NP completeness of historical consensus｜ MIP* of effective universe computational complexity sequence

Decrypt history, Encrypt future™

NP completeness of historical consensus｜ MIP* of effective universe computational complexity sequence

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Theme

No.1 : Which historical topology do we choice to reside?

historical tautology:Thales, Pytagoras, Euclid, Diophantus, Fermat,Pascal,Newton,Eular,Hamilton,Abel, Galois,Boole, Riemann, Maxwell, Cantor, Frege, Hilbert, Plank, Russell, Einstein, Markov, Noether, Dickson, Zermelo, Fraenkel, Banach, Tarski , Szilard, G ̈odel̈, Kolmogorov, Church, Turing, Oppenheimer, Von Neumann,Shannon, Callen-Welton, Landauer,Leech, Howard, Bennett, etc.

P vs NP : Cook, Levin, Chaitin, Martin-Löf,….Voevodsky, Lurie, Wigderson, etc.

No.2 how I descrive that topological manifold by algebraic geometry

Diophantine consensus: Byzantine→Probablistic(Arthur-Merlin)→pseudo randomness→randomness of RE

No.3 how I truncate that tautological topology into Diophantine fomula(or simple equation, geodesic)

topology class:, P vs NP, NP→3-SAT, NP→ZKP, Byzantine Fault Tolerance (BFT), NP=PCP(log n, 1), IP=AMPoly=PSPACE,MIP=NEXP, MIP*=RE,BPP ⊆ BQP ⊆P#P

physical reduction: homotopy→topology→E∞→E8→Diophantine fomula or geodesic

Draft Proposition(direction)

There possibly exists a mathematically effective:

  1. Proof construction method: Homotopy↔︎Topology↔︎Geometry. Possibly there is mathematical and logical efficient model in finding approximations/adjunctions between algebraic topology and algebraic geometry. Does a “2IP* prover-verifier role-switching probabilistic approach” equal RE, or is it strictly more powerful than RE?→similar to conjecture by Laszlo Babai and partially proved AMpoly (n)=AM
  2. Proof distribution method: A “Tiki-Taka” (football metaphor) approach to Zero-Knowledge Proofs (ZKP).

Draft Conjecture

1.Proof construction method

Human civilization can be viewed as a potential “NP” entity that is evolutionary proceeding 3-SAT step toward “P”. Since almost all mathematical consensus currently points to P ≠ NP, any claim regarding the possibility of P = NP must be made with caution. However, certain elite mathematicians or exceptional individuals operate as if P = NP were already realized.

Hardness within this framework is processed by randomness (via Probablistic Arthur-Merlin, Bounded-error Probabilistic Polynomial-time BPP, etc.). Since most operational randomness is merely pseudorandomness, it is true randomness that may ultimately bridge the gap to realize P = NP.

For example, historical hardness—such as the impossibility of finding a general algebraic solution for the quintic equation (Abel’s impossibility theorem)—is now categorized under an easy, accessible P class for modern computation. Similarly, for the Four Color Theorem, while the search space for a proof was immense, verifying a specific solution is a highly efficient process. Crucially, this does not mean the intrinsic nature of these problems changed from NP to P. Structurally, P ≠ NP remains the baseline reality.

2.Proof distribution method
Human civilization possesses the distinct capability to distribute efficient solutions—problems in NP or harder that are not yet formally proved but are verified via Interactive Proof(IP) or Zero-Knowledge Proof (ZKP)—without requiring knowledge of the underlying proof steps. This inherent human trait fosters innovation by allowing society to utilize breakthrough results without needing to explain why or how they work fundamentally.

For instance, imagine if Fermat’s Last Theorem had proliferated and been integrated into mathematics even while Fermat’s own “marvelous proof” remained unrevealed. Similarly, Bitcoin succeeded because people instinctively decided that its probabilistic consensus algorithm was superior to classical Byzantine Fault Tolerance (BFT), long before the general public understood the underlying cryptographic and game-theoretic machinery.

