Gödel Prize

https://sigact.org/prizes/gödel.html

20年以上前、Gödel の論文を読もうという身内の企画があった。計算機の仕事で時間が取れずに参加できなかった。

企画そのものが空中分解したのかもしれない。

読んでないという負い目をおいながら仕事をしてきた。

Gödel Prizeがあることを知った。

何一つ読んでいない。これらをすべて読みながら、Gödel の論文も読もうと思った。

Gödel, Kurt (1931), “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I.”, Monatshefte für Mathematik und Physik 38: 173–198, doi:10.1007/BF01700692.

英語じゃない。

Gödel, Kurt (1986), Feferman, Solomon, ed., Kurt Gödel: Collected Works: Volume I: Publications 1929-1936, Oxford University Press, pp. 144-195, ISBN 978-0-19-503964-1

Gödel, Kurt Meltzer, B.訳 (1992), On Formally

Undecidable Propositions of Principia Mathematica and Related Systems, Dover Books on Mathematics, Dover Publications, ISBN 978-0-486-66980-9

Gödel, Kurt; Hirzel, Martin (2000-11-27), “On formally undecidable propositions of Principia Mathematica and related systems I” (PDF), Boulder: 173-196

http://hirzels.com/martin/papers/canon00-goedel.pdf

Gödel, Kurt (2002), “Some metamathematical results on completeness and consistency, On formally undecidable propositions of Principia mathematica and related systems I, and On completeness and consistency”, in van Heijenoort, Jean, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Source Books in the History of the Sciences (Fourth Printing ed.), Harvard University Press, pp. 592-617, ISBN 978-0-674-32449-7

Gödel, Kurt (2004), “On Formally Undecidable Propositions of the Principia Mathematica and Related Systems. I.”, in Davis, Martin, The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Dover Books on Mathematics, Courier Corporation, pp. 4-38, ISBN 978-0-486-43228-1

https://books.google.co.jp/books?id=qW8x7sQ4JXgC&pg=PA4&redir_esc=y#v=onepage&q&f=false

英訳をまず確認中。

ゲーデル賞の方は、2013, 2014, 2015年の詳細記事がリンク切れになっていた。人名等を元に、論文を探して記録する。

http://citeseerx.ist.psu.edu/

https://arxiv.org

http://researchgate.net

で検索。

- Requested としたのはhttp://researchgate.net で関係者に著作物送達依頼した文献。

論文が揃ったら単語帳を作る予定。Wikipediaの記述も直接PDF等のURLを記載していないものもある。

該当する論文がWEBで無償では見当たらず、原型がArxivに掲載のあるものもある。論文URLの後ろに年号を記載した。

https://en.wikipedia.org/wiki/Gödel_Prize#cite_note-36

原文と同一のものはWikipediaに加筆する予定。

#人名一覧

2020: Robin A. Moser and Gábor Tardos

2019: Irit Dinur

2018: Oded Regev

2017: Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith

2016: Stephen Brookes and Peter W. O’Hearn

2015: Daniel A. Spielman and Shang-Hua Teng

2014: Ronald Fagin, Amnon Lotem, and Moni Naor

2013: Antoine Joux, Dan Boneh, and Matthew K. Franklin

2012: Elias Koutsoupias, Christos H. Papadimitriou, Tim Roughgarden, Éva Tardos, Noam Nisan, and Amir Ronen

2011: Johan T. Håstad

2010: Sanjeev Arora and Joseph S. B. Mitchell

2009: Omer Reingold, Salil Vadhan, and Avi Wigderson

2008: Dan Spielman and Shang-Hua Teng

2007: Alexander A. Razborov and Steven Rudich

2006: Manindra Agrawal, Neeraj Kayal, and Nitin Saxena

2005: Noga Alon, Yossi Matias and Mario Szegedy

2004: Maurice Herlihy and Nir Shavit / Michael Saks and Fotios Zaharoglou

2003: Yoav Freund and Robert Schapire

2002: Géraud Sénizergues

2001: Sanjeev Arora, Uriel Feige, Shafi Goldwasser, Carsten Lund, László Lovász, Rajeev Motwani, Shmuel Safra, Madhu Sudan, and Mario Szegedy

2000: Moshe Vardi and Pierre Wolper

1999: Peter W. Shor

1998: Seinosuke Toda

1997: Joseph Halpern and Yoram Moses

1996: Mark Jerrum and Alistair Sinclair

1995: Neil Immerman and Róbert Szelepcsényi

1994: Johan Håstad

1993: László Babai, Shafi Goldwasser, Silvio Micali, Shlomo Moran, and Charles Rackoff

# 論文一覧

#### 2020: Robin A. Moser and Gábor Tardos

The 2020 Gödel Prize is awarded to Robin A. Moser and Gábor Tardos for their algorithmic version of the Lovász Local Lemma in the paper:

“A constructive proof of the general Lovász Local Lemma,” Journal of the ACM 57(2): 11:1-11:15 (2010).

https://arxiv.org/pdf/0903.0544.pdf, 2009

#### References

[Knu69] Donald E. Knuth. The Art of Computer Programming, Vol. I, Addison Wesley, London, 1969, p. 396 (Exercise 11).

[EL75] Paul Erdo ̋s and La ́szl ́o Lov ́asz. Problems and results on 3-chromatic hypergraphs and some related ques- tions. In A. Hajnal, R. Rado and V.T. So ́s, editors, Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdo ̋s on his 60th birthday), volume II, pages 609–627. North-Holland, 1975.

[ES91] Paul Erdo ̋s and Joel Spencer. Lopsided Lova ́sz Local Lemma and Latin Transversals. Discrete Applied Mathematics, 30:151-154, 1991.

[Bec91] J ́oszef Beck. An Algorithmic Approach to the Lova ́sz Local Lemma. Random Structures and Algorithms, 2(4):343–365, 1991.

[Alo91] Noga Alon. A parallel algorithmic version of the local lemma. Random Structures and Algorithms, 2(4):367–378, 1991.

[KST93] JanKratochv ́ılandPetrSavicky ́andZsoltTuza.Onemoreoccurrenceofvariablesmakessatisfiability jump from trivial to NP-complete. SIAM J. Comput., Vol. 22, No. 1, pp. 203–210, 1993.

