Turing Machines Alan Turing 1912 1954 Kings College

Turing Machines Alan Turing (1912 -1954) Kings College 1933 mathematics graduate student at Princeton University n Turing Machines Church-Turing thesis n Bletchley Park (German code (Enigma) breaking) Bombe (electronic computer) n Manchester University, Word problem n

n 011000#011000. . . . n x 11000#x 11000. . n xxxxxx#xxxxxx …. accept

Church-Turing Thesis This is a mathematically unprovable belief that a reasonable intuitive definition of "computable" is equivalent to the list provably equivalent formal models of computation: n http: //plato. stanford. edu/entries/churchturing/ n http: //www. loebner. net/Prizef/Turing. Article. html n

Decision problem http: //en. wikipedia. org/wiki/Entscheidungsproblem "On Computable Numbers, with an Application to the Entscheidungsproblem" (튜링 1936), he proved that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is unsolvable: n 즉 어떤 튜링 기계든지 줄때 그 튜링기계가 멈출지 안 멈출지 알려주는 튜링기계는 없다. n ATM = { <M, w> | M is a TM and M accepts w} is Turing recognizable, but Turing undecidable n

ATM is undecidable n n n H를 그런 TM(decider)라고 하자. H(<M, w>)=accept if M accepts w, reject if M rejects w. D=“On input <M> 1. Run H on <M, <M>> 2. Reject if H accepts, accept if H rejects D(<M>) = accept if D does not accept <M>, reject if D accepts <M>. D(<D>) =accept if D does not accept <D>, reject if D accept <D>. contradiction

Cellular automata 간단한 룰만 가지고 만들었으나 설명되지 않는 여러가지 행동을 보임 n Conway’s game of life n Wolfram http: //www. stephenwolfram. com/ “A New kind of science” http: //www. wolframscience. com/ n