Anything says is false 0 1 2 3 0 4 4 3 6378 9 3 3 1 5 5 0 2 0 How exactly might such a script work 4 0 3 8 6 3 1 7 2 3 5 A 0 5 5 5 3 5 0 5 5 B 5 9 9 9 5 C D C 3 0 C 36 7 8 6 7 5 5 5 9 9 5 C D C 3 0 C 36 7 8 6 7 5 5 5 5 5 C 6 9 C D E F 0 0 C 7 5 C 0 0 6 7 8 6 7 8 5 5 C D E F 0 0 C 7 C 6 C G 5 5 5 5 C D E F 0 C 0 C E F 5 0 5 5 0 5 5 Fix a single programming language 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 3637 3637 0 0 5 3 0 0 3637 B 0 4 4 9 H H 8E 3 3637 3 I 3637 H F 3 6378 6378 3637 3637 5 3 H 8 E3 I 3637 6378 6378 3 H 3 H 0 H 5 F 5 JK7 6 5 6378 6378 0 0 3637 3637 5 9 9 637 637 L L 6 7 M 5 D 9 6 9 7 N 6 9 7 8 9 6 9 7 0 9 6 6 9 7 8 9 7 9 637 637 L L 6 7 M 5 D 9 6 9 7 N 6 9 7 8 9 6 9 7 0 CONTRADICTION 9 6 6 9 7 8 9 7 O 3 3 3 B 3P B 3 3 B 3 B YES if Pi Pj halts No otherwise 3 3 3 3 3 B 3P B B B 3 C O N F U S E di HALT Pi B B 9 63 7 9 8 0 6 Q7 5 5 5 5 5 5 0 3 H 5 5 C 5 3 36 78 36 78 4 0 H 5 5 5 C 5 3 3 6 7 5 6 5 7 H 5 9 8 E F 08 E 0607 H M 0F K is NOT decidable but it is enumerable Let K java P P P does not halt Is K enumerable Now that we have established that the Halting Set is undecidable we can use it for a jumping off points for more natural undecidability results 9 Is x S YES NO Oracle for S 8 4 No 81 Yes Oracle for S K0 the set of programs that take no input and halt H 5 4 0 0 H GIVEN Oracle for K0 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 GIVEN Oracle for K Oracle for K0 Thus if K0 were decidable then K would be as well We already know K is not decidable hence K0 is not decidable HELLO the set of program that print hello and halt Does P halt Let P be P with all print statements removed P print HELLO is a hello program BUILD GIVEN Oracle for K0 HELLO Oracle HELLO is not decidable EQUAL All P Q such that P and Q have identical output behavior on all inputs Does P equal HELLO Let HI print HELLO Are P and HI equal BUILD GIVEN HELLO Oracle EQUAL Oracle Halting with input Halting without input Hello and EQUAL are not decidable PHILOSOPHICAL INTERLUDE 0 5 5 5 0 5 5 0 N 0 5 5 5 0 5 5 0 5 5 0 5 5 5 5 5 5 5 5 0 5 5 R 5 5 S 5 5 0 0 0S There are many other viewpoints you might have concerning the Church Turing Thesis But this ain t philosophy class 3 GG 5 4 5 4 9 9 6 9 7 4 5 4 0 5 637 8 0 5 9 6 9 7 4 9 4 3 0 9 A Q T N T N 8 0 G 5 8E I3 9 U F 0 5 5 V 5 5 A 3 3 5 9 4 3 3 3 9 637 D D the set of all integers polynomials with integer roots Does program P halt F P BUILD GIVEN HALTI NG Oracle D 9 6 3 That Nasty Polynomial 3 7 5 Problems that have no obvious relation to halting or even to computation can encode the Halting Problem is non obvious ways Do these theorems about the limitations of computation tell us something about the limitations of human thought
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