[MR98] Michael Molloy and Bruce Reed. Further Algorithmic Aspects of the Local Lemma. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing, pages 524–529, 1998.

[CS00] Artur Czumaj and Christian Scheideler. Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lova ́sz local lemma. Symposium on Discrete Algorithms, 30–39, 2000.

[BKS03] Piotr Berman, Marek Karpinski, and Alexander D. Scott. Approximation hardness and satisfiability of bounded occurrence instances of SAT. Electronic Colloquium on Computational Complexity (ECCC), 10(022), 2003.

[Mos06] Robin A. Moser. On the Search for Solutions to Bounded Occurrence Instances of SAT. Not published. Semester Thesis, ETH Zu ̈rich. 2006.

[PT09] J ́anos Pach and G ́abor Tardos. Conflict-free colorings of graphs and hypergraphs. Manuscript, 2009.

[Sri08] Aravind Srinivasan. Improved algorithmic versions of the Lova ́sz Local Lemma. Proceedings of the nine- teenth annual ACM-SIAM symposium on Discrete algorithms (SODA), San Francisco, California, pp. 611–620, 2008.

[Wel08] Emo Welzl. Boolean Satisfiability - Combinatorics and Algorithms. Lecture notes, Version Fall 2008.

[Mos08] Robin A. Moser. Derandomizing the Lova ́sz Local Lemma more Effectively. Eprint, arXiv:0807.2120v2,

2008.

[Mos09] Robin A. Moser. A constructive proof of the Lova ́sz Local Lemma. Proceedings of the 41st annual ACM Symposium on Theory of Computing (STOC 2009), to appear. Available at arXiv:0810.4812v2, 2008.

####2019: Irit Dinur

https://sigact.org/prizes/gödel/citation2019.pdf

The 2019 Gödel Prize is awarded to Irit Dinur for her proof of the PCP Theorem in the paper:

The PCP theorem by gap amplification,

Journal of the ACM, Vol 54 (3), Article 12, 2007.

(preliminary version in the proceedings of the 38th Symposium on Theory of Computing, STOC 2006).

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=CC1C896ABF24E8D7EA957CD543AAACBB?doi=10.1.1.103.2644&rep=rep1&type=pdf

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#### 2018: Oded Regev

The 2018 Gödel Prize is awarded to Professor Oded Regev for his paper:

On lattices, learning with errors, random linear codes, and cryptography Journal of the ACM, volume 56, issue 6, 2009 (preliminary version in the 37th annual Symposium on Theory of Computing, STOC 2005.)

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.215.3543&rep=rep1&type=pdf

#### 2017: Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith

The 2017 Gödel Prize is awarded to Cynthia Dwork, Frank McSherry, Kobbi Nissim and Adam Smith for their work on Differential Privacy in the paper:

Calibrating Noise to Sensitivity in Private Data Analysis, Journal of Privacy and Confidentiality, Volume 7, Issue 3, 2016 (preliminary version in Theory of Cryptography, TCC 2006).

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=CC1C896ABF24E8D7EA957CD543AAACBB?doi=10.1.1.131.9267&rep=rep1&type=pdf

(Calibrating Data to Sensitivity in Private Data Analysis https://arxiv.org/pdf/1203.3453.pdf, 2014)

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[2] Dakshi Agrawal and Charu C. Aggarwal. On the design and quantification of

privacy preserving data mining algorithms. In Proceedings of the Twentieth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Sys- tems. ACM, 2001.

[3] Rakesh Agrawal and Ramakrishnan Srikant. Privacy-preserving data mining. In Weidong Chen, Jeffrey F. Naughton, and Philip A. Bernstein, editors, SIGMOD Conference, pages 439–450. ACM, 2000.

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[5] Web page for the Bertinoro CS-Statistics workshop on privacy and confidentiality. Available from http://www.stat.cmu.edu/~hwainer, July 2005.

[6] Avrim Blum, Cynthia Dwork, Frank McSherry, and Kobbi Nissim. Practical privacy: The sulq framework. In PODS, 2005.

[7] Shuchi Chawla, Cynthia Dwork, Frank McSherry, Adam Smith, and Hoeteck Wee. Toward privacy in public databases. In Theory of Cryptography Confer- ence (TCC), pages 363–385, 2005.

[8] Shuchi Chawla, Cynthia Dwork, Frank McSherry, and Kunal Talwar. On the utility of privacy-preserving histograms. In 21st Conference on Uncertainty in Artificial Intelligence (UAI), 2005.

[9] Dorothy E. Denning. Secure statistical databases with random sample queries. ACM Transactions on Database Systems, 5(3):291–315, September 1980.

[10] Irit Dinur and Kobbi Nissim. Revealing information while preserving privacy. In

Proceedings of the Twenty-Second ACM SIGACT-SIGMOD-SIGART Symposium

on Principles of Database Systems, pages 202–210, 2003.

[11] Cynthia Dwork and Kobbi Nissim. Privacy-preserving datamining on vertically

partitioned databases. In Matthew K. Franklin, editor, CRYPTO, volume 3152

of Lecture Notes in Computer Science, pages 528–544. Springer, 2004.

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####2016: Stephen Brookes and Peter W. O’Hearn

https://eatcs.org/index.php/component/content/article/1-news/2280-2016-godel-prize

The 2016 Gödel Prize is awarded to Stephen Brookes and Peter W. O'Hearn for their invention of Concurrent Separation Logic, as described in the following two papers:

S. Brookes, A Semantics for Concurrent Separation Logic. Theoretical Computer Science 375(1-3): 227-270 (2007)

https://www.cs.cmu.edu/~brookes/papers/seplogicrevisedfinal.pdf

#### References

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Benjamin/Cummings, 1991.

[2] L. Birkedal, N. Torp-Smith, and J.C. Reynolds. Local Reasoning about a Copying Garbage Collector. Proc. POPL Conference, Venice, pp. 220- 231, January 2004.

[3] R. Bornat, C. Calcagno, P. W. O’Hearn, and M. Parkinson. Permission accounting in separation logic. Proc. POPL 2005, pp. 59-70.

