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 i@bK'(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 PKHALT(P) = no, if PKHALT 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
Unlocking...