Given the language
$\{ <M> \mid\:$ M is a Turing machine and there is some w ∈ Σ* for which the computation M(w) takes more than 10 transitions$\}$
How can one prove that this language is decidable?
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babou
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Chad Dingle
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1What language is supposed to be decidable? I cannot understand the statement of your problem. All you have is a single Turing machne M, that takes more than 10 steps to halt at least once. Where is the language? May be you should clarify the question. – babou Oct 05 '14 at 11:51
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If you're referring to the language {
| M is a Turing machine and there is some w ∈ Σ* for which the computation M(w) takes more than 10 transitions}, than this language is NOT decidable – Roi Divon Oct 05 '14 at 11:53 -
@babou Ah sorry, I made a bad typo. Roi Divon is right, I meant to say {
| M ...}. Which is why I'm confused, it looks non-decidable to me, but the question clearly says "show that the language is decidable". – Chad Dingle Oct 05 '14 at 12:29 -
1http://cs.stackexchange.com/questions/3101/is-the-set-of-turing-machines-which-stop-in-at-most-50-steps-on-all-inputs-deci – d'alar'cop Oct 05 '14 at 12:40
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@d'alar'cop Thankyou! Very helpful, I'm surprised I couldn't find this. – Chad Dingle Oct 05 '14 at 12:44
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No worries :) it's actually quite a popular problem – d'alar'cop Oct 05 '14 at 12:44
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@d'alarcop But, it seems to be the opposite to my question-- my question is about a language where there is a word that takes more than x steps. How could I relate the two? – Chad Dingle Oct 05 '14 at 12:46
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The complement of a decidable language is decidable. CC @d'alar'cop. – babou Oct 05 '14 at 14:11
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BTW why was http://cs.stackexchange.com/questions/3101/ upvoted +10 while this question is downvoted. Missing a previous instance of a question is frequent, even after searching, and it is hardly a crime. – babou Oct 05 '14 at 14:13
1 Answers
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Given a Turing Machine M (with its finite number of tape symbols, finite number of states, etc...), how hard is it to prove that there is an input on which it will run for more than 10 steps? How much of the input will be read in 10 steps at most? How many different computations have to be checked to see whether one takes more than 10 steps?
added later: I was editing this when the reference to question Is the set of Turing machines which stop in at most 50 steps on all inputs, decideable? was given as comment.