[4] R. Bornat, C. Calcagno, and P. W. O’Hearn. Local reasoning, separa- tion, and aliasing. Proc. 2nd ACM/SIGPLAN Workshop on Semantics, Program Analysis, and Computing Environments for Memory Manage- ment, SPACE 2004, January 2004.

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P. W. O’Hearn, Resources, Concurrency, and Local Reasoning. Theoretical Computer Science 375(1-3): 271-307 (2007)

http://www0.cs.ucl.ac.uk/staff/p.ohearn/papers/concurrency.pdf

#### References

[1] G. Andrews. Concurrent programming: principles and practice. Benjamin/Cummings, 1991.

[2] A. Banerjee and D. A. Naumann. Representation independence, confinement and access con- trol. In 29th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, 2002.

[3] J. Berdine, C. Calcagno, and P. O’Hearn. A decidable fragment of separation logic. In Proceedings of FSTTCS, pages 97–109, 2004. LNCS 3328.

[4] R. Bornat. Variables as resource in Separation Logic. Proceedings of the 21st Conference on Mathematical Foundations of Computer Science, Springer LNCS, to appear, 2005.

[5] R. Bornat, C. Calcagno, P. O’Hearn, and M. Parkinson. Permission accounting in separation logic. In 32nd POPL, pages 59–70, 2005.

[6] C. Boyapati, R. Lee, and M. Rinard. Ownership types for safe programming: Preventing data races and deadlocks. OOPSLA, 2002.

[7] J. Boyland. Checking interference with fractional permissions. In R. Cousot, editor, Static Analysis: 10th International Symposium, volume 2694 of Lecture Notes in Computer Science, pages 55–72, Berlin, Heidelberg, New York, 2003. Springer.

[8] P. Brinch Hansen. The nucleus of a multiprogramming system. Comm. ACM, 13(4):238–250, 1970.

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[10] P. Brinch Hansen. Operating System Principles. Prentice Hall, 1973.

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#### 2015: Daniel A. Spielman and Shang-Hua Teng

The 2015 Gödel Prize is awarded to Daniel A. Spielman and Shang-Hua Teng for their series of papers on nearly-linear-time Laplacian solvers:

Spectral sparsification of graphs. SIAM J. Computing 40:981-1025, 2011.

https://arxiv.org/pdf/0808.4134.pdf

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Dengyong Zhou, Olivier Bousquet, Thomas Navin Lal, Jason Weston, and Bernhard Scho ̈lkopf. Learning with local and global consistency. In Adv. in Neural Inf. Proc. Sys. 16, pages 321–328, 2003.

Xiaojin Zhu, Zoubin Ghahramani, and John D. Lafferty. Semi-supervised learning using Gaussian fields and harmonic functions. In Proc. 20th Int. Conf. on Mach. Learn., 2003.

Fuzhen Zhang, editor. The Schur Complement and its Applications, volume 4 of Numerical Methods and Algorithms. Springer, 2005.

####2014: Ronald Fagin, Amnon Lotem, and Moni Naor

Optimal Aggregation Algorithms for Middleware

https://arxiv.org/pdf/cs/0204046.pdf

####2013: Antoine Joux, Dan Boneh, and Matthew K. Franklin

A One Round Protocol for Tripartite Diffie–Hellman

Antoine Joux, Journal of Cryptology volume 17, pages263–276(2004)

https://link.springer.com/content/pdf/10.1007/s00145-004-0312-y.pdf

Identity-Based Encryption from the Weil Pairing

Dan Boneh∗ Matthew Franklin

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.66.1131&rep=rep1&type=pdf

Nisan, Noam; Ronen, Amir (2001). "Algorithmic Mechanism Design". Games and Economic Behavior. 35 (1–2): 166–196.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.21.1731&rep=rep1&type=pdf

####2012: Elias Koutsoupias, Christos H.Papadimitriou, Tim Roughgarden, Éva Tardos, Noam Nisan, and Amir Ronen

Koutsoupias, Elias; Papadimitriou, Christos (2009). "Worst-case equilibria". Computer Science Review. 3 (2): 65–69. doi:10.1016/j.cosrev.2009.04.003.

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=08343A18C75A6F5A0E57B25562499E5C?doi=10.1.1.102.6991&rep=rep1&type=pdf

Roughgarden, Tim; Tardos, Éva (2002). "How bad is selfish routing?". Journal of the ACM. 49 (2): 236–259. CiteSeerX 10.1.1.147.1081. doi:10.1145/506147.506153.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.147.1081&rep=rep1&type=pdf

Nisan, Noam; Ronen, Amir (2001). "Algorithmic Mechanism Design". Games and Economic Behavior. 35 (1–2): 166–196. CiteSeerX 10.1.1.21.1731. doi:10.1006/game.1999.0790.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.21.1731&rep=rep1&type=pdf

####2011: Johan T. Håstad

https://sigact.org/prizes/gödel/2011.html

The 2011 Gödel Prize is awarded to Johan T. Håstad for his paper:

Some optimal inapproximability results, Journal of the ACM, 48: 798--859, 2001.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.638.2808&rep=rep1&type=pdf

####2010: Sanjeev Arora and Joseph S. B. Mitchell

https://sigact.org/prizes/gödel/2010.html

The 2010 Gödel Prize is awarded to

Sanjeev Arora and Joseph S.B. Mitchell for their concurrent discovery of a polynomial-time approximation scheme (PTAS) for the Euclidean Travelling Salesman Problem (ETSP):

Sanjeev Arora. (1998). Polynomial-time approximation schemes for Euclidean TSP and other geometric problems, Journal ACM 45(5), 753-782.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.23.6765&rep=rep1&type=pdf

Joseph S.B. Mitchell (1999). Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems, SIAM J. Computing 28(4), 1298-1309.

