15 251 Great Theoretical Ideas in Computer Science What 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 Computation Lecture 25 November 18 2007 Anything I say say is false This lecture will change the way you think about computer programs Many questions which appear easy at first glance are impossible to solve in general The HELLO assignment Write 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 Script 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 How exactly might such a script work 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 Program n 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 the Riemann 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 Are all reals describable NO Are all reals computable NO We saw that computable describable but do we also have describable computable The grading function we just described is not computable We ll see a proof soon Computable Function 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 is computable if there is a program P that when executed on an ideal computer computes f That is for all strings x in 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 Functions The 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 0 Hence the set of all f 0 1 has the same size as the power set of which is uncountable Countably many computable functions Uncountably many functions 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 Conventions Fix a single programming language Java When we write program P we are talking about the text of the source code for P P 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 x The 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 The Halting Set K Definition K is the set of all programs P such that P P halts K Java P P P halts The Halting Problem Is there a program HALT such that HALT P HALT P halt yes if P P halts no if P P does not 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 HALT P yes if P P halts no if P P does not halt We will call HALT as a subroutine in a new program called CONFUSE CONFUSE CONFUSE P if HALT P then loop forever else exit text of HALT goes here i e we dont halt i e we halt Does CONFUSE CONFUSE halt CONFUSE CONFUSE P if HALT P then loop forever else exit text of HALT goes here i e we dont halt i e we halt Suppose CONFUSE CONFUSE halts then HALT CONFUSE TRUE CONFUSE will loop forever on input CONFUSE Suppose CONFUSE CONFUSE does not halt then HALT CONFUSE FALSE CONTRADICTIO CONFUSE will halt on input CONFUSE N Alan Turing 1912 1954 Theorem 1937 There is no program to solve the halting problem Turing s argument is essentially the reincarnation of Cantor s Diagonalization argument that we saw in the previous lecture All Programs the input All Programs P0 P1 P2 Pj P0 P1 Pi Programs computable functions are countable so we can put them in a countably long list All Programs the input All Programs P0 P1 P2 Pj P0 P1 Pi YES if Pi Pj halts No otherwise All Programs the input All Programs P0 P0 P1 Pi P1 P2 Pj Let di HALT Pi d0 d1 di CONFUSE Pi halts iff di no The CONFUSE function is the negation of the diagonal 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 computed Suppose it is and program FRED computes it then consider the following program MYSTERY program text P for j 0 to forever do if P Pj then use FRED to compute jth bit of R return YES if bit 1 NO if bit 0 MYSTERY solves the halting problem From the last lecture Are all reals describable NO Are all reals computable NO We saw that computable describable but do we also have describable computable Computability A Vocabulary lesson Computability Theory Vocabulary Lesson We call a set S decidable or recursive if there is a program P such that P x yes if x S P x no if x S We already know the halting set K is undecidable Decidable and Computable Subset S of Function fS x in S fS x 1 x not in S fS x 0 Set S is decidable function fS is computable Sets are decidable or undecidable whereas functions are computable or not Oracles and Reductions Oracle For Set S Is x S YES NO Oracle for S Example Oracle S Odd Naturals 4 No 81 Yes Oracle for S K0 the set of programs that take no input and halt Hey I ordered an oracle for the famous halting set K but when I opened the package it was an oracle for the different set K0 GIVEN Oracle for K0 But you can use this oracle for K0 to build an oracle for K K0 the set of programs that take no input and halt P input I Q Does P P halt Does I P Q halt BUILD Oracle for K GIVEN Oracle for K0 We ve reduced the problem of deciding membership in K to the problem of deciding membership in K0 Hence deciding membership for …
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