew C. B. Zaremsky
<span title="2014-03-05">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n.  ...  This result is useful, for example to calculate stability bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem is global in nature.  ...  Parts of this paper are based on results in the first named author's Ph.D. thesis [McE10] , done at the University of Virginia.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0907.4642v4">arXiv:0907.4642v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/apuvfviuevfbxb4thf4ojuajpe">fatcat:apuvfviuevfbxb4thf4ojuajpe</a> </span>
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A simple, combinatorial proof of Gowers Dichotomy for Banach Spaces [article]

Ryszard Frankiewicz, Sławomir Kusiński
<span title="2022-01-19">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we present a simple proof of Gowers Dichotomy which states that every infinite dimensional Banach Space has a subspace which either contains an unconditional basic sequence or is hereditarily  ...  Our approach is purely combinatorial and mainly based on work of Ellentuck, Galvin and Prikry in infinite Ramsey theory.  ...  A simple, combinatorial proof of Gowers Dichotomy for Banach Spaces Une preuve combinatoire simple de la dichotomie  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.10728v3">arXiv:2106.10728v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7w5fdjrd2bfqzoilmyxnrtyrfa">fatcat:7w5fdjrd2bfqzoilmyxnrtyrfa</a> </span>
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A proof of uniqueness of the Gurariĭ space

Wiesław Kubiś, Sławomir Solecki
<span title="2012-10-11">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/whoel4vntnhddbjlrtj3w2jgra" style="color: black;">Israel Journal of Mathematics</a> </i> &nbsp;
We present a short and elementary proof of isometric uniqueness of the Gurarii space.  ...  Proof. Use Lemma 2.1 together with the definition of a Gurarii space. Proof of Theorem 1.1.  ...  The Gurarii space is unique up to a bijective isometry. 2 Proof of uniqueness of the Gurarii space Lemma 2.1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11856-012-0134-9">doi:10.1007/s11856-012-0134-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cjxopllt7fgn7bhuxihntkzp2e">fatcat:cjxopllt7fgn7bhuxihntkzp2e</a> </span>
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Computer aided Unirationality Proofs of Moduli Spaces [article]

Frank-Olaf Schreyer
<span title="2011-09-21">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We illustrate the use of the computer algebra system Macaulay2 for simplifications of classical unirationality proofs. We explicitly treat the moduli spaces of curves of genus g=10, 12 and 14.  ...  S Space Curves The proof of the unirationality of the moduli space M g for g = 11, 12 and 13 by Sernesi and Chang-Ran is based on models of these curves in P 3 , see [Ser81, CR84] .  ...  Random Plane Curves Let M g denote the moduli space of curves of genus g.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1109.4600v1">arXiv:1109.4600v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gewohvyy6bag7hwl2ot4uo63uq">fatcat:gewohvyy6bag7hwl2ot4uo63uq</a> </span>
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Algebraic proof of the symmetric space theorem

C. Daboul
<span title="">1996</span> <i title="AIP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nwwheyjjk5hnlmj6q5mtpqmmhi" style="color: black;">Journal of Mathematical Physics</a> </i> &nbsp;
I give a relatively elementary proof of the symmetric space theorem, due to Goddard, Nahm and Olive [1].  ...  Unlike their original proof, which involves the quark-model construction, I only use elementary algebraic techniques.  ...  It relates the vanishing of certain coset Virasoro algebras to the existence of symmetric spaces.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1063/1.531583">doi:10.1063/1.531583</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3ohvqz34ivcmldz23yjmvwm464">fatcat:3ohvqz34ivcmldz23yjmvwm464</a> </span>
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A Proof of the HRT Conjecture for Widely Spaced Sets [article]

Michael Kreisel
<span title="2018-09-08">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
to the decay of f.  ...  Given f ∈ C_0(R^n) and Λ⊂R^2n a finite set we demonstrate the linear independence of the set of time-frequency translates G(f, Λ) = {π(λ)f}_λ∈Λ when the time coordinates of points in Λ are far apart relative  ...  We denote by C 0 pR n q the space of continuous functions for which |f ptq| Ñ 0 as ||t|| Ñ 8. Theorem 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.06116v2">arXiv:1805.06116v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4vyrphz5ufhexbhmbz76gtupqu">fatcat:4vyrphz5ufhexbhmbz76gtupqu</a> </span>
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A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces [article]

Manor Mendel
<span title="2022-02-28">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal's Ramsey decompositions [Bartal 2021].  ...  The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of "nearly ultrametric" subsets of compact spaces having full Hausdorff dimension.  ...  Bartal's proof of Lemma 5 is written for finite spaces, but applies to compact spaces with only minor adaptations. We include the proof here for completeness. We also include a proof of Lemma 15.  ... 
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A Rice method proof of the Null-Space Property over the Grassmannian [article]

J.-M Azaïs
<span title="2015-10-08">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this article, we provide the first proof of NSP using random processes theory and the Rice method.  ...  The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations.  ...  to the case S ∅ in the space of dimension m − k instead of m and with p − k vectors.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1405.6417v3">arXiv:1405.6417v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wjzb4f3nebdnpg4hkosbis7r4m">fatcat:wjzb4f3nebdnpg4hkosbis7r4m</a> </span>
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