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=EBC1BCCD86A8ECD59B018C11236ECA89?doi=10.1.1.56.4805&rep=rep1&type=pdf

####2009: Omer Reingold, Salil Vadhan, and Avi Wigderson

https://sigact.org/prizes/gödel/2009.html

The 2009 Gödel Prize for outstanding papers in the area of theoretical computer science is awarded to

(1) Entropy waves, the zig-zag graph product and new constant degree expanders. Omer Reingold, Salil Vadhan and Avi Wigderson, Annals of Mathematics, Vol 155, (2002), 157-187

https://arxiv.org/pdf/math/0406038.pdf

and

(2) Undirected connectivity in Log-Space. Omer Reingold, Journal of ACM, Vol 55 (4) (2007).

- Requested.

####2008: Dan Spielman and Shang-Hua Teng

https://sigact.org/prizes/gödel/2008.html

The 2008 Gödel Prize for outstanding papers in the area of theoretical computer science is awarded to

Daniel A. Spielman and Shang-Hua Teng for their paper: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time by Daniel A. Spielman and Shang-Hua Teng, Journal of the ACM (JACM), 51(3), May 2004, 385-463.

First presented at the Annual ACM Symposium on the Theory of Computing (STOC 01), 2001, 296-305.

https://arxiv.org/pdf/math/0212413.pdf

####2007: Alexander A. Razborov and Steven Rudich

https://sigact.org/prizes/gödel/2007.html

The 2007 Gödel Prize for outstanding papers in the area of theoretical computer science is awarded to Alexander A. Razborov and Steven Rudich for their paper:

Natural Proofs, Journal of Computer and System Sciences, Vol. 55, No. 1, 1997, pp. 24-35.

First presented at the Twenty-sixth Annual ACM Symposium on Theory of computing, Montreal, Quebec, Canada. 1994, pp. 204-213.

https://reader.elsevier.com/reader/sd/pii/S002200009791494X?token=9E8D5C0FAA5F8E450DFB32BDEDED9B07CACDC4BD40EA21B35A33BF9E74A24E840B6AF2E6FF1C20F6ED02778CA1FC109C

####2006: Manindra Agrawal, Neeraj Kayal, and Nitin Saxena

https://sigact.org/prizes/gödel/2006.html

The 2006 Gödel Prize for outstanding papers in the area of theoretical computer science is awarded to Manindra Agrawal, Neeraj Kayal, and Nitin Saxena for their paper:

PRIMES is in P. Annals of Mathematics 160(2), 781-793, 2004.

https://web.archive.org/web/20110607101302/http://math.berkeley.edu/~coleman/Courses/Fall08/Cryptography/primality_v6.pdf

#### 2005: Noga Alon, Yossi Matias and Mario Szegedy

The 2005 Gödel Prize for outstanding papers in the area of theoretical computer science is awarded to

Noga Alon, Yossi Matias and Mario Szegedy for their paper: The space complexity of approximating the frequency moments. Journal of Computer and System Sciences 58 (1999), 137-147 First presented at the 28th ACM Symposium on Theory of Computing, 1996.

http://www.math.tau.ac.il/~nogaa/PDFS/amsz4.pdf

#### 2004: Maurice Herlihy and Nir Shavit / Michael Saks and Fotios Zaharoglou

The 2004 Gödel Prize for outstanding journal articles in theoretical computer science is shared between the papers:

"The Topological Structure of Asynchronous Computation" by Maurice Herlihy and Nir Shavit, Journal of the ACM, Vol. 46 (1999), 858-923,

http://cs.brown.edu/~mph/HerlihyS99/p858-herlihy.pdf

and

"Wait-Free k-Set Agreement Is Impossible: The Topology of Public Knowledge" by Michael Saks and Fotios Zaharoglou, SIAM J. on Computing, Vol. 29 (2000), 1449-1483.

- Requested.

#### 2003: Yoav Freund and Robert Schapire

The prize was awarded to

Yoav Freund and Robert Schapire for their paper "A Decision Theoretic Generalization of On-Line Learning and an Application to Boosting," Journal of Computer and System Sciences 55 (1997), pp. 119-139.

https://www-ai.cs.tu-dortmund.de/LEHRE/PG/PG445/literatur/freund_schapire_97a.pdf

#### 2002: Géraud Sénizergues

Geraud Senizergues, for "L(A)=L(B)? Decidability results from complete formal systems", Theoretical Computer Science 251 (2001), pp.1-166.

https://reader.elsevier.com/reader/sd/pii/S0304397500002851?token=64A2B37155D426438B12D3E16D9DA7D657C01E4E4944CA307F3319EB3724727119FF108137109C77E438B12E754C1179

#### 2001: Sanjeev Arora, Uriel Feige, Shafi Goldwasser, Carsten Lund, László Lovász, Rajeev Motwani, Shmuel Safra, Madhu Sudan, and Mario Szegedy

The 2001 Gödel Prize for outstanding journal article in the area of theoretical computer science is awarded to

Sanjeev Arora, Uriel Feige, Shafi Goldwasser, Carsten Lund, Laszlo Lovász, R. Motwani, Shmuel Safra, Madhu Sudan, and Mario Szegedy for the following series of papers:

"Interactive Proofs and the Hardness of Approximating Cliques". Uriel Feige, Shafi Goldwasser, Laszlo Lovász, Shmuel Safra, and Mario Szegedy. Journal of the ACM 43 (1996), 268-292.

https://www.researchgate.net/profile/Mario_Szegedy/publication/234783217_Interactive_proofs_and_the_hardness_of_approximating_cliques/links/09e41514878e8c2f78000000/Interactive-proofs-and-the-hardness-of-approximating-cliques.pdf

"Probabilistic Checking of Proofs: A New Characterization of NP". Sanjeev Arora and Shmuel Safra. Journal of the ACM 45 (1998), 70-122.

https://web.archive.org/web/20110610101051/http://www.cs.umd.edu/~gasarch/pcp/AS.pdf

"Proof Verification and the Hardness of Approximation Problems". Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Journal of the ACM 45 (1998), 501-555.

https://web.archive.org/web/20110610101241/https://www.cs.umd.edu/~gasarch/pcp/ALMSS.pdf

#### 2000: Moshe Vardi and Pierre Wolper

The 2000 Gödel Prize is awarded to

Moshe Vardi and Pierre Wolper for their paper:

Reasoning about infinite computations. Moshe Y. Vardi, Pierre Wolper. Information and Computation 115 (1994), 1-37.

