哥德尔奖

维基百科,自由的百科全书

哥德尔奖(英語:Gödel Prize)是一個頒發給理論計算機科學領域傑出論文的年度獎項,由歐洲理論計算機科學協會英语European Association for Theoretical Computer Science(EATCS)和美國計算機協會算法和計算理論特別興趣小組(計算機協會算法和計算理論特別興趣小組英语ACM SIGACT)聯合頒發。該獎項是為紀念庫爾特·哥德爾而命名的。哥德爾是第一個提出P/NP問題的人,在1956年寫給約翰·馮·諾伊曼的信中,哥德爾問某個NP完全的問題是否可以用二次或是線性時間來解決[1]

哥德爾獎於1993年開始在STOC(ACM計算理論研討會英语Symposium on Theory of Computing,北美理論計算機科學的主要會議之一)或ICALP(自動機、語言和編程國際座談會英语International Colloquium on Automata, Languages and Programming,該領域的主要歐洲會議之一)上頒發。獲獎論文必須在理論計算機領域具有開創性重大貢獻,同時需在獲獎前14年內在學術期刊上正式發表。該獎項包括5,000美元的獎金[2]

哥德爾獎的評審委員會由6名成員組成,分別由EATCS主席和SIGACT主席各提名三名成員,任期三年並交錯進行。委員會由EATCS和SIGACT的代表輪流擔任主席。

获奖者[编辑]

年份 獲獎者 原因 發表時間
1993 拉斯洛·鮑鮑伊英语László Babai莎菲·戈德瓦塞爾希爾維奧·米卡利什洛莫·莫蘭英语Shlomo Moran查爾斯·拉克福英语Charles Rackoff 表彰其開發交互式證明系統 1988[paper 1]
1989[paper 2]
1994 約翰·哈斯塔德英语Johan Håstad 表彰其證明恆深布林電路英语Boolean circuit大小的指數下界(對於奇偶函數英语Parity function 1989[paper 3]
1995 尼爾·伊莫曼英语Neil Immerman羅伯特·塞萊普切尼英语Róbert Szelepcsényi 表彰其證明非決定性空間複雜性的伊莫曼-塞萊普切尼定理英语Immerman–Szelepcsényi theorem 1988[paper 4]
1988[paper 5]
1996 馬克·傑魯姆英语Mark Jerrum阿利斯泰爾·辛克萊爾 表彰其在馬可夫鏈和矩陣積和式的近似方面的工作 1989[paper 6]
1989[paper 7]
1997 約瑟夫·哈爾彭英语Joseph Halpern約拉姆·莫塞斯英语Yoram Moses 表彰其為分佈式環境中的「知識」定義一個正式的概念 1990[paper 8]
1998 戶田誠之助日语戸田誠之助 表彰其證明顯示計數解(PP)和量詞交替(PH)之間的關聯的戶田定理 1991[paper 9]
1999 彼得·秀爾 表彰其開發在量子計算機上以多項式時間計算整數分解秀爾算法 1997[paper 10]
2000 摩西·瓦迪英语Moshe Vardi皮埃爾·沃爾珀英语Pierre Wolper 表彰其在有限狀態機時間邏輯方面的工作 1994[paper 11]
2001 桑吉夫·阿羅拉英语Sanjeev Arora烏列爾·費奇英语Uriel Feige莎菲·戈德瓦塞爾卡斯坦·隆德英语Carsten Lund洛瓦茲·拉茲洛拉耶夫·莫特瓦尼英语Rajeev Motwani什穆埃爾·沙夫拉英语Shmuel Safra邁度·蘇丹英语Madhu Sudan馬里奧·塞格德英语Mario Szegedy 表彰其證明PCP定理英语PCP theorem以及在近似硬度方面的應用 1996[paper 12]
1998[paper 13]
1998[paper 14]
2002 傑霍·賽尼澤格英语Géraud Sénizergues 表彰其證明確定下推自動機的等價性是可決定英语Decidability (logic) 2001[paper 15]
2003 約阿夫·弗羅因德羅伯特·沙皮爾 表彰其開發機器學習中的AdaBoost算法 1997[paper 16]
2004 莫里斯·赫利希英语Maurice Herlihy麥可·薩克斯英语Michael Saks (mathematician)尼爾·沙維特英语Nir Shavit弗蒂奧斯·札哈羅格羅英语Fotios Zaharoglou 表彰其在拓撲學分佈式計算理論中的應用 1999[paper 17]
2000[paper 18]
2005 諾加·阿隆約西·馬蒂亞斯英语Yossi Matias馬里奧·塞格德英语Mario Szegedy 表彰其對Streaming algorithm英语串流演算法的基礎性貢獻 1999[paper 19]
2006 曼寧德拉·阿格拉瓦爾英语Manindra Agrawal尼拉吉·卡亞爾英语Neeraj Kayal尼汀·沙克謝納英语Nitin Saxena 表彰其開發AKS質數測試 2004[paper 20]
2007 亞歷山大·拉茲波洛夫英语Alexander Razborov史蒂芬·魯迪奇英语Steven Rudich 表彰其提出 自然證明英语Natural proof 1997[paper 21]
2008 丹尼爾·斯皮爾曼英语Daniel Spielman滕尚华 表彰其開發算法的平滑分析英语Smoothed analysis 2004[paper 22]
2009 奧馬爾·萊因戈爾德英语Omer Reingold薩利爾·瓦德漢英语Salil Vadhan阿維·威格森 表彰其證明鋸齒乘積英语Zig-zag product對數空間中的無定向連接性 2002[paper 23]
2008[paper 24]
2010 桑吉夫·阿羅拉英语Sanjeev Arora約瑟夫·S·B·米切爾英语Joseph S. B. Mitchell 表彰其發現歐幾里得旅行推銷員問題(ETSP)的多項式時間近似算法(PTAS) 1998[paper 25]
1999[paper 26]