Reference

Thales of Miletus(B.C.624~)

Pythagoras(B.C.582~)

Euclid(B.C.325~)

Diophantus(200)

Pierre de Fermat(1607~)

Blaise Pascal(1623~)

Isaac Newton(1642~)

Leonhard Euler(1707~)

William Rowan Hamilton(1805~)

Niels Henrik Abel(1824) Mémoire sur les équations algébriques, où l’on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré

Évariste Galois(1829/1846) Mémoire sur les conditions de résolubilité des équations par radicaux

https://www.numdam.org/item/JMPA_1846_1_11__381_0.pdf

George Boole(1847)The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning

https://ia802801.us.archive.org/5/items/mathematicalanal00booluoft/mathematicalanal00booluoft.pdf

Bernhard Riemann Über die Hypothesen, welche der Geometrie zu Grunde liegen” (1854/1867) （On the Hypotheses which lie at the Bases of Geometry ）

https://www.emis.de/classics/Riemann/Geom.pdf

Alexandre-Émile Béguyer de Chancourtois (France)

John Alexander Reina Newlands (United Kingdom)

James Clerk Maxwell

https://www.math.ucdavis.edu/~temple/MAT22C/MaxwellOnPhysicalLinesOfForce.pdf

“A Dynamical Theory of the Electromagnetic Field” (1865)Philosophical Transactions of the Royal Society of London

On Physical Lines of Force https://www.bem.fi/library/1865-001.pdf

Theory of heat https://academicweb.nd.edu/~powers/ame.20231/maxwell1872.pdf

Dmitri Mendeleev (Russian Empire)

Ludwig Boltzmann (Austria)

Felix Klein (Germany)

Georg Cantor(1874): Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen.
(On a property of the class of all real algebraic numbers)Journal für die Reine und Angewandte Mathematik
https://jamesrmeyer.com/pdfs/cantor-1874-ueber-eine-eigenschaft-des-inbegriffes.pdf

Sophus Lie(1874) Ueber Gruppen von Transformationen.” Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1874

https://link.springer.com/chapter/10.1007/978-3-663-01390-7_1

Gottlob Frege(1879), Begriflsschriit, a formula language, modeled uponthat of arithmetic, for pure thought
https://logic-teaching.github.io/pred/texts/Frege%201967%20-%20Begriffsschrift.pdf

Killing, Wilhelm 1888 Germany“Die Zusammensetzung der stetigen endlichen Transformationsgruppen. Zweiter Theil”https://ia800805.us.archive.org/view_archive.php?archive=/13/items/crossref-pre-1909-scholarly-works/10.1007%252Fbf01442663.zip&file=10.1007%252Fbf01444109.pdf

Cartan, Élie 1894 Francehttps://www.numdam.org/article/ASENS_1904_3_21__153_0.pdf

David Hilbert(1900) Mathematical ProblemsLecture delivered before the International Congress of
Mathematicians at Paris
https://www.aemea.org/math/Hilbert_23_Mathematical_Problems_1900.pdf

Max Plank(1900) Ueber eine Verbesserung der Wien’schen Spectralgleichung
https://www.ub.edu/hcub/hfq/sites/default/files/1900_Planck_VPG.pdf

Bertrand Russell(1903) Principles of Mathematics
https://www.finophd.eu/wp-content/uploads/2018/11/Russell-Principles-of-Mathematics.pdf

Albert Einstein(1905)Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt
https://onlinelibrary.wiley.com/doi/epdf/10.1002/andp.19053220607

Andrey Markov “Распространение закона больших чисел на величины, зависящие друг от друга” (1906) （Extension of the limit theorems of probability theory to a sum of variables connected in a chain）https://dspace.spbu.ru/items/088210fb-8460-489c-808c-05b4103f9963

Henry Moseley (United Kingdom)

Niels Bohr (Denmark)

“An Example of Statistical Investigation of the Text of ‘Eugene Onegin’ Illustrating the Dependence of Samples in a Chain” (1913) Bulletin de l’Académie Impériale des Sciences de St.-Pétersbourg

https://alpha60.de/research/markov/DavidLink_AnExampleOfStatistical_MarkovTrans_2007.pdf

Invariante Variationsprobleme” (1918) （Invariant Variation Problems ）Emmy Noether

https://stanford.edu/~cantwell/AA218_Course_Material/Resources/Invariant_Variation_Problems_Emmy_Noether.pdf

Leonard Eugene Dickson(1919)On Quaternions and Their Generalization and the History of the Eight Square Theorem

https://www.jstor.org/stable/1967865?seq=17

Ludwig Wittgenstein(1921)　Tractatus Logico-Philosophicus

https://www.gutenberg.org/files/5740/5740-pdf.pdf

Zermelo–Fraenkel set theory(1922)
https://user.math.uzh.ch/halbeisen/publications/pdf/brussels.pdf

