09ma6cHW8

09ma6cHW8 - Ma/CS 6c Assignment#9 Due Thursday June 2 at 1...

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Unformatted text preview: Ma/CS 6c Assignment #9 Due Thursday, June 2 at 1 p.m. (30%) 1. Consider Turing machines on the alphabet { 1 , } . Let T n be the set of Turing machines on this alphabet that have at most n states. Clearly T n is finite. For each TM M on { 1 , } , let P M be defined by P M =the number of 1’s appearing in the output, when the input is M is θ (the empty word) and M stops on that input; 0 otherwise. So P M ∈ N . Let B ( n ) = max { P M : M ∈ T n } . So B : N → N . (By convention B (0) = 0.) So for each n ≥ 1 ,B ( n ) is the maximum number of 1’s that a TM with at most n states can print in the output, assuming it stops, when started at the empty input. We call B the busy beaver function . Prove that there is no TM on the alphabet { 1 , } which computes B , i.e., when pre- sented with input * * 1 1 ... 1 | {z } n ** produces as output * * 1 1 ... 1 | {z } B ( n ) ** You can take for granted that we can construct a Turing machine M on { 1 , } that on input * * * 1 1 ... 1 | {z }...
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This note was uploaded on 07/19/2010 for the course MA 6 taught by Professor Inessaepstein during the Spring '09 term at Caltech.

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09ma6cHW8 - Ma/CS 6c Assignment#9 Due Thursday June 2 at 1...

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