https://web.archive.org/web/20110825210914/http://reference.kfupm.edu.sa/content/r/e/reasoning_about_infinite_computations__102167.pdf

#### 1999: Peter W. Shor

The 1999 Gödel Prize for outstanding journal article in the area of theoretical computer science is awarded to

Peter W. Shor for his paper "Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer", SIAM Journal on Computing 26 (1997), 1484-1509.

https://arxiv.org/pdf/quant-ph/9508027.pdf

#### 1998: Seinosuke Toda

The 1998 Gödel Prize for outstanding journal article in the area of theoretical computer science will be awarded to

Seinosuke Toda for his paper "PP is as Hard as the Polynomial-Time Hierarchy," SIAM Journal on Computing 20 (1991), 865-877.

https://people.engr.tamu.edu/j-chen3/courses/637/2008/pres/korben.pdf

#### 1997: Joseph Halpern and Yoram Moses

The 1997 Gödel Prize for outstanding journal articles in the area of theoretical computer science will be awarded to

Joseph Halpern and Yoram Moses for their paper "Knowledge and Common Knowledge in a Distributed Environment," J. ACM 37} (1990), 549-587.

https://www.cs.cornell.edu/home/halpern/papers/common_knowledge.pdf

#### 1996: Mark Jerrum and Alistair Sinclair

The 1996 Gödel Prize for outstanding journal articles in the area of theoretical computer science was awarded on May 23, 1996 jointly to

Mark Jerrum and Alistair Sinclair for their papers "Approximate counting, unform generation and rapidly mixing Markov chains," Information and Computation 82 (1989), 93-133, by Sinclair and Jerrum,

https://reader.elsevier.com/reader/sd/pii/0890540189900679?token=518EF4273B015187831CB0A883A601D699224964F2F448061BEE0B856FC0FE6A24FCDD1567B81553115DC281EB962793

and

"Approximating the permanent," SIAM Journal on Computing 18 (1989), 1149-1178, by Jerrum and Sinclair.

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.431.4190&rep=rep1&type=pdf

#### 1995: Neil Immerman and Róbert Szelepcsényi

The Gödel prize committee has selected the co-recipients of the Gödel Prize for 1995. The prize is awarded for the following two papers:

Neil Immerman, "Nondeterministic space is closed under complementation," SIAM Journal on Computing 17 (1988), 935-938

https://people.cs.umass.edu/~immerman/pub/space.pdf

and

Róbert Szelepcsényi, "The method of forced enumeration for nondeterministic automata," Acta Informatica 26 (1988), 279-284.

https://dml.cz/bitstream/handle/10338.dmlcz/120489/ActaOstrav_03-1995-1_10.pdf

#### 1994: Johan Håstad

Johan Håstad, "Almost optimal lower bounds for small depth circuits," Advances in Computing Research 5 (1989), 143-170.

https://web.archive.org/web/20120222163102/http://reference.kfupm.edu.sa/content/a/l/almost_optimal_lower_bounds_for_small_de_134215.pdf

#### 1993: László Babai, Shafi Goldwasser, Silvio Micali, Shlomo Moran, and Charles Rackoff

The 1993 Gödel Prize was shared by two papers,

László Babai and Shlomo Moran, "Arthur-Merlin games: a randomized proof system and a hierarchy of complexity classes," Journal of Computer and System Sciences, 36 (1988), 254-276,

http://crypto.cs.mcgill.ca/~crepeau/COMP647/2007/TOPIC01/AMgames-Babai-Moran.pdf

and

Shafi Goldwasser, Silvio Micali, and Charles Rackoff, "The knowledge complexity of interactive proof systems," SIAM Journal on Computing 18 (1989), 186-208.

http://crypto.cs.mcgill.ca/~crepeau/COMP647/2007/TOPIC02/GMR89.pdf

#作業場(work space)

まずゲーテルの論文の単語帳。

自分のPCにあるフォルダ名が

/Users/administrator/Downloads/godel

の場合

```
$ docker run -v /Users/administrator/Downloads/godel:/tmp/godel -it kaizenjapan/godel /bin/bash
```