Stefan Banach, Alfred Tarski “Sur la décomposition des ensembles de points en parties respectivement congruentes” (1924) （英訳: On the decomposition of sets of points into respectively congruent parts）Fundamenta Mathematicae

Wolfgang Pauli (Austria)

  • 1925 Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren (On the Connexion between the Completion of Electron Groups in an Atom and the Complex Structure of Spectra / The Pauli Exclusion Principle)
  • http://www.fisicafundamental.net/relicario/doc/Pauli_1925.pdf

http://kielich.amu.edu.pl/Stefan_Banach/pdf/oeuvres1/12.pdf

Leo Szilard“Über die Entropieverminderung in einem thermodynamischen System durch Eingriffe intelligenter Wesen” (1929) （On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings ）


Kurt G ̈odel̈(1931)UBER FORMAL UNENTSCHEIDBARE S ̈ATZE DER “PRINCIPIA MATHEMATICA” UND VERWANDTER SYSTEME I
https://www.w-k-essler.de/pdfs/goedel.pdf

Andray Kolmogorov”Grundbegriffe der Wahrscheinlichkeitsrechnung” (1933) （Foundations of the Theory of Probability）https://archive.org/details/kolmogoroff-1933-grundbegriffe-der-wahrscheinlichkeitsrechnung

“Three approaches to the quantitative definition of information” (1965) Problems of Information Transmission

https://knowen-production.s3.amazonaws.com/uploads/attachment/file/3520/Three%2Bapproaches%2Bto%2Bthe%2Bquantitative%2Bdefinition%2Bof%2Binformation.pdf

Alonzo Church(1936) An Unsolvable Problem of Elementary Number Theory

https://ics.uci.edu/~lopes/teaching/inf212W12/readings/church.pdf

Alan Turing(1936) ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM
https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

Otto Hahn & Fritz Strassmann (Germany)

Jan 1939 Über den Nachweis und das Verhalten der bei der Bestrahlung des Urans mittels Neutronen entstehenden Erdalkalimetalle (On the Detection and Characteristics of the Alkaline Earth Metals Formed by Irradiation of Uranium with Neutrons)

https://www.nssp.uni-saarland.de/lehre/Vorlesung/Kernphysik_SS19/History/Papers/Hahn_Strassmann.pdf

Lise Meitner (Austria) & Otto Robert Frisch (Austria)

Feb 1939 Disintegration of Uranium by Neutrons: a New Type of Nuclear Reaction (The Physical Mechanism and Energy Formula of Nuclear Fission)

https://germanhistory-intersections.org/en/knowledge-and-education/ghis:document-160.pdf

J. R. OPPENHEIMER AND H. SNYDER(1939)On Continued Gravitational Contraction
https://the-center-of-gravity.com/documents/140/Oppenheimer-Snyder_On-continued-Gravitational-Contraction.pdf

Jon Von Neumann(1945) First draft of a report on the EDVAC
https://web.mit.edu/sts.035/www/PDFs/edvac.pdf

Claude Shannon(1948) A Mathematical Theory of Communication）
https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf

Herbert B. Callen, Theodore A. Welton“Irreversiblity and generalized noise” (1951)

https://123.physics.ucdavis.edu/week_3_files/callen_irreversibility_and_noise_1951.pdf

Rolf Landauer“Irreversibility and Heat Generation in the Computing Process” (1961)

https://worrydream.com/refs/Landauer_1961_-_Irreversibility_and_Heat_Generation_in_the_Computing_Process.pdf

Paul Baran (United States)

  • 1964 On Distributed Communications Networks (The Mathematical Foundation of Packet Switching and Survivable Decentralized Network Topologies)

Peter J. Freyd

Abelian Categories: An Introduction to the Theory of Functors” (1964）

https://webhomes.maths.ed.ac.uk/~v1ranick/papers/freydab.pdf

Ilya Prigogine, Paul Glansdorff“Thermodynamics of Irreversible Processes” (1967) / “Thermodynamic Theory of Structure, Stability and Fluctuations” (1971)

https://scispace.com/pdf/thermodynamics-of-irreversible-processes-the-experimental-tmzd1qteux.pdf

John Leech（1967）Notes on SPHERE PACKINGS

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/E926AB3EA0CB315FB5FAEFDE65719E79/S0008414X00054250a.pdf/notes-on-sphere-packings.pdf