2011 約翰·哈斯塔德英语Johan Håstad 表彰其證明各種組合問題的最佳非近似性結果 2001[paper 27]
2012 埃利亞斯·庫特索皮亞斯英语Elias Koutsoupias赫里斯托斯·帕帕季米特里烏諾姆·尼散、阿米爾·羅能(Amir Ronen)、提姆·羅加登英语Tim Roughgarden陶爾多什·埃娃 表彰其奠定算法博弈論英语Algorithmic game theory的基礎[3] 2009[paper 28]
2001[paper 29]
2002[paper 30]
2013 丹·博内馬修·K·富蘭克林英语Matthew K. Franklin安托萬·朱斯英语Antoine Joux 表彰其開發多方迪菲-赫爾曼密鑰交換和密碼學中的博內-富蘭克林法英语Boneh–Franklin scheme[4] 2003[paper 31]
2004[paper 32]
2014 羅納德·法金英语Ronald Fagin、阿姆農·洛特姆(Amnon Lotem)、莫尼·瑙爾英语Moni Naor 表彰其開發中介軟體的最佳集合算法[5] 2003[paper 33]
2015 丹尼爾·斯皮爾曼英语Daniel Spielman滕尚华 表彰其開發近線性時間的拉普拉斯求解器[6] 2011[paper 34]
2013[paper 35]
2014[paper 36]
2016 史蒂芬·布魯克斯(Stephen Brookes)、彼得·歐赫恩英语Peter O'Hearn 表彰其開發並行分離邏輯英语Separation logic 2007[paper 37]
2007[paper 38]
2017[2] 辛西婭·德沃克英语Cynthia Dwork弗蘭克·麥克謝里英语Frank McSherry科比·尼西姆英语Kobbi Nissim亞當·D·史密斯英语Adam D. Smith 表彰其開發差分隱私 2006[paper 39]
2018[7] 歐德德·瑞格夫英语Oded Regev (computer scientist) 表彰其介紹容錯學習問題 2009[paper 40]
2019[8] 伊里特·迪努爾英语Irit Dinur 表彰其利用間隙放大法重新證明PCP定理英语PCP theorem 2007[paper 41]
2020[9] 羅賓·莫瑟(Robin Moser)、陶爾多什·加博爾英语Gábor Tardos 表彰其對拉茲洛局部定理英语Lovász local lemma建設性證明英语Algorithmic Lovász local lemma 2010[paper 42]
2021[10] 安德烈·布拉托夫(Andrei Bulatov)、蔡进一英语Jin-Yi Cai陈汐英语Xi Chen馬丁·戴爾英语Martin Dyer、大衛·里切爾比(David Richerby) 表彰其在約束滿足問題計數複雜性英语Counting problem (complexity)分類方面的工作 2013[paper 43]
2013[paper 44]
2017[paper 45]
2022[11] 茲維卡·布拉克斯基(Zvika Brakerski)、克雷格·金特里英语Craig Gentry (computer scientist)、維諾德·維昆塔森(Vinod Vaikuntanathan) 表彰其通過構建高效的全同態加密(FHE)法對密碼學的變革性貢獻 2014[paper 46]
2014[paper 47]

獲獎論文[编辑]

  1. ^ Babai, László; Moran, Shlomo, Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity class (PDF), Journal of Computer and System Sciences, 1988, 36 (2): 254–276 [2022-10-31], ISSN 0022-0000, doi:10.1016/0022-0000(88)90028-1可免费查阅, (原始内容存档 (PDF)于2011-07-06) 
  2. ^ Goldwasser, S.; Micali, S.; Rackoff, C., The knowledge complexity of interactive proof systems (PDF), SIAM Journal on Computing, 1989, 18 (1): 186–208 [2022-10-31], CiteSeerX 10.1.1.397.4002可免费查阅, ISSN 1095-7111, doi:10.1137/0218012, (原始内容存档 (PDF)于2011-09-27) 
  3. ^ Håstad, Johan, Almost Optimal Lower Bounds for Small Depth Circuits (PDF), Micali, Silvio (编), Randomness and Computation, Advances in Computing Research 5, JAI Press: 6–20, 1989, ISBN 978-0-89232-896-3, (原始内容 (PDF)存档于2012-02-22) 
  4. ^ Immerman, Neil, Nondeterministic space is closed under complementation (PDF), SIAM Journal on Computing, 1988, 17 (5): 935–938 [2022-10-31], CiteSeerX 10.1.1.54.5941可免费查阅, ISSN 1095-7111, doi:10.1137/0217058, (原始内容存档 (PDF)于2012-02-07) 
  5. ^ Szelepcsényi, R., The method of forced enumeration for nondeterministic automata (PDF), Acta Informatica, 1988, 26 (3): 279–284 [2022-10-31], S2CID 10838178, doi:10.1007/BF00299636, hdl:10338.dmlcz/120489, (原始内容存档 (PDF)于2022-08-01) 
