Final answer:
The language L = a²ⁿbaⁿ is regular and can be recognized by a deterministic finite automaton. It is not context-free, proven using the pumping lemma. The determinability of whether a Turing machine can recognize L requires more information.
Step-by-step explanation:
The language L = a2nban can be determined to be regular by providing a deterministic finite automaton (DFA) that recognizes it. The DFA would have three states: one for reading 'a', one for reading 'b', and one for accepting the string if it satisfies the given condition. The transitions between the states will depend on the input symbols. For example, the state for reading 'a' will transition to the state for reading 'b' after receiving an 'a' input, and the state for reading 'b' will transition back to the state for reading 'a' after receiving a 'b' input.
The language L is not context-free. One way to prove this is by using the pumping lemma for context-free languages. Suppose L is context-free, then there exists a pumping length p such that any string s in L, with |s| ≥ p, can be divided into five parts, u, v, x, y, and z, satisfying certain conditions. Using the string s = a2pbap, we can show that no matter how we divide s, one of the conditions of the pumping lemma will not hold.
It cannot be determined whether the language L can be recognized by a Turing machine without further information. The regularity of a language implies that it can be recognized by a finite automaton, but it does not guarantee that it can be recognized by a Turing machine. To determine if a language is Turing-recognizable, additional information about the language and its properties is required.