William A. Howard “The formulae-as-types notion of construction” (1969 / 1980) Essays on Combinatory Logic, Lambda Calculus and Formalisms

https://www.cs.cmu.edu/~crary/819-f09/Howard80.pdf

Hervert A . Simon(1970)Human Problem Solving

https://worrydream.com/refs/Simon_1970_-_Human_Problem_Solving,_The_State_of_the_Theory_in_1970.pdf

DPRM theorem

Vinton G. Cerf & Robert E. Kahn (United States)

Charles H. Bennett“The Thermodynamics of Computation—A Review” (1982)

https://www.cpt.univ-mrs.fr/~verga/pdfs/Bennett-1982fk.pdf

Planar Formulae and Their Uses1982

https://dl.acm.org/doi/10.1137/0211025

Tim Berners-Lee (United Kingdom)

Christopher Jarzynski A nonequilibrium equality for free energy differences (1997)

https://arxiv.org/pdf/cond-mat/9610209

Other modern mathematicians

81 Turing Award

36 Abel Laureates

64 Fields Medalists

Proof theory

P vs NP

Cook, S. A. (1971). “The complexity of theorem-proving procedures.” In Proceedings of the third annual ACM symposium on Theory of computing

https://dl.acm.org/doi/epdf/10.1145/800157.805047

May, J. Peter 1972 USA“The Geometry of Iterated Loop Spaces(E∞)”https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf

Levin, L. A. (1973). “Universal search problems.” Problemy Peredachi Informatsii, 9(3)

https://www.karlin.mff.cuni.cz/~krajicek/levin.pdf

Gregory Chaitin“Information-Theoretic Computational Complexity” (1974)

https://www.cs.ox.ac.uk/activities/ieg/e-library/sources/ieee74.pdf

Efficient Planetary Testing 1974

J O H N H O P C R O F T and R O B E R T TAR JAN

https://dl.acm.org/doi/epdf/10.1145/321850.321852

Per Martin-Löf“An Intuitionistic Theory of Types” (1975)

https://archive-pml.github.io/martin-lof/pdfs/An-Intuitionistic-Theory-of-Types-1972.pdf

Garey, M. R., & Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co.

https://perso.limos.fr/~palafour/PAPERS/PDF/Garey-Johnson79.pdf

Gill (1977)Computational Complexity of Probabilistic Turing Machines

https://www.proquest.com/openview/ae9a6e6c19e75acb825b3b0ed2601d7a/1?pq-origsite=gscholar&cbl=666313

Lamport, L., Shostak, R., & Pease, M. (1982). “The Byzantine Generals Problem.” ACM Transactions on Programming Languages and Systems (TOPLAS), 4(3),

https://lamport.azurewebsites.net/pubs/byz.pdf

Sipser (1983)

Heterotic string theory (I). The free heterotic string

Gross, David J. ; Harvey, Jeffrey A. ; Martinec, Emil ; Rohm, Ryan Nuclear Physics, Section B, Volume 256, p. 253-284. Pub Date: 1985 Heterotic string theory (I). The free heterotic string

https://www.sciencedirect.com/science/article/abs/pii/0550321385903943

László Babai“Trading Group Theory for Randomness” (1985)

https://crypto.cs.mcgill.ca/~crepeau/COMP647/2007/TOPIC01/babai.pdf

ZKP (1985/1989) GMR85 Shafi Goldwasser,Silvio Micali, Charles Rackoff The Knowledge Complexity of Interactive Proof-Systems
https://people.csail.mit.edu/silvio/Selected%20Scientific%20Papers/Proof%20Systems/The_Knowledge_Complexity_Of_Interactive_Proof_Systems.pdf

NP=ZKP(1986) GMW86 O. Goldreich, S. Micali, A. Wigderson How to Prove All NP-Statements in Zero-Knowledge

https://dl.acm.org/doi/10.5555/36664.36675

MIP=NEXP

Babai, L., Fortnow, L., & Lund, C. (1991). “Non-deterministic exponential time has two-prover interactive protocols.” Computational Complexity,

https://lance.fortnow.com/papers/files/mip2.pdf

IP=PSPACE

Shamir, A. (1992). “IP = PSPACE.” Journal of the ACM (JACM), 39(4), 865-877.

https://dl.acm.org/doi/epdf/10.1145/146585.146609

Lund, C., Fortnow, L., Karloff, H., & Nisan, N. (1992). “Algebraic methods for interactive proof systems.” Journal of the ACM

https://dl.acm.org/doi/epdf/10.1145/146585.146605

Bernstein and Vazirani (1993 / 1997)