いつもは上位１００語を表示する。

数式が多く、１語は式でしか現れないかもしれない。

１０語以上だと上位１３６位まで。

整列は表計算ソフトに読み込んで並べ替えをした。

work | count |
---|---|

the | 413 |

x | 395 |

of | 255 |

a | 237 |

is | 211 |

n | 182 |

y | 159 |

and | 146 |

to | 120 |

that | 119 |

for | 112 |

v | 106 |

in | 102 |

r | 80 |

we | 77 |

by | 75 |

i | 73 |

z | 71 |

not | 69 |

are | 66 |

number | 63 |

this | 59 |

p | 58 |

formula | 55 |

it | 51 |

provable | 49 |

q | 48 |

primitive | 47 |

one | 46 |

as | 44 |

be | 43 |

if | 43 |

with | 43 |

forall | 41 |

sign | 39 |

axiom | 37 |

length | 37 |

class | 36 |

k | 36 |

free | 35 |

item | 35 |

or | 35 |

recursive | 35 |

from | 34 |

subst | 34 |

which | 34 |

there | 33 |

an | 32 |

proof | 32 |

relation | 31 |

type | 31 |

system | 30 |

theorem | 30 |

formulae | 29 |

variable | 29 |

all | 28 |

numbers | 28 |

where | 28 |

will | 28 |

can | 27 |

e | 27 |

you | 27 |

axioms | 26 |

on | 26 |

s | 25 |

xn | 25 |

b | 24 |

have | 23 |

every | 22 |

natural | 22 |

paper | 22 |

succ | 22 |

other | 21 |

such | 21 |

but | 19 |

following | 19 |

sequence | 19 |

go | 18 |

translation | 18 |

would | 18 |

also | 17 |

rq | 17 |

u | 17 |

consistency | 16 |

consistent | 16 |

del | 16 |

then | 16 |

basic | 15 |

finite | 15 |

text | 15 |

undecidable | 15 |

variables | 15 |

any | 14 |

defined | 14 |

function | 14 |

has | 14 |

pm | 14 |

seq | 14 |

signs | 14 |

argmin | 13 |

because | 13 |

call | 13 |

concepts | 13 |

does | 13 |

now | 13 |

only | 13 |

first | 12 |

formal | 12 |

holds | 12 |

more | 12 |

no | 12 |

prooffor | 12 |

used | 12 |

was | 12 |

at | 11 |

decision | 11 |

define | 11 |

fact | 11 |

functions | 11 |

its | 11 |

let | 11 |

schema | 11 |

set | 11 |

so | 11 |

systems | 11 |

these | 11 |

c | 10 |

example | 10 |

hand | 10 |

my | 10 |

obtained | 10 |

proposition | 10 |

quantor | 10 |

relations | 10 |

some | 10 |

well | 10 |

## 受賞論文

２論文が未入手、２論文が事前投稿論文。

式の中に２文字の記号を含むと仮定すると

676語は式中の記号の可能性がある。

ひとまず出現頻度が100までの683語を示す。

残りは、最後に記載したのDocker Hub上にファイルあり。

```
./ptt.sh
cat text/* > all.txt
awk -f wc.awk all.txt > wc.txt
```

