Great Theoretical Ideas In Computer Science Anupam Gupta CS 15 251 Lecture 26 Nov 29 2005 Fall 2005 Carnegie Mellon University Turing s Legacy The Limits Of Computation Anything says is false 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 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 nuwlocaO m vpbks fxntdCeghiry a 1 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 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 From Lecture 25 Are all reals describable Are all reals computable We saw that computable describable but do we also have describable computable NO NO The grading function we just described is not computable We ll see a proof soon Infinite RAM Model Platonic 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 Function Fix 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 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 Computable Function Fix 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 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 x not in S fS x 1 fS x 0 Uncountably many functions The functions f 0 1 are in 1 1 onto correspondence with the subsets of the powerset of Hence the set of all f 0 1 has the same size as the power set of And since is countably infinite its power set is uncountably infinite 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 incomputable function Can we describe an interesting incomputable 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 P P So that s what I look like 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 The Halting Problem K P P P halts Is there a program HALT such that HALT P HALT P yes if P K no if P K HALT 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 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 i e we halt text of HALT goes here i e we dont halt Suppose CONFUSE CONFUSE halts then HALT CONFUSE TRUE CONFUSE will loop forever on input CONFUSE Suppose CONFUSE CONFUSE does not halt CONTRADICTIO then HALT CONFUSE FALSE N CONFUSE will halt on input CONFUSE 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 All Programs the input P0 P1 P2 Pj P0 P1 Pi Programs computable functions are countable so we can put them in a countably long list All Programs All Programs the input P0 P1 P2 Pj P0 P1 Pi YES if Pi Pj halts No otherwise All Programs All Programs the input P0 P1 P2 Pj P0 P1 Pi d0 d1 di Let di HALT Pi CONFUSE Pi halts iff di no The CONFUSE function is the negation of the diagonal Alan Turing 1912 1954 Theorem 1937 There is no program to solve the halting problem From last lecture Is …
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