Nisan and Wigderson (1994)

Shor (1994 / 1997)

Impagliazzo and Wigderson (1997)

Johan Håstad “Some Optimal Inapproximability Results” (STOC 1997 / J.ACM 2001)

https://www.cs.umd.edu/~gasarch/TOPICS/pcp/hastadopt.pdf

PCP定理における「定数（1）」を、「エラー確率を半分強に抑えながら、たったの3ビットをチェックするだけでよい」PCP1-ε, 1/2+ε(log n, 3)

Impagliazzo and Wigderson (1998)

NP=PCP(log n, 1)

Arora, S., Lund, C., Motwani, R., Sudan, M., & Szegedy, M. (1998). “Proof verification and the hardness of approximation problems.”

https://dl.acm.org/doi/epdf/10.1145/278298.278306

Arora, S., & Safra, S. (1998). “Probabilistic checking of proofs: A new characterization of NP.” Journal of the ACM (JACM)

https://dl.acm.org/doi/epdf/10.1145/273865.273901

Janzing, Wocjan, and Beth (2002)

https://arxiv.org/pdf/quant-ph/0605181v2

Miltersen and Vinodchandran (2005)

Shaltiel and Umans (2005)

https://dl.acm.org/doi/epdf/10.1145/3055399.3055414

Gutfreund, Shaltiel, and Ta-Shma (2003 / 2007)

Aharonov, Jones, and Landau (2006)

https://arxiv.org/abs/quant-ph/0511096

Fokko du Cloux / Focalized by the Atlas of Lie Groups and Representations Project (Jeffrey Adams, et al.)2007 International (USA / France / etc.)“The Character Table of E8”https://math.mit.edu/~dav/notices07.pdf

Coldea, Radu, et al., 2010, UK / Germany, “Quantum Criticality in an Ising Chain: Experimental Evidence for E8 Symmetry”　https://arxiv.org/pdf/1103.3694

Aaronson (2011)

Arvind, Gopal, Joglekar, and Kulkarni (2011) / Santhanam

Hohm, Olaf & Samtleben, Henning, 2014, USA / Germany, Exceptional Field Theory. III. E8

https://arxiv.org/pdf/1406.3348

Raz and Tal (2019)

Vladimir Voevodsky(2013) Homotopy Type Theory
https://www.cs.uoregon.edu/research/summerschool/summer14/rwh_notes/hott-book.pdf

Jacob Lurie(2017) Higher Topos Theory
https://www.math.ias.edu/~lurie/papers/HTT.pdf

Cohn, Henry / Kumar, Abhinav / Miller, Stephen D. / Radchenko, Danylo / Viazovska, Maryn USA / India / Ukraine 2017 “The sphere packing problem in dimension 24”

https://arxiv.org/pdf/1603.06518

Avi Wigderson(2019) Mathematics and Computation
https://www.math.ias.edu/files/Book-online-Aug0619.pdf

Font, Anamaría / Fraiman, Bernardo / Graña, Mariana / Núñez, Carmen A.　Venezuela / Argentina　 2020 (Published in JHEP) “Exploring the landscape of heterotic strings on T^d

https://arxiv.org/pdf/2007.10358

Dray, Tevian & Manogue, Corinne A., et al. 2024 “A new division algebra representation of E7 from E8”

https://arxiv.org/pdf/2401.10534

MIP*=RE Ji, Z., Natarajan, A., Vidick, T., Wright, J., & Yuen, H. (2021). “MIP* = RE.” Communications of the ACM

https://arxiv.org/pdf/2001.04383

Dieter van Melkebeek and Nicollas Mocelin Sdroievski (2023 / 2024)

Conjectures

正十七角形の作図：Constructibility of the regular heptadecagon

Origin

Solution

ケプラー予想：The Kepler Conjecture

Origin

Solution

フェルマーの最終定理：Fermat’s Last Theorem

Origin

Solution

4色問題：The Four Color Conjecture

Origin

Solution

The four color theorem 1997

https://thomas.math.gatech.edu/PAP/fc.pdf

Georges Gonthier“A computer-checked proof of the Four Colour Theorem”2008

https://www.cse.chalmers.se/~abela/lehre/WS05-06/CAFR/4colproof.pdf

ポアンカレ予想：The Poincaré Conjecture

Origin

Solution

  • Author: Perelman, Grigori
  • Year: 2002
  • Country: Russia
  • Title: “The entropy formula for the Ricci flow and its geometric applications”
  • https://arxiv.org/pdf/math/0211159

