What's a trivial property?

Question

I have to show a property P is trivial.

This problem has to do with Rice's Theorem, which I do not completely understand. Can someone explain the difference between trivial and non-trivial properties?

Answer 1

A "trivial" property is one that holds either for all languages or for none.

Answer 2

A "property" is simply a subset of languages in $RE$ -- the set of all the languages that "satisfy" that property. A non-trivial property $P$ is a non-empty set $P$ which is strictly contained in $RE$, that is $\emptyset \subset P\subset RE$. It means there is at least one RE language that doesn't satisfy $P$, and at least one RE language that does satisfy it.

Examples:

The property: "the language is finite." Formal definition: $P = \{ L \mid |L|<\infty\}$. This property is non-trivial.

The property: "The language can be recognized by a TM". Formally: $P=\{ L \mid \exists M \text{ s.t. } L=L(M)\}$. This property is trivial: in fact $P=RE$, that is, ALL RE languages satisfy it.

The property: "The language has a reduction from the non-halting problem". Formally: $P = \{ L \mid \overline{HP} \le L\} \cap RE$. This property is trivial: in fact, $P=\emptyset$.