  6. ^ Sinclair, A.; Jerrum, M., Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation, 1989, 82 (1): 93–133, ISSN 0890-5401, doi:10.1016/0890-5401(89)90067-9 
  7. ^ Jerrum, M.; Sinclair, Alistair, Approximating the permanent, SIAM Journal on Computing, 1989, 18 (6): 1149–1178, CiteSeerX 10.1.1.431.4190可免费查阅, ISSN 1095-7111, doi:10.1137/0218077 
  8. ^ Halpern, Joseph; Moses, Yoram, Knowledge and common knowledge in a distributed environment (PDF), Journal of the ACM, 1990, 37 (3): 549–587 [2022-10-31], S2CID 52151232, arXiv:cs/0006009可免费查阅, doi:10.1145/79147.79161, (原始内容存档 (PDF)于2019-02-02) 
  9. ^ Toda, Seinosuke, PP is as hard as the polynomial-time hierarchy (PDF), SIAM Journal on Computing, 1991, 20 (5): 865–877 [2022-10-31], CiteSeerX 10.1.1.121.1246可免费查阅, ISSN 1095-7111, doi:10.1137/0220053, (原始内容 (PDF)存档于2016-03-03) 
  10. ^ Shor, Peter W., Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing, 1997, 26 (5): 1484–1509, ISSN 1095-7111, S2CID 2337707, arXiv:quant-ph/9508027可免费查阅, doi:10.1137/S0097539795293172 
  11. ^ Vardi, Moshe Y.; Wolper, Pierre, Reasoning about infinite computations (PDF), Information and Computation, 1994, 115 (1): 1–37, ISSN 0890-5401, doi:10.1006/inco.1994.1092, (原始内容 (PDF)存档于2011-08-25) 
  12. ^ Feige, Uriel; Goldwasser, Shafi; Lovász, Laszlo; Safra, Shmuel; Szegedy, Mario, Interactive proofs and the hardness of approximating cliques (PDF), Journal of the ACM, 1996, 43 (2): 268–292 [2022-10-31], ISSN 0004-5411, doi:10.1145/226643.226652可免费查阅, (原始内容存档 (PDF)于2011-06-10) 
  13. ^ Arora, Sanjeev; Safra, Shmuel, Probabilistic checking of proofs: a new characterization of NP (PDF), Journal of the ACM, 1998, 45 (1): 70–122, ISSN 0004-5411, S2CID 751563, doi:10.1145/273865.273901, (原始内容 (PDF)存档于2011-06-10) 
  14. ^ Arora, Sanjeev; Lund, Carsten; Motwani, Rajeev; Sudan, Madhu; Szegedy, Mario, Proof verification and the hardness of approximation problems (PDF), Journal of the ACM, 1998, 45 (3): 501–555, CiteSeerX 10.1.1.145.4652可免费查阅, ISSN 0004-5411, S2CID 8561542, doi:10.1145/278298.278306, (原始内容 (PDF)存档于2011-06-10) 
  15. ^ Sénizergues, Géraud, L(A) = L(B)? decidability results from complete formal systems, Theor. Comput. Sci., 2001, 251 (1): 1–166, ISSN 0304-3975, doi:10.1016/S0304-3975(00)00285-1可免费查阅 
  16. ^ Freund, Y.; Schapire, R.E., A decision-theoretic generalization of on-line learning and an application to boosting (PDF), Journal of Computer and System Sciences, 1997, 55 (1): 119–139 [2022-10-31], ISSN 1090-2724, doi:10.1006/jcss.1997.1504, (原始内容存档 (PDF)于2011-07-19) 
