New version page

CMU CS 15251 - lecture

Documents in this Course
lecture

lecture

66 pages

lecture

lecture

79 pages

lecture

lecture

111 pages

lecture

lecture

85 pages

lecture17

lecture17

64 pages

Lecture

Lecture

85 pages

Lecture

Lecture

71 pages

Lecture

Lecture

70 pages

Lecture

Lecture

11 pages

Lecture

Lecture

45 pages

Lecture

Lecture

50 pages

Lecture

Lecture

93 pages

Lecture

Lecture

93 pages

Lecture

Lecture

35 pages

Lecture

Lecture

98 pages

Lecture

Lecture

74 pages

Lecture

Lecture

13 pages

Lecture

Lecture

15 pages

Lecture

Lecture

66 pages

Lecture

Lecture

82 pages

Lecture

Lecture

15 pages

Lecture

Lecture

47 pages

Lecture

Lecture

69 pages

Lecture

Lecture

13 pages

Lecture

Lecture

67 pages

Lecture

Lecture

68 pages

Lecture

Lecture

69 pages

lecture03

lecture03

44 pages

Lecture

Lecture

69 pages

Lecture

Lecture

68 pages

Lecture

Lecture

55 pages

Lecture

Lecture

79 pages

Lecture

Lecture

85 pages

Lecture

Lecture

87 pages

Lecture

Lecture

85 pages

Lecture

Lecture

103 pages

Lecture

Lecture

9 pages

Lecture

Lecture

83 pages

Lecture

Lecture

8 pages

lecture03

lecture03

68 pages

lecture24

lecture24

78 pages

lecture03

lecture03

72 pages

Thales

Thales

129 pages

lecture13

lecture13

81 pages

Lecture

Lecture

64 pages

lecture01

lecture01

59 pages

lecture11

lecture11

105 pages

Lecture

Lecture

89 pages

Lecture

Lecture

74 pages

lecture25

lecture25

57 pages

Lecture

Lecture

99 pages

lecture

lecture

50 pages

lecture

lecture

14 pages

Lecture

Lecture

78 pages

lecture

lecture

8 pages

Lecture

Lecture

98 pages

lecture

lecture

83 pages

lecture23

lecture23

88 pages

lecture

lecture

64 pages

lecture

lecture

72 pages

Lecture

Lecture

88 pages

lecture

lecture

79 pages

Lecture

Lecture

60 pages

lecture

lecture

74 pages

lecture19

lecture19

72 pages

lecture25

lecture25

86 pages

lecture

lecture

13 pages

lecture17

lecture17

79 pages

lecture

lecture

91 pages

lecture

lecture

78 pages

Lecture

Lecture

11 pages

Lecture

Lecture

54 pages

lecture

lecture

119 pages

lecture

lecture

167 pages

lecture

lecture

73 pages

lecture

lecture

73 pages

lecture

lecture

83 pages

lecture

lecture

49 pages

lecture

lecture

16 pages

lecture

lecture

67 pages

lecture

lecture

81 pages

lecture

lecture

72 pages

lecture

lecture

57 pages

lecture16

lecture16

82 pages

lecture21

lecture21

46 pages

Lecture

Lecture

92 pages

Lecture

Lecture

14 pages

Lecture

Lecture

49 pages

Lecture

Lecture

132 pages

Lecture

Lecture

101 pages

Lecture

Lecture

98 pages

Lecture

Lecture

59 pages

Lecture

Lecture

64 pages

Lecture

Lecture

106 pages

Lecture

Lecture

70 pages

Lecture

Lecture

80 pages

Lecture

Lecture

76 pages

Lecture

Lecture

91 pages

Lecture

Lecture

112 pages

Lecture

Lecture

91 pages

Lecture

Lecture

10 pages

Lecture

Lecture

39 pages

Lecture

Lecture

79 pages

Lecture

Lecture

74 pages

Lecture

Lecture

44 pages

Lecture

Lecture

39 pages

Lecture

Lecture

99 pages

Lecture

Lecture

44 pages

Lecture

Lecture

59 pages

Lecture

Lecture

36 pages

lecture17

lecture17

36 pages

lecture

lecture

71 pages

lecture

lecture

79 pages

lecture

lecture

12 pages

lecture

lecture

43 pages

lecture

lecture

87 pages

lecture

lecture

35 pages

lecture03

lecture03

23 pages

lecture

lecture

68 pages

lecture

lecture

74 pages

lecture

lecture

21 pages

lecture

lecture

