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Rice's theorem: Any nontrivial property about the language recognized by a Turing machine is undecidable.

I know the proof but I am not able to understand the theorem. I think trivial properties are like :language recognised by Turing machine contain at substring "aab". where as nontrivial property means is language accepted by Turing machine is finite etc.

My question : Give at least two example of trivial properties and two non-trivial property about the language recognized by a Turing machine.

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In the context of Rice's theorem, a trivial property has a very specific meaning.

Here, we define a property to be a set of Turing-recognizable languages. A property is non-trivial unless it contains no languages, or contains all Turing-recognizable languages.

We can clearly recognize these with a Turing Machine, since we can just make a machine that always says "YES" for the first case, or make a machine that always says "NO" for the second.

So both the properties you have identified are non-trivial properties, and are thus undecidable by Rice's theorem.

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