Slide 1What does this do?Turing’s Legacy: The Limits Of ComputationThis lecture will change the way you think about computer programs…The HELLO assignmentGrading ScriptSlide 7Slide 8Nasty ProgramSlide 10Slide 11Slide 12Computable FunctionSlide 14Uncountably Many FunctionsSlide 16Slide 17Notation And ConventionsThe meaning of P(P)The Halting Set KThe Halting ProblemTHEOREM: There is no program to solve the halting problem (Alan Turing 1937)CONFUSESlide 24Alan Turing (1912-1954)Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Proof that R cannot be computedSlide 33Computability A Vocabulary lessonComputability Theory: Vocabulary LessonDecidable and ComputableOracles and ReductionsOracle For Set SExample Oracle S = Odd NaturalsSlide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Diophantine EquationsResolution of Hilbert’s 10th Problem: Dramatis PersonaePolynomials can Encode ProgramsSlide 51Slide 52CHURCH-TURING THESISThe Church-Turing Thesis is NOT a theorem. It is a statement of belief concerning the universe we live in.Empirical IntuitionMechanical IntuitionQuantum Intuition15-251Great Theoretical Ideas in Computer ScienceWhat does this do?_(__,___,____){___/__<=1?_(__,___+1,____):!(___%__)?_(__,___+1,0):___%__==___/ __&&!____?(printf("%d\t",___/__),_(__,___+1,0)):___%__>1&&___%__<___/__?_(__,1+ ___,____+!(___/__%(___%__))):___<__*__?_(__,___+1,____):0;}main(){_(100,0,0);}Turing’s Legacy: The Limits Of ComputationAnything I say say is false!Lecture 25 (November 18, 2007)This lecture will change the way you think about computer programs…Many questions which appear easy at first glance are impossible to solve in generalThe 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 ScriptHow exactly might such a script work?The grading script G must be able to take any Java program P and grade it.G(P)=Pass, if P prints only the word “HELLO” and halts.Fail, otherwise.What does this do?_(__,___,____){___/__<=1?_(__,___+1,____):!(___%__)?_(__,___+1,0):___%__==___/ __&&!____?(printf("%d\t",___/__),_(__,___+1,0)):___%__>1&&___%__<___/__?_(__,1+ ___,____+!(___/__%(___%__))):___<__*__?_(__,___+1,____):0;}main(){_(100,0,0);}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 false.A TA nightmare: 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.From the last lecture:The “grading function” we just describedis not computable! (We’ll see a proof soon.)Are all reals describable?Are all reals computable?NONOWe saw that computable describablebut do we also have describable computable?Computable FunctionHence: countably many computable functions!Fix a finite set of symbols, Fix a 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).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 fSx in S fS(x) = 1x not in S fS(x) = 0Hence, the set of all f:Σ* {0,1} has the same size as the power set of Σ*, which is uncountable.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 uncomputable function? Can we describe an interesting uncomputable 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 codeThe 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 haltTHEOREM: There is no program to solve the halting problem(Alan Turing 1937)Suppose a program HALT existed that solved the halting problem.HALT(P) = yes, if P(P) haltsHALT(P) = no, if P(P) does not haltWe will call HALT as a subroutine in a new program called CONFUSE.CONFUSEDoes CONFUSE(CONFUSE) halt?CONFUSE(P){ if (HALT(P)) then loop forever; //i.e., we dont halt else exit; //i.e., we halt // text of HALT goes here}CONFUSECONFUSE(P){ if (HALT(P)) then loop forever; //i.e., we dont halt else exit; //i.e., we halt // text of HALT goes here }Suppose CONFUSE(CONFUSE) halts:then HALT(CONFUSE) = TRUE CONFUSE will loop forever on input CONFUSESuppose CONFUSE(CONFUSE) does not haltthen HALT(CONFUSE) = FALSE CONFUSE will halt on input CONFUSECONTRADICTIONAlan Turing (1912-1954)Theorem: [1937]There is no program to solve the halting problemTuring’s argument is essentially the reincarnation of Cantor’s Diagonalization argument that we saw in the previous lecture.P0P1P2…Pj…P0P1…Pi…All ProgramsAll Programs (the input)Programs (computable functions) are countable,so we can put them in a (countably long) listP0P1P2…Pj…P0P1…Pi…All ProgramsAll Programs (the input)YES, if Pi(Pj) haltsNo, otherwiseP0P1P2…Pj…P0d0 P1d1……Pidi……All ProgramsAll Programs (the input)Let di = HALT(Pi) CONFUSE(Pi) halts iff di = no(The CONFUSE function is the negation of the diagonal.)Hence CONFUSE cannot be on this list.Is there a real number that can be described, but not computed?Consider the real number R whose binary expansion has a 1 in the jth position iff the jth program halts.Proof that R cannot be computedMYSTERY solves the halting problem!Suppose it is,
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