79 pages

lecture

lecture

15 pages

lecture

lecture

83 pages

lecture

lecture

13 pages

Lecture

Lecture

53 pages

lecture

lecture

55 pages

lecture

lecture

49 pages

lecture

lecture

10 pages

lecture

lecture

70 pages

lecture

lecture

12 pages

Lecture

Lecture

105 pages

Lecture

Lecture

9 pages

Lecture

Lecture

72 pages

Lecture

Lecture

66 pages

Lecture

Lecture

54 pages

Lecture

Lecture

98 pages

Lecture

Lecture

57 pages

Lecture

Lecture

75 pages

Lecture

Lecture

48 pages

lecture

lecture

53 pages

Lecture

Lecture

72 pages

Lecture

Lecture

53 pages

Lecture

Lecture

84 pages

Lecture

Lecture

55 pages

Lecture

Lecture

15 pages

Lecture

Lecture

6 pages

Lecture

Lecture

38 pages

Lecture

Lecture

71 pages

Lecture

Lecture

110 pages

Lecture

Lecture

70 pages

lecture

lecture

48 pages

lecture

lecture

76 pages

lecture

lecture

48 pages

lecture

lecture

52 pages

lecture

lecture

43 pages

lecture

lecture

81 pages

lecture

lecture

82 pages

lecture

lecture

83 pages

lecture

lecture

64 pages

lecture

lecture

71 pages

lecture

lecture

65 pages

lecture

lecture

56 pages

lecture

lecture

12 pages

lecture

lecture

66 pages

lecture

lecture

50 pages

lecture

lecture

86 pages

lecture

lecture

70 pages

Lecture

Lecture

74 pages

Lecture

Lecture

54 pages

Lecture

Lecture

90 pages

lecture

lecture

78 pages

lecture

lecture

87 pages

Lecture

Lecture

55 pages

Lecture

Lecture

12 pages

lecture21

lecture21

66 pages

Lecture

Lecture

11 pages

lecture

lecture

83 pages

Lecture

Lecture

53 pages

Lecture

Lecture

69 pages

Lecture

Lecture

12 pages

lecture04

lecture04

97 pages

Lecture

Lecture

14 pages

lecture

lecture

75 pages

Lecture

Lecture

74 pages

graphs2

graphs2

8 pages

lecture

lecture

82 pages

Lecture

Lecture

8 pages

lecture

lecture

47 pages

lecture

lecture

91 pages

lecture

lecture

76 pages

lecture

lecture

73 pages

lecture

lecture

10 pages

lecture

lecture

63 pages

lecture

lecture

91 pages

lecture

lecture

79 pages

lecture

lecture

9 pages

lecture

lecture

70 pages

lecture

lecture

86 pages

lecture

lecture

102 pages

lecture

lecture

145 pages

lecture

lecture

91 pages

Lecture

Lecture

87 pages

lecture

lecture

87 pages

Notes

Notes

19 pages

Lecture

Lecture

50 pages

Lecture

Lecture

13 pages

Lecture

Lecture

97 pages

Lecture

Lecture

98 pages

Lecture

Lecture

83 pages

Lecture

Lecture

77 pages

Lecture

Lecture

102 pages

Lecture

Lecture

63 pages

Lecture

Lecture

104 pages

lecture

lecture

41 pages

lecture

lecture

14 pages

Lecture

Lecture

87 pages

Lecture

Lecture

94 pages

lecture

lecture

9 pages

Lecture

Lecture

96 pages

Lecture

Lecture

72 pages

Lecture

Lecture

35 pages

Lecture

Lecture

77 pages

Lecture

Lecture

98 pages

Lecture

Lecture

48 pages

Lecture

Lecture

66 pages

Lecture

Lecture

53 pages

lecture18

lecture18

101 pages

Lecture

Lecture

10 pages

Lecture

Lecture

70 pages

Lecture

Lecture

12 pages

Lecture

Lecture

74 pages

graphs

graphs

10 pages

Lecture

Lecture

62 pages

Lecture

Lecture

11 pages

Lecture

Lecture

71 pages

Lecture

Lecture

42 pages

lecture15

lecture15

72 pages

Lecture

Lecture

82 pages

Load more
Upgrade to remove ads

This preview shows page 1-2-3-4-5-34-35-36-37-68-69-70-71-72 out of 72 pages.

Save
View Full Document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience

Upgrade to remove ads
Unformatted text preview:

