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With only using our thinking. What do I have to think about when finding a complement of a Turing machine for example.

L={M∣M is a TM that halts on empty tape after even transition steps} What's the complement of L would it be:

  1. L={M∣M is a TM that does not halt on empty tape after even transition steps}
  2. L={M∣M is a TM that halts on empty tape after odd transition steps}

Please give me your train of thought when coming up with a complement for an automaton.

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    $\begingroup$ You're not being asked to find the complement of a Turing machine. You're being asked to find the complement of a language. $\endgroup$ – Tom van der Zanden Oct 3 '19 at 12:22
  • $\begingroup$ @TomvanderZanden Oh okay, I see where I made a mistake. $\endgroup$ – alexW Oct 3 '19 at 23:38
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You cant find the complement of a $TM$ for undecidable languages. A decidable language is such that a $TM$ which recognizes language membership, always halts with a yes. In this case finding the complement of the machine is simple, just reverse the yes with the no, obtaining the complement of the original decision problem. I leave to you the task of determining whether the language of your example is decidable or not.

HINT: you can compare it with the Halting problem or $ATM$

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  • $\begingroup$ What about a language that was co-recognizable? $\endgroup$ – Rick Decker Oct 3 '19 at 11:32
  • $\begingroup$ @RickDecker well the complement of co-recognizable is recognizable so yes, you can do it. $\endgroup$ – Yamar69 Oct 3 '19 at 12:02
  • $\begingroup$ The TM in my question is semi-decidable. When the answer is a YES (that means the machine is in L) the machine will halt but when it's a NO the machine does not halt and therefore we can't find out. However, I'm just confused how to find the complement (U - L) and U(all possible turing machines) is huge. So is compl of L option 1 or 2? or is it both? $\endgroup$ – alexW Oct 3 '19 at 23:45

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