  17. ^ Herlihy, Maurice; Shavit, Nir, The topological structure of asynchronous computability (PDF), Journal of the ACM, 1999, 46 (6): 858–923 [2022-10-31], CiteSeerX 10.1.1.78.1455可免费查阅, S2CID 5797174, doi:10.1145/331524.331529, (原始内容存档 (PDF)于2011-06-05) . Gödel prize lecture页面存档备份,存于互联网档案馆
  18. ^ Saks, Michael; Zaharoglou, Fotios, Wait-free k-set agreement is impossible: The topology of public knowledge, SIAM Journal on Computing, 2000, 29 (5): 1449–1483, doi:10.1137/S0097539796307698 
  19. ^ Alon, Noga; Matias, Yossi; Szegedy, Mario, The space complexity of approximating the frequency moments (PDF), Journal of Computer and System Sciences, 1999, 58 (1): 137–147, doi:10.1006/jcss.1997.1545 . First presented at the Symposium on Theory of Computing (STOC) in 1996.
  20. ^ Agrawal, M.; Kayal, N.; Saxena, N., PRIMES is in P, Annals of Mathematics, 2004, 160 (2): 781–793, ISSN 0003-486X, doi:10.4007/annals.2004.160.781可免费查阅 
  21. ^ Razborov, Alexander A.; Rudich, Steven, Natural proofs, Journal of Computer and System Sciences, 1997, 55 (1): 24–35, ISSN 0022-0000, doi:10.1006/jcss.1997.1494可免费查阅 
  22. ^ Spielman, Daniel A.; Teng, Shang-Hua, Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time, J. ACM, 2004, 51 (3): 385–463, ISSN 0004-5411, arXiv:math/0212413可免费查阅, doi:10.1145/990308.990310 
  23. ^ Reingold, Omer; Vadhan, Salil; Wigderson, Avi, Entropy waves, the zig-zag graph product, and new constant-degree expanders, Annals of Mathematics, 2002, 155 (1): 157–187, CiteSeerX 10.1.1.236.8669可免费查阅, ISSN 0003-486X, JSTOR 3062153, MR 1888797, S2CID 120739405, doi:10.2307/3062153 
  24. ^ Reingold, Omer, Undirected connectivity in log-space, J. ACM, 2008, 55 (4): 1–24 [2022-10-31], ISSN 0004-5411, S2CID 207168478, doi:10.1145/1391289.1391291, (原始内容存档于2011-06-12) 
  25. ^ Arora, Sanjeev, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, Journal of the ACM, 1998, 45 (5): 753–782, CiteSeerX 10.1.1.23.6765可免费查阅, ISSN 0004-5411, S2CID 3023351, doi:10.1145/290179.290180 
  26. ^ Mitchell, Joseph S. B., Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems, SIAM Journal on Computing, 1999, 28 (4): 1298–1309, ISSN 1095-7111, doi:10.1137/S0097539796309764 
  27. ^ Håstad, Johan, Some optimal inapproximability results (PDF), Journal of the ACM, 2001, 48 (4): 798–859 [2022-10-31], CiteSeerX 10.1.1.638.2808可免费查阅, ISSN 0004-5411, S2CID 5120748, doi:10.1145/502090.502098, (原始内容存档 (PDF)于2012-03-12) 
  28. ^ Koutsoupias, Elias; Papadimitriou, Christos. Worst-case equilibria. Computer Science Review. 2009, 3 (2): 65–69. doi:10.1016/j.cosrev.2009.04.003. 