Turing’s Legacy: The Limits Of Computation.The HELLO assignmentGrading ScriptWhat does this do?Slide 5Nasty ProgramSlide 7Slide 8Slide 9Infinite RAM ModelComputable FunctionSlide 12Slide 13Uncountably many functionsSlide 15Slide 16Slide 17Notation And ConventionsThe meaning of P(P)P(P) … So that’s what I look likeThe Halting Set KThe Halting ProblemThe Halting Problem K = {P | P(P) halts }THEOREM: There is no program to solve the halting problem (Alan Turing 1937)CONFUSESlide 26Alan Turing (1912-1954)Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Proof that RK cannot be computedComputability Theory: Vocabulary LessonDecidable and ComputableOracles and ReductionsOracle For Set SExample Oracle S = Odd NaturalsSlide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Diophantine EquationsResolution of Hilbert’s 10th Problem: Dramatis PersonaeGood old Fibonaccimal…Polynomials can encode programs.Slide 53Slide 54Self-Reference PuzzleHW12: Auto Cannibal MakerSuppose HALT with no input was programmable in JAVA.Slide 58CHURCH-TURING THESISThe Church-Turing Thesis is NOT a theorem. It is a statement of belief concerning the universe we live in.Empirical IntuitionMechanical IntuitionSpiritual IntuitionQuantum IntuitionSlide 65Another important notionSlide 67Slide 68Enumerating KSlide 70Slide 71Slide 72Turing’s Legacy: The Limits Of Computation.Great Theoretical Ideas In Computer ScienceAnupam GuptaCS 15-251 Fall 2005Lecture 26 Nov 29, 2005 Carnegie Mellon UniversityAnything says is false!The HELLO assignmentWrite a JAVA program to output the word “HELLO” on the screen and halt.Space and time are not an issue. The program is for an ideal computer. PASS for any working HELLO program, no partial credit.Grading ScriptThe grading script G must be able to take any Java program P and grade it. Pass, if P prints only the word G(P)= “HELLO” and halts. Fail, otherwise.How exactly might such a script work?What does this do?#include <stdio.h> main(t,_,a)char *a;{return!0<t?t<3?main(-79,-13,a+main(-87,1-_, main(-86,0,a+1)+a)):1,t<_?main(t+1,_,a):3,main(-94,-27+t,a)&&t==2?_<13? main(2,_+1,"%s %d %d\n"):9:16:t<0?t<-72?main(_,t, "@n'+,#'/*{}w+/w#cdnr/+,{}r/*de}+,/*{*+,/w{%+,/w#q#n+,/#{l,+,/n{n+,/+#n+,/#\ ;#q#n+,/+k#;*+,/'r :'d*'3,}{w+K w'K:'+}e#';dq#'l \ q#'+d'K#!/+k#;q#'r}eKK#}w'r}eKK{nl]'/#;#q#n'){)#}w'){){nl]'/+#n';d}rw' i;# \ ){nl]!/n{n#'; r{#w'r nc{nl]'/#{l,+'K {rw' iK{;[{nl]'/w#q#n'wk nw' \ iwk{KK{nl]!/w{%'l##w#' i; :{nl]'/*{q#'ld;r'}{nlwb!/*de}'c \ ;;{nl'-{}rw]'/+,}##'*}#nc,',#nw]'/+kd'+e}+;#'rdq#w! nr'/ ') }+}{rl#'{n' ')# \ }'+}##(!!/") :t<-50?_==*a?putchar(31[a]):main(-65,_,a+1):main((*a=='/')+t,_,a+1) :0<t?main(2,2,"%s"):*a=='/'||main(0,main(-61,*a, "!ek;dc [email protected]'(q)-[w]*%n+r3#l,{}:\nuwloca-O;m .vpbks,fxntdCeghiry"),a+1);}What kind of program could a student who hated his/her TA hand in?Nasty Programn:=0;while (n is not a counter-example to the Riemann Hypothesis) {n++;}print “Hello”;The nasty program is a PASS if and only if theRiemann Hypothesis is true.Despite the simplicity of the HELLO assignment, there is no program to correctly grade it! And we will prove this.The theory of what can and can’t be computed by an ideal computer is called Computability Theory or Recursion Theory.Are all reals describable?Are