How to determine which class these languages belong to?

Question

{WW} - Decidable but not Context free
{WW^R} - Context Free, but not in Regular
Σ* - Regular language
How can you determine which class they belong to?

Answer 1

May be my answer helpful to you:

L₁ = {ww | w ∈ {a, b}^* }

is not context Free Language because a (Push down Automata) PDA is not possible (even Non-Deterministic-PDA ). Why? suppose you push first w in stack. To match second w with first w you have to push first w in reverse order (either you need to match second w in reverse order with stack content) that is not possible with stack (and we can't read input in reverse order). Although its decidable because be can draw a Turing Machine for L₁ that always half after finite number of steps.

L₃ = {ww^R | w ∈ {a, b}^* }

Language L₃ is a Non-Deterministic Context Free Language, because n-PDA is possible but Finite Automate is not possible for L₃. you can also proof this using Pumping Lemma for Regular Languages.

Σ* - Regular Language(RL)

Σ* is Regular Expression (RE) e.g if Σ = {a, b} then RE is (a + b)* RE is possible only for RLs.

The examples in my question may be more helpful to you.