  29. ^ Nisan, Noam; Ronen, Amir. Algorithmic Mechanism Design. Games and Economic Behavior. 2001, 35 (1–2): 166–196. CiteSeerX 10.1.1.21.1731可免费查阅. doi:10.1006/game.1999.0790. 
  30. ^ Roughgarden, Tim; Tardos, Éva. How bad is selfish routing?. Journal of the ACM. 2002, 49 (2): 236–259. CiteSeerX 10.1.1.147.1081可免费查阅. S2CID 207638789. doi:10.1145/506147.506153. 
  31. ^ Boneh, Dan; Franklin, Matthew. Identity-based encryption from the Weil pairing. SIAM Journal on Computing. 2003, 32 (3): 586–615. CiteSeerX 10.1.1.66.1131可免费查阅. MR 2001745. doi:10.1137/S0097539701398521. 
  32. ^ Joux, Antoine. A one round protocol for tripartite Diffie-Hellman. Journal of Cryptology. 2004, 17 (4): 263–276. MR 2090557. S2CID 3350730. doi:10.1007/s00145-004-0312-y. 
  33. ^ Fagin, Ronald; Lotem, Amnon; Naor, Moni. Optimal aggregation algorithms for middleware. Journal of Computer and System Sciences. 2003, 66 (4): 614–656. arXiv:cs/0204046可免费查阅. doi:10.1016/S0022-0000(03)00026-6. 
  34. ^ Spielman, Daniel A.; Teng, Shang-Hua. Spectral Sparsification of Graphs. SIAM Journal on Computing. 2011, 40 (4): 981–1025. ISSN 0097-5397. S2CID 9646279. arXiv:0808.4134可免费查阅. doi:10.1137/08074489X. 
  35. ^ Spielman, Daniel A.; Teng, Shang-Hua. A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning. SIAM Journal on Computing. 2013, 42 (1): 1–26. ISSN 0097-5397. S2CID 9151077. arXiv:0809.3232可免费查阅. doi:10.1137/080744888. 
  36. ^ Spielman, Daniel A.; Teng, Shang-Hua. Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems. SIAM Journal on Matrix Analysis and Applications. 2014, 35 (3): 835–885. ISSN 0895-4798. S2CID 1750944. arXiv:cs/0607105可免费查阅. doi:10.1137/090771430. 
  37. ^ Brookes, Stephen. A Semantics for Concurrent Separation Logic (PDF). Theoretical Computer Science. 2007, 375 (1–3): 227–270 [2022-10-31]. doi:10.1016/j.tcs.2006.12.034. (原始内容存档 (PDF)于2021-05-09). 
  38. ^ O'Hearn, Peter. Resources, Concurrency and Local Reasoning (PDF). Theoretical Computer Science. 2007, 375 (1–3): 271–307 [2022-10-31]. doi:10.1016/j.tcs.2006.12.035可免费查阅. (原始内容存档 (PDF)于2021-03-04). 
