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Typical examples of formal languages would be the set of all words over {a,b}, the set {an|n is prime}, the set of syntactically correct programs in some programming language, or the set of inputs upon which a certain Turing machine halts.
Formal languages can be defined in a great variety of ways, such as being produced by some formal grammar, being accepted by some Turing machine or some sort of automaton, or being some suitable encoding of a combinatiorial problem.
A typical questions asked about a formal language is how difficult it is to decide whether a given word belongs to the language. This is the domain of [computability theory]? and [complexity theory]?.