all reals computable?We saw thatcomputable  describable, but do we also havedescribable  computable?NONOFrom Lecture 25:The “grading function” we just describedis not computable! (We’ll see a proof soon.)Infinite RAM ModelPlatonic Version: One memory location for each natural number 0, 1, 2, …Aristotelian Version: Whenever you run out of memory, the computer contacts the factory. A maintenance person is flown by helicopter and attaches 100 Gig of RAM and all programs resume their computations, as if they had never been interrupted.Computable FunctionFix any finite set of symbols, . Fix any precise programming language, e.g., Java. A program is any finite string of characters that is syntactically valid.A function f : Σ*Σ* is computable if there is a program P that when executed on an ideal computer, computes f. That is, for all strings x in Σ*, f(x) = P(x).Computable FunctionFix any finite set of symbols, . Fix any precise programming language, e.g., Java. A program is any finite string of characters that is syntactically valid.A function f : Σ*Σ* is computable if there is a program P that when executed on an ideal computer, computes f. That is, for all strings x in Σ*, f(x) = P(x).Hence: countably many computable functions!There are only countably many Java programs. Hence, there are only countably many computable functions.Uncountably many functionsThe functions f: *  {0,1} are in 1-1 onto correspondence with the subsets of * (the powerset of * ). Subset S of *  Function fS x in S  fS(x) = 1 x not in S  fS(x) = 0Uncountably many functionsThe functions f: *  {0,1} are in 1-1 onto correspondence with the subsets of * (the powerset of * ).Hence, the set of all f: Σ*  {0,1} has thesame size as the power set of Σ*. And since Σ* is countably infinite, its power set is uncountably infinite.Countably many computable functions.Uncountably manyfunctions from * to {0,1}.Thus, most functions from * to {0,1} are not computable.Can we explicitly describe an incomputable function? Can we describe an interesting incomputable function?Notation And ConventionsFix a single programming language (Java)When we write program P we are talking about the text of the source code for PP(x) means the output that arises from running program P on input x, assuming that P eventually halts.P(x) =  means P did not halt on xThe meaning of P(P)It follows from our conventions that P(P) means the output obtained when we run P on the text of its own source code.P(P) … So that’s what I look likeThe Halting Set KDefinition:K is the set of all programs P such that P(P) halts.K = { Java P | P(P) halts }The Halting ProblemIs there a program HALT such that:HALT(P) = yes, if P(P) haltsHALT(P) = no, if P(P) does not haltThe Halting ProblemK = {P | P(P) halts }Is there a program HALT such that:HALT(P) = yes, if PKHALT(P) = no, if PKHALT decides whether or not any given program is in K.THEOREM: There is no program to solve the halting problem(Alan Turing 1937)Suppose a program HALT existed that solved the halting problem.HALT(P) =


View Full Document
Download lecture
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view lecture and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view lecture 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?