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L = {M | M is a TM and there exists an input that the TM M accepts in less than 50 steps}

I need to find a minimal class it belongs to between R/ RE/ co-RE/ not in RE∪co-RE. I managed to show that it is in RE with a TM. I think its not in co-RE, because it has to check every input to know wheter a TM M belongs to L. and there are an infinite amount of inputs. I tried to use mapping recursion with ATM, but that failed. Would appreciate any advice, thanks.

Raphael
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Bar
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3 Answers3

3

Maybe that will help ?

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-045j-automata-computability-and-complexity-spring-2011/lecture-notes/MIT6_045JS11_lec09.pdf

Page 42 Credits : Nancy Lynch MIT

Applications of Rice’s Theorem

• Example 3: Another property that isn’t a language property and is decidable

{ M |M is a TM that runs for at most 37 steps on input 01 }

– Not a language property, not a property of a machine’s structure.

– Rice doesn’t apply.

– Obviously decidable, since, given the TM description, we can just simulate it for 37 steps.

Gal Rettig
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Your problem is decidable, since a Turing machine accepts some input in $t$ steps iff it accepts some input of length at most $t$ in $t$ steps, and the latter property can be easily checked.

Repeating my comment, if you aimed at proving a negative answer, one avenue would be to prove that the language is not recursive; since you already know that the language is recursively enumerable, it would follow that it is not co-recursively enumerable.

Stated succinctly, an r.e. language is recursive iff it is co-r.e.

Yuval Filmus
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This is my solution:

reduction from ATM to L

On input M,w: return a machine M', that on input x: runs w on M, and if M accepts, M' accepts(in a single step) if M decline or loops, decline

if M accepts w, L(M')=Sigma* and therefore is y in Sigma* that M' accepts in a single step(in fact it's true for all y in Sigma*)-that's less 50. if M loops or declines, L(M')=emptyset and therefore there is no word the M' accepts in less than 50 steps, because M' rejects all words.

good luck on your hw!

jon
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