1 2
1 1 n ¨ 1 ¨ 2
2
3
a , a (a ≧a )
4 1 n (Turing Machine) 1936 Alan Turing Alan Turing 1912-1954
Σ: K: • • Σ • K • • 5 2 RAM (Random Access Machine)
( )
¨ ¨ ¨
6
M=(K,Σ,δ,s) K s∈K Σ K∩Σ= `␣’ ` ’ `␣’ `␣’ δ K Σ → K∪{h,``yes’’,``no’’}) Σ {←,→, }, h: halting state ( ) yes: accepting state ( no: rejecting state ( ←: →: : δ(q, )=(p, ,→) 7 0 1
M=(K,Σ,δ,s), K={s,q,q0,q1}, Σ={0,1, ␣, } δ(s,0)=(s,0,→) δ(s,1)=(s,1,→) δ(s, ␣)=(q, ␣,←) 010 010 010 010 δ(s, )=(s, ,→) s s s s δ(q,0)=(q0, ␣,→) δ(q,1)=(q , ␣,→) 1 010␣ 010␣ 01␣␣ 01␣0 δ(q, ␣)=(q, ␣, ) δ(q, )=(h, ,→) s q q0 s δ(q0,0)=(s,0,←) δ(q0,1)=(s,0,←) 01␣0 0␣␣0 0␣10 0␣10 δ(q0, ␣)=(s,0,←) q q1 s q δ(q0, )=(h, ␣,→) δ(q1,0)=(s,1,←) 10 010 010 010 δ(q1,1)=(s,1,←) ␣␣ ␣ ␣ ␣ δ(q , )=(s,1,←) 1 ␣ q0 s q h δ(q1, )=(h, ␣,→) 8 Σ* Σ
def Σ* f ⇔ x∈Σ* M x f(x) M
δ K Σk → K∪{h,``yes’’,``no’’}) Σk {←,→, }k
δ K Σ → 2 K∪{h,``yes’’,``no’’}) Σ {←,→, }
2X X X 9 Church Church-Turing) (1936)
Alonzo Church, 1903-1995, f ⇔ f λ ⇔ f Gödel Church Turing Church, Kleene Turing
λ
(λ )
10
• →YES
• M p
M p x (p,x) M x
11
M p x (p,x) M x halt
halt(p,x)= 1 if p x 0 otherwise
halt
12 halt
halt halt M
M M p if(halt(u,u)=1) { while(true){ u←u;} M (u=)p } else { ⇒halt(p,p)=1 return(0); ⇒M p } ( )
M (u=p) ⇒halt(p,p)=0 ⇒M p
halt
13
14 1966 (ACM) 2007 Edmund M. Clarke 2003 Alan Kay E. Allen Emerson Joseph Sifakis LSI 2004 Vinton G. Cerf Robert E. Kahn TCP/IP
2005 Peter Naur Algol 60 2008 Barbara Liskov( ) 2006 Frances E Allen ( CLU Argus 2010 Leslie G. Valiant 2009 Charles P. Thacker 2011 Judea Pearl Alto PC 2012 Silvio Micali, Shafi Goldwasser 2013 Leslie Lamport URL: https://amturing.acm.org/byyear.cfm 2014 Michael Stonebraker 2015 Whitfield Diffie, Martin Hellman 2016 Berners-Lee, Tim World Wide Web (WWW) 2017 John L. Hennessy, David Patterson 2018 Yoshua Bengio, Geoffrey E Hinton, Yann LeCun 2019 Edwin E. Catmull (*1), Patrick M. Hanrahan (CG) 15 (*1 ACM EATCS 5,000
2 Kurt Gödel, 2004 Maurice Herlihy, Michael Saks, Nir Shavit and Fotios Zaharoglou: applications of topology to the theory of distributed computing 1906-1978, 2005 Noga Alon, Yossi Matias and Mario Szegedy: their foundational contribution to streaming algorithms 2006 Manindra Agrawal, Neeraj Kayal, Nitin Saxena: the AKS primality test 2007 Alexander Razborov, Steven Rudich: natural proofs 2008 Daniel Spielman, Shanghua Teng: smoothed analysis of algorithms 2009 Omer Reingold, Salil Vadhan, Avi Wigderson: zig-zag product of graphs and undirected connectivity in log space 2010 Sanjeev Arora, Joseph S. B. Mitchell: their concurrent discovery of a polynomial-time approximation scheme (PTAS) for the Euclidean Travelling Salesman Problem (ETSP) 2011 Johan Håstad: proving optimal inapproximability result for various combinatorial problems 2012 Elias Koutsoupias, Christos Papadimitriou, Noam Nisan, Amir Ronen, Tim Roughgarden and Éva Tardos: laying the foundations of algorithmic game theory 2013 Dan Boneh, Matthew K. Franklin, and Antoine Joux: for multi-party Diffie–Hellman key exchange and the Boneh–Franklin scheme in cryptography 2014 Ronald Fagin, Amnon Lotem, and Moni Naor: for Optimal Aggregation Algorithms for Middleware 2015 Daniel Spielman, Shanghua Teng or their series of papers on nearly-linear-time Laplacian solvers #P 1998 16 Fields Medal(1936-) Carl Friedrich Gauss Prize(2006-) 4 40 15,000 IMU ) Rolf Nevanlinna, 1895-1980, Year Laureate Nationality 1982 Robert Tarjan United States 1986 Leslie Valiant United Kingdom 1990 Alexander Razborov Russia 1994 Avi Wigderson Israel 1998 Peter Shor United States 2002 Madhu Sudan India/ United States 2006 Jon Kleinberg United States 2010 Daniel Spielman United States 2014 Subhash Khot india/ United States John von Neumann (1992-) 4000 (IEEE) 1990 3 von Neumann 17