  39. ^ Dwork, Cynthia; McSherry, Frank; Nissim, Kobbi; Smith, Adam. Halevi, Shai; Rabin, Tal , 编. Calibrating Noise to Sensitivity in Private Data Analysis. Theory of Cryptography (TCC). Lecture Notes in Computer Science 3876. Springer-Verlag: 265–284. 2006. ISBN 978-3-540-32731-8. doi:10.1007/11681878_14可免费查阅. 
  40. ^ Regev, Oded. On lattices, learning with errors, random linear codes, and cryptography. Journal of the ACM. 2009, 56 (6): 1–40. CiteSeerX 10.1.1.215.3543可免费查阅. S2CID 207156623. doi:10.1145/1568318.1568324. 
  41. ^ Dinur, Irit. The PCP theorem by gap amplification. Journal of the ACM. 2007, 54 (3): 12–es. S2CID 53244523. doi:10.1145/1236457.1236459. 
  42. ^ A constructive proof of the general Lovász Local Lemma. Journal of the ACM. 2010, 57 (2). ISSN 0004-5411. doi:10.1145/1667053. 
  43. ^ Bulatov, Andrei A. The complexity of the counting constraint satisfaction problem. Journal of the ACM (Association for Computing Machinery (ACM)). 2013, 60 (5): 1–41. ISSN 0004-5411. S2CID 8964233. doi:10.1145/2528400. 
  44. ^ Dyer, Martin; Richerby, David. An Effective Dichotomy for the Counting Constraint Satisfaction Problem. SIAM Journal on Computing (Society for Industrial & Applied Mathematics (SIAM)). 2013, 42 (3): 1245–1274. ISSN 0097-5397. S2CID 1247279. arXiv:1003.3879可免费查阅. doi:10.1137/100811258. 
  45. ^ Cai, Jin-Yi; Chen, Xi. Complexity of Counting CSP with Complex Weights. Journal of the ACM (Association for Computing Machinery (ACM)). 2017-06-22, 64 (3): 1–39. ISSN 0004-5411. S2CID 1053684. arXiv:1111.2384可免费查阅. doi:10.1145/2822891. 
  46. ^ Brakerski, Zvika; Vaikuntanathan, Vinod. Efficient Fully Homomorphic Encryption from (Standard) $\mathsf{LWE}$. SIAM Journal on Computing. January 2014, 43 (2): 831–871. ISSN 0097-5397. doi:10.1137/120868669. 
  47. ^ Brakerski, Zvika; Gentry, Craig; Vaikuntanathan, Vinod. (Leveled) fully homomorphic encryption without bootstrapping. Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS '12 (New York, New York, USA: ACM Press). 2012. doi:10.1145/2090236.2090262. 

参考文献[编辑]

  1. ^ The Gödel Letter. 2009-02-12 [2022-10-31]. (原始内容存档于2019-01-30). 
  2. ^ 2.0 2.1 2017 Gödel Prize. European Association for Theoretical Computer Science. EATCS. [29 March 2017]. (原始内容存档于2019-04-16). 
  3. ^ Three Papers Cited for Laying Foundation of Growth in Algorithmic Game Theory. 16 May 2012 [16 May 2012]. (原始内容存档于18 July 2013). 
  4. ^ ACM Group Presents Gödel Prize for Advances in Cryptography: Three Computer Scientists Cited for Innovations that Improve Security 互联网档案馆存檔,存档日期2013-06-01., Association for Computing Machinery, May 29, 2013.
  5. ^ Recipients Achieved Groundbreaking Results for Aggregating Data from Multiple Sources页面存档备份,存于互联网档案馆), Association for Computing Machinery, April 30, 2014.
  6. ^ 2015 Gödel Prize announcement 互联网档案馆存檔,存档日期2017-12-09. by Association for Computing Machinery.
  7. ^ 2018 Gödel Prize citation. [2020-05-17]. (原始内容存档于2018-10-05). 
  8. ^ 2019 Gödel Prize citation. [2020-05-17]. (原始内容存档于2020-07-28). 
  9. ^ 2020 Gödel Prize citation. [2020-05-17]. (原始内容存档于2020-07-16). 
  10. ^ 2021 Gödel Prize citation. [2022-10-31]. (原始内容存档于2022-06-04). 
  11. ^ 2022 Gödel Prize citation. [2022-10-31]. (原始内容存档于2022-10-31).