Computational Models. Nachum Dershowitz. Jenny (Evgenia) Falkovich

Transcription

1 Computational Models Nachum Dershowitz Jenny (Evgenia) Falkovich

2 Main Issues What can a program do? What can t a program do? What is hard for a program to do? What if memory is limited?

3 Time Magazine Put the right kind of software into a computer, and it will do whatever you want it to. There may be limits on what you can do with the machines themselves, but there are no limits on what you can do with software.

4 Sonnet Take all my loves, my love, yea take them all; What hast thou then more than thou hadst before? No love, my love, that thou mayst true love call; All mine was thine, before thou hadst this more. Then, if for my love, thou my love receivest, I cannot blame thee, for my love thou usest; But yet be blam'd, if thou thy self deceivest By wilful taste of what thyself refusest. I do forgive thy robbery, gentle thief, Although thou steal thee all my poverty: And yet, love knows it is a greater grief To bear love's wrong, than hate's known injury. Lascivious grace, in whom all ill well shows, Kill me with spites yet we must not be foes.

5 Bloomfield Science Museum

6 Science Museum

7 Man vs. Machine David Levy vs. Deep Thought (1989) Gary Kasparov vs. Deep Blue (1996) Gary Kasparov vs. Deep Junior (2003) Vladimir Kramnik vs. Deep Fritz (2006)

8 Chess Games: Positions: Positions: 20, 400, 5362, 71852, (?), (?),...

9 Quiz What s the smallest number that cannot be defined in twelve words?

10 Leopold Kronecker God created the natural numbers, and all the rest is the work of man.

11 21 January 2013

12 99!

13 Busy Beaver

14 Busy Beaver maximum integer computed by inputless program of length at most n bb(n) = max{ p() : p in L, p() N, p n }

15 Busy Scheme bb Scheme (11) = (expt 9 99) =

16 Busy Beaver bb(n) = max{ p() : p in L, p() N, p n } Seems easy: generate all programs, run them, take maximum

17 Proof by Contradiction Theorem: 2 is irrational Proof: Suppose not, that is 2=m/n for some m,n that are relatively prime. Then n 2=m and so 2n 2 =m 2. So m is even, say m=2k. Then 2k 2 =n 2, in which case n is also even. Contradiction.

18 Busy Beaver bb(n) = max{ p() : p in L, p() N, p n } Suppose b(n) = bb(n) for all n, N = b Let c() := b(5n)+1 where b... c = b + N + 27 N + N + 3N = 5N bb(5n) c()= b(5n)+1 Oops!

19 Turing ( )

20 ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM By A. M. TURING. [Received 28 May, Read 12 November, 1936.] Computing is normally done by writing certain symbols on paper. "We may suppose this paper is divided into squares like a child's arithmetic book. In elementary arithmetic the two-dimensional character of the paper is sometimes used. But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation. I assume then that the computation is carried out on

21 Turing pioneered the Turing test (imitation game) for artificial intelligence

22 Alan Turing Turing machine Halting problem Turing test Bombe...

23 Easy Termination 1. a(m,n) := 2. if m=0 then n+1 3. else if n=0 then a(m-1,1) 4. else a(m-1,a(m,n-1))

24 Hard Termination b(n,x) := if x=0 then 0 else b(n+1,d(n,x)) d(n,x) := if 2 x then r(x) else if 2 q(x) then f(n,r(q(x)),r(x)) f(n,x,y) := if n=0 then y else 2 x 3 f(n-1,x,y) q(x) := if 2 x then q(x/2)+1 else 0 r(x) := if 3 x then r(x/3)+1 else 0

25 Unknown Termination 1. c(n) := 2. if n=1 then 1 3. else if 2 n then c(n/2) 4. else c(3n+1)

26 A 2-MINUTE PROOF OF THE 2nd-MOST IMPORTANT THEOREM OF THE 2nd MILLENUIUM by Doron Zeilberger Written: Oct. 4, 1998

27 2-MINUTE PROOF Theorem: (Turing's No-Halting Theorem adapted to Maple) There is no Maple procedure, A(m,n), that inputs positive integers m and n, and outputs true iff the m-th Maple program (in length-then-lex. order) outputs at least n bits. Proof (Turing, shrunk by DZ): Denote by P(m) the m-th Maple program, and by P(m)[n] its n-th bit of output (if it exists). If A(m,n) exists then the following is a valid Maple program: T:=proc() local m: for m do if A(m,m) then print(1-p(m)[m]): else print(0):fi: od:end: T(); Let it be the m0-th Maple program. Then, its m0-th output bit is both P(m0)[m0] and 1-P(m0)[m0]. Contradiction. QED.

28 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(p) = if halt(p,p) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan).

29 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan, alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan): yes; no; nothing

30 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan, alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan): yes

31 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan, alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan): no

32 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan, alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan): no answer

33 Epimenides (-550) All Cretans are liars... One of their own poets has said so.

34 Eubulides (-400) I am lying.

35 Counting How many programs are there? How many functions are there?

36 Diagonalization N p o e ^2 pr :

37 Diagonalization N p o e ^2 pr : c = {1,3,4,5,..}

38 Diagonalization id e / ^ ! :

39 Diagonalization id e / ^ ! : c

40 Busy B k bb(n) = max{ p() : p in L, p() N, p n } Suppose b k (n) = bb(n) for all n k, N = b k Let c() := b k (5N)+1 where b... c = b k + N + 27 N + N + 3N = 5N bb(5n) c()= b k (5N)+1 5 b k = 5N > k, or b k > k/5

41 Don Knuth (1966) Algorithms are concepts which have existence apart from any programming language Algorithms were present long before Turing et al. formulated them, just as the concept of the number two was in existence long before the writers of first grade textbooks and other mathematical logicians gave it a certain precise definition.

42 Algorithm P r o g r a m State

43 Euclid Image courtesy of the Clay Mathematics Institute

44 Euclid s Elements Finitely describable in terms of basic compass operations

45 Antenaresis Δύο ἀριθμῶν ἀνίσων ἐκκειμένων, ἀνθυφαιρουμένου δὲ ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, ἐὰν ὁ λειπόμενος μηδέποτε καταμετρῇ τὸν πρὸ ἑαυτοῦ, ἕως οὗ λειφθῇ μονάς, οἱ ἐξ ἀρχῆς ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ἔσονται When two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime.

46 A Neolithic Algorithm

47 Eve s Algorithm If something s left and it s my turn Put one in my pile Now it s his turn If something s left and it s his turn -Put one in his pile -Now it s my turn

48 Lego Turing Machine

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50 Turing s Premises Sequential symbol manipulation Deterministic Finite internal states Finite symbol space Finite observability and local action Linear external memory

51 Turing s Operations Move (one square) Look (and decide how to proceed) Write (or erase) Quit

52 2D Turing Machines

53 2D INPUT/OUTPUT

54 Instructions Move Paint #" #" #" Look!"!"!" Go Done!!

55 !" #"!"!" #" #"!"!" #"!!!

56 !" #"!"!" #" #"!"!" #"!!!

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58 Business Attendance required Individual (bi/weekly) homeworks (10%) Midterm quiz (8-16%) Final exam (74-82%) Bonus surprise quizzes (5%)