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I was wondering how I can show that the language $\{a^n b^n c^n \mid n \geq 0 \}$ is Turing-recognizable. Also, if it is Turing-decidable?

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    $\begingroup$ You can prove it by describing a Turing machine that accepts the language. $\endgroup$ – siracusa Oct 22 at 4:24
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A language is Turing decidable if you can write a C program (replace C with your favorite programming language) that outputs YES if the input belongs to the language and outputs NO otherwise. It is Turing recognizable if in the latter case the C program simply never halts.

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