word | count | note |
---|---|---|

the | 29214 | |

a | 19144 | |

of | 18139 | |

and | 11769 | |

is | 11715 | |

in | 10220 | |

y | 9933 | |

f | 9700 | |

i | 9139 | |

to | 9012 | |

x | 8913 | |

that | 8803 | |

b | 8657 | |

s | 8465 | |

n | 8129 | |

h | 7348 | |

we | 7335 | |

u | 7322 | |

for | 6999 | |

q | 6720 | |

p | 5916 | |

c | 5906 | |

g | 5442 | |

k | 5207 | |

m | 4554 | |

e | 4407 | |

v | 4307 | |

by | 4258 | |

t | 4257 | |

d | 4090 | |

be | 4043 | |

j | 4022 | |

this | 3882 | |

r | 3542 | |

l | 3330 | |

as | 3196 | |

with | 3119 | |

on | 3094 | |

an | 2935 | |

if | 2741 | |

it | 2565 | |

are | 2553 | |

let | 2399 | |

then | 2198 | |

at | 2137 | |

w | 2051 | |

proof | 2000 | |

from | 1929 | |

can | 1921 | |

z | 1873 | |

lemma | 1717 | |

which | 1700 | |

o | 1692 | |

not | 1646 | |

all | 1577 | |

such | 1562 | |

have | 1509 | |

each | 1419 | |

log | 1411 | |

set | 1407 | |

theorem | 1394 | |

where | 1315 | |

there | 1278 | |

algorithm | 1201 | |

some | 1187 | |

one | 1186 | |

random | 1172 | |

time | 1143 | |

any | 1132 | |

will | 1130 | |

our | 1090 | |

probability | 1053 | |

or | 1041 | |

every | 1019 | |

has | 998 | |

_ | 993 | |

polynomial | 959 | |

number | 957 | |

case | 920 | |

two | 895 | |

given | 887 | |

only | 883 | |

bi | 856 | |

since | 844 | |

graph | 825 | |

so | 823 | |

function | 821 | |

most | 802 | |

problem | 776 | |

these | 743 | |

when | 739 | |

using | 722 | |

section | 719 | |

also | 715 | |

its | 714 | |

follows | 710 | |

de | 696 | |

first | 683 | |

now | 682 | |

use | 677 | |

following | 661 | |

size | 649 | |

least | 645 | |

us | 639 | |

computer | 635 | |

input | 624 | |

ii | 616 | |

than | 615 | |

thus | 611 | |

value | 611 | |

system | 599 | |

more | 594 | |

over | 594 | |

but | 592 | |

no | 589 | |

show | 588 | |

see | 587 | |

linear | 575 | |

hence | 548 | |

prove | 543 | |

state | 543 | |

other | 529 | |

definition | 518 | |

same | 512 | |

pp | 511 | |

note | 509 | |

xu | 507 | |

science | 505 | |

np | 504 | |

above | 503 | |

may | 503 | |

free | 495 | |

ca | 477 | |

do | 475 | |

variables | 459 | |

vertex | 459 | |

constant | 457 | |

result | 448 | |

bound | 446 | |

consider | 442 | |

new | 441 | |

problems | 435 | |

algorithms | 433 | |

TRUE | 429 | |

approximation | 428 | |

verifier | 425 | |

paper | 424 | |

does | 421 | |

protocol | 421 | |

assume | 413 | |

deterministic | 411 | |

was | 409 | |

step | 407 | |

vol | 407 | |

exists | 398 | |

between | 397 | |

degree | 397 | |

th | 395 | |

complexity | 394 | |

denote | 391 | |

point | 390 | |

xi | 390 | |

output | 389 | |

define | 384 | |

pr | 384 | |

si | 383 | |

edges | 381 | |

example | 380 | |

edge | 379 | |

resource | 379 | |

used | 376 | |

fact | 373 | |

theory | 371 | |

values | 371 | |

must | 366 | |

graphs | 361 | |

bits | 357 | |

oracle | 357 | |

both | 351 | |

max | 351 | |

length | 349 | |

assignment | 345 | |

functions | 343 | |

their | 342 | |

form | 339 | |

key | 339 | |

would | 336 | |

they | 335 | |

process | 334 | |

results | 331 | |

computing | 328 | |

matrix | 327 | |

work | 327 | |

defined | 325 | |

acm | 324 | |

condition | 323 | |

under | 318 | |

list | 316 | |

proofs | 316 | |

test | 316 | |

possible | 315 | |

way | 314 | |

min | 313 | |

lower | 311 | |

systems | 311 | |

suppose | 307 | |

mod | 306 | |

property | 306 | |

here | 304 | |

cost | 303 | |

small | 303 | |

order | 301 | |

instance | 300 | |

into | 300 | |

get | 298 | |

sequence | 297 | |

tree | 296 | |

how | 294 | |

distribution | 292 | |

second | 291 | |

well | 291 | |

whose | 289 | |

un | 288 | |

emp | 287 | |

wi | 287 | |

non | 286 | |

however | 285 | |

obtain | 283 | |

ai | 281 | |

implies | 281 | |

ui | 280 | |

within | 280 | |

been | 279 | |

vi | 279 | |

need | 278 | |

vertices | 277 | |

map | 275 | |

factor | 273 | |

quantum | 273 | |

based | 272 | |

access | 271 | |

complex | 271 | |

part | 270 | |

optimal | 269 | |

model | 267 | |

points | 266 | |

even | 265 | |

independent | 265 | |

analysis | 264 | |

make | 261 | |

nite | 261 | |

symposium | 261 | |

about | 260 | |

simplex | 260 | |

vector | 260 | |

al | 259 | |

because | 259 | |

bounded | 256 | |

path | 253 | |

proceedings | 253 | |

computation | 252 | |

sum | 252 | |

yjv | 252 | |

say | 251 | |

sets | 251 | |

string | 251 | |

variable | 251 | |

class | 250 | |

program | 248 | |

corresponding | 246 | |

large | 246 | |

main | 245 | |

while | 245 | |

hard | 244 | |

ud | 244 | |

udc | 244 | |

depth | 243 | |

automata | 241 | |

known | 241 | |

procedure | 241 | |

automaton | 240 | |

construction | 239 | |

natural | 236 | |

sj | 236 | |

accepts | 235 | |

bit | 235 | |

bounds | 235 | |

either | 234 | |

formula | 234 | |

ta | 234 | |

give | 233 | |

theoretical | 233 | |

particular | 232 | |

logic | 231 | |

object | 231 | |

objects | 228 | |

general | 227 | |

check | 226 | |

nition | 225 | |

last | 223 | |

many | 221 | |

chosen | 220 | |

space | 220 | |

proposition | 218 | |

udy | 218 | |

otherwise | 217 | |

simple | 217 | |

zero | 216 | |

et | 215 | |

notion | 213 | |

states | 213 | |

ratio | 212 | |

sorted | 212 | |

properties | 211 | |

local | 210 | |

product | 210 | |

interactive | 209 | |

parallel | 209 | |

call | 208 | |

approximate | 207 | |

equations | 207 | |

complete | 206 | |

full | 206 | |

claim | 204 | |

di | 204 | |

holds | 203 | |

integer | 203 | |

ti | 203 | |

find | 202 | |

finite | 202 | |

next | 202 | |

ne | 201 | |

satisfying | 200 | |

inequality | 199 | |

pi | 199 | |

similar | 199 | |

up | 199 | |

xud | 199 | |

ia | 198 | |

close | 197 | |

just | 197 | |

mechanism | 197 | |

poly | 196 | |

annual | 195 | |

hhv | 195 | |

processes | 195 | |

recall | 195 | |

circuit | 193 | |

gives | 193 | |

unsat | 193 | |

left | 192 | |

weight | 192 | |

choose | 191 | |

conditions | 191 | |

task | 191 | |

therefore | 190 | |

very | 190 | |

flow | 188 | |

ned | 188 | |

corollary | 187 | |

times | 187 | |

arbitrary | 186 | |

called | 185 | |

line | 185 | |

standard | 185 | |

subdivision | 185 | |

top | 185 | |

proc | 183 | |

compute | 182 | |

cannot | 181 | |

queries | 181 | |

uniform | 181 | |

contains | 180 | |

expected | 180 | |

structure | 180 | |

ieee | 179 | Institute of Electrical and Electronics Engineers |

probabilistic | 179 | |

running | 179 | |

chain | 178 | |

languages | 178 | |

enizergues | 177 | human name |

induction | 177 | |

language | 177 | |

group | 176 | |

shown | 176 | |

reduction | 175 | |

type | 175 | |

code | 174 | |

distance | 174 | |