ヴァイユ予想：Weil Conjectures

Origin

Solution（最終的な解決・リーマン予想部分の証明）

Diophantine Equation=Turing Machine

Origin

  • 1900年：ヒルベルトの23の問題（デヴィット・ヒルベルト）
    • 著者: David Hilbert
    • 論文名: Mathematische Probleme（数学諸問題）
    • 掲載誌: Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 253–297 (1900).
    • ※翌1901年にフランス語訳、1902年に英語訳（Bulletin of the American Mathematical Society）が公開され世界に広まりました。この中の「Problem 10」が該当の問題です。

Solution

  • 1931年：不完全性定理（クルト・ゲーデル）
    • 著者: Kurt Gödel
    • 論文名: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I（プリンキピア・マテマティカおよび関連する体系の形式的に決定不能な命題について I）
    • 掲載誌: Monatshefte für Mathematik und Physik, Vol. 38, pp. 173–198 (1931).
  • 1936年：チューリングマシンの提示と停止性問題（アラン・チューリング）
    • 著者: Alan M. Turing
    • 論文名: On Computable Numbers, with an Application to the Entscheidungsproblem（計算可能数について、附・決定問題への応用）
    • 掲載誌: Proceedings of the London Mathematical Society, Series 2, Vol. 42, pp. 230–265 (1936年受付、1937年発行).
  • 1953年：ディオファントス集合の先駆的研究（マーティン・デーヴィス）
    • 著者: Martin Davis
    • 論文名: Arithmetical Problems and Recursively Enumerable Predicates（算術的問題と帰納的可算述語）
    • 掲載誌: Journal of Symbolic Logic, Vol. 18, No. 1, pp. 33–41 (1953).
    • ※すべての帰納的可算集合が数式（の形の1つであるガジ・プレディケート）で表せることを示し、DPRM定理への最初のレンガを敷きました。
  • 1952年：指数関数の増大度の研究（ジュリア・ロビンソン）
    • 著者: Julia Robinson
    • 論文名: Existential Definability in Arithmetic（算術における存在記号による定義可能性）
    • 掲載誌: Transactions of the American Mathematical Society, Vol. 72, No. 3, pp. 437–449 (1952).
    • ※のちに「ロビンソンの仮説」と呼ばれる、指数関数を多項式で縛るための基礎理論を確立しました。
  • 1961年：DPR定理の確立（デーヴィス、パトナム、ロビンソン）
    • 著者: Martin Davis, Hilary Putnam, Julia Robinson
    • 論文名: The Decision Problem for Exponential Diophantine Equations（指数ディオファントス方程式の決定問題）
    • 掲載誌: Annals of Mathematics, Second Series, Vol. 74, No. 3, pp. 425–436 (1961).
    • ※「指数関数が使えればチューリングマシンと等価である」ことを証明した、DPRMの「DPR」部分にあたる論文です。
  • 1970年：マティヤセヴィチの定理 / DPRM定理の完成（ユーリ・マティヤセヴィチ）
    • 著者: Yuri V. Matiyasevich (Ю. В. Матиясевич)
    • 論文名: Диофантовость перечислимых множеств（英訳題: Enumerable sets are Diophantine ／ 和訳: 帰納的可算集合はディオファントス的である）
    • 掲載誌: Doklady Akademii Nauk SSSR, Vol. 191, pp. 279–282 (1970).
    • 英訳版: Soviet Mathematics. Doklady, Vol. 11, No. 2, pp. 354–358 (1970).
    • ※フィボナッチ数列を用いて「指数関数は多項式で表現できる」ことを証明し、ヒルベルトの第10問題を完全に否定的に解決（不可能であることを証明）した決定打の論文です。

Unsolved Conjectures

リーマン予想：The Riemann Hypothesis

Origin

Solution

  • 未解決（Unsolved）

Conjecture

conjecture by Shoichiro Tanaka, TANAAKK

  1. Is P vs NP is basically P≠NP but almost similary duplicate as P=NP by randomness to NP sequence?(controllable decidability by undecidability resource, P = BPP conjecture)
  2. Is this universe Diophantine equation = Turing machine(continuum vs discrete)
  3. Any of human desire is about materializing computation?(finding ideal NTM by DTM, finding ideal P=NP by P≠NP lower bound detection)

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