information | 174 | |

obtained | 174 | |

required | 174 | |

rule | 173 | |

agent | 172 | |

describe | 172 | |

without | 171 | |

after | 170 | |

construct | 170 | |

figure | 170 | |

exactly | 169 | |

should | 169 | |

assumption | 168 | |

cau | 168 | |

could | 168 | |

hypothesis | 168 | |

satis | 168 | |

what | 168 | |

almost | 167 | |

positive | 167 | |

matrices | 166 | |

three | 166 | |

uses | 166 | |

grade | 165 | |

satisfies | 165 | |

described | 164 | |

below | 162 | |

fixed | 162 | |

good | 162 | |

nu | 162 | |

out | 162 | |

ri | 162 | |

solution | 162 | |

version | 162 | |

knowledge | 160 | |

pair | 160 | |

terms | 160 | |

counting | 159 | |

different | 159 | |

whether | 159 | |

alphabet | 158 | |

connected | 158 | |

constraint | 158 | |

ji | 158 | |

siam | 158 | Society for Industrial and Applied Mathematics |

vectors | 158 | |

choice | 157 | |

clearly | 157 | |

efficient | 157 | |

sat | 157 | |

steps | 157 | |

agents | 156 | |

cs | 156 | computer science |

inputs | 156 | |

pages | 156 | |

programs | 156 | |

those | 156 | |

public | 154 | |

shows | 153 | |

special | 152 | |

them | 152 | |

easy | 151 | |

fraction | 151 | |

less | 151 | |

proved | 151 | |

approximating | 150 | |

pcp | 150 | |

apply | 149 | |

decision | 149 | |

high | 149 | |

notation | 149 | |

right | 149 | |

scheme | 149 | |

similarly | 149 | |

am | 147 | |

before | 147 | |

prover | 147 | |

abort | 146 | |

cases | 146 | |

were | 146 | |

another | 145 | |

cut | 145 | |

finally | 145 | |

know | 145 | |

through | 145 | |

discrete | 144 | |

id | 144 | identifier |

method | 144 | |

arora | 143 | |

au | 142 | |

outputs | 142 | |

constraints | 141 | |

ei | 141 | |

error | 141 | |

view | 141 | |

elements | 140 | |

makes | 140 | |

rst | 140 | |

regular | 139 | |

rules | 139 | |

subset | 139 | |

takes | 139 | |

fp | 138 | |

lattice | 138 | |

memory | 138 | |

optimality | 138 | |

prime | 138 | |

much | 137 | |

relation | 137 | |

requires | 136 | |

single | 136 | |

again | 135 | |

being | 135 | |

equivalent | 135 | |

series | 135 | |

hand | 134 | |

satisfy | 134 | |

nodes | 133 | |

partition | 133 | |

programming | 133 | |

boolean | 132 | |

gf | 132 | |

observe | 132 | |

st | 132 | |

total | 132 | |

vb | 132 | |

words | 132 | |

field | 131 | |

game | 131 | |

machine | 131 | |

self | 131 | |

write | 131 | |

answer | 130 | |

numbers | 130 | |

power | 130 | |

real | 130 | |

argument | 129 | |

basic | 129 | |

database | 129 | |

done | 129 | |

might | 129 | |

owned | 129 | |

run | 129 | |

take | 129 | |

able | 128 | |

accept | 128 | |

curve | 128 | |

seen | 128 | |

springer | 128 | |

start | 128 | |

useful | 128 | |

circuits | 127 | |

equal | 127 | |

moreover | 127 | |

present | 127 | |

always | 126 | |

pairing | 126 | |

entries | 125 | |

markov | 125 | |

put | 125 | |

accesses | 124 | |

comput | 124 | |

level | 124 | |

merlin | 124 | |

original | 124 | |

remark | 124 | |

context | 123 | |

combinatorial | 122 | |

dom | 122 | |

further | 122 | |

idea | 122 | |

nd | 122 | |

tsp | 122 | |

ci | 121 | |

cr | 121 | |

execution | 121 | |

notes | 121 | |

simplicial | 121 | |

xk | 121 | |

aw | 120 | |

exist | 120 | |

low | 120 | |

randomized | 120 | |

rather | 120 | |

aggregation | 119 | |

applications | 119 | |

formal | 119 | |

notice | 119 | |

operations | 119 | |

exp | 118 | |

family | 118 | |

inv | 118 | |

perfect | 118 | |

query | 118 | |

sense | 118 | |

upper | 118 | |

aq | 117 | |

bk | 117 | |

restriction | 117 | |

square | 117 | |

trace | 117 | |

transition | 117 | |

approach | 116 | |

equation | 116 | |

polynomials | 116 | |

root | 116 | |

heap | 115 | |

word | 115 | |

conclude | 113 | |

encoding | 113 | |

hardness | 113 | |

made | 113 | |

maximum | 113 | |

previous | 113 | |

techniques | 113 | |

uj | 113 | |

element | 112 | |

private | 112 | |

separation | 112 | |

simply | 112 | |

strategy | 112 | |

design | 111 | |

event | 111 | |

journal | 111 | |

lists | 111 | |

nc | 111 | |

writes | 111 | |

arthur | 110 | |

encryption | 110 | |

er | 110 | |

instances | 110 | |

ny | 110 | |

soundness | 110 | |

yj | 110 | |

chromatic | 109 | |

equivalence | 109 | |

generator | 109 | |

li | 109 | |

models | 109 | |

nonzero | 109 | |

respectively | 109 | |

completeness | 108 | |

dl | 108 | |

lecture | 108 | |

security | 108 | |

acceptance | 107 | |

buf | 107 | |

bw | 107 | |

open | 107 | |

checking | 106 | |

classes | 106 | |

iff | 106 | |

minimum | 106 | |

tb | 106 | |

formulas | 105 | |

important | 105 | |

initial | 105 | |

read | 105 | |

side | 105 | |

unique | 105 | |

xn | 105 | |

conductance | 104 | |

latency | 104 | |

paths | 104 | |

appear | 103 | |

bn | 103 | |

boundary | 103 | |

distinct | 103 | |

oracles | 103 | |

sequential | 103 | |

still | 103 | |

strings | 103 | |

accepting | 102 | |

ap | 102 | |

computed | 102 | |

long | 102 | |

node | 102 | |

question | 102 | |

associated | 101 | |

easily | 101 | |

im | 101 | |

udn | 101 | |

gate | 100 | |

like | 100 | |

matching | 100 | |

means | 100 | |

ownership | 100 | |

partial | 100 | |

provided | 100 | |

row | 100 | |

symmetric | 100 | |

weighted | 100 |

## tools

単語出現頻度計数

```
# Print list of word frequencies
# https://researchmap.jp/jomd7nobo-45644/
# https://qiita.com/kaizen_nagoya/items/670d4d332e07fd2e5fc2
# This is for papers, not for programming.
# remove _ from gsub list 20200101 @kaizen_nagoya
{
$0 = tolower($0) #
gsub(/[^a-z \t]/, " ", $0) #
for (i = 1; i <= NF; i++)
freq[$i]++
}
END {
for (word in freq)
printf "%s\t%d\n", word, freq[word]
}
```

一括処理

```
#!/bin/bash
# https://news.mynavi.jp/article/bashonwindows-17/
# https://qiita.com/kaizen_nagoya/items/319672853519990cee42
cd ../pdf
for File in *; do
case ${File##*.} in
pdf|PDF )
echo "Convert the PDF:" ${File}
pdftotext -q ${File} ../text/${File}.txt ;;
*) ;;
esac
done
```

# docker hub登録

docker 起動（最初）

```
$ docker run -v /Users/administrator/Downloads/godel:/tmp/godel -it ubuntu /bin/bash
```

dockerでの作業

```
# apt update; apt -y upgrade
# apt install -y poppler-utils vim wget
# wget http://hirzels.com/martin/papers/canon00-goedel.pdf
# pdftotext -q canon00-goedel.pdf canon00-goedel.txt
# awk -f wc.awk canon00-goedel.txt >godelwc.txt
```

docker hub 登録

```
$ docker commit 7aa98d94d8da kaizenjapan/godel
$ docker push kaizenjapan/godel
```

次回起動

```
$ docker run -v /Users/administrator/Downloads/godel:/tmp/godel -it kaizenjapan/godel /bin/bash
```

# 文書履歴(document history)

ver. 0.01 初稿 20201020

ver. 0.02 ゲーデル論文URL記載 20201024午前

ver. 0.03 ゲーテル論文単語帳 20201024午後

ver. 0.04 単語帳作成、docker登録 20201024 夜

ver. 0.05 Reference 追記 20201025