I think the answer is no, and that the main difference between human language and other languages is its grammatical structure.

The languages of other creatures seem to be context free, or just regular expressions of the type "if your hear sound X replace that with concept or behavior Y". Human language is a lot more generative and context dependent, allowing nested or recursive structures. A Turing Complete language allows arbitrary types of machines to be supported within the culture. Of course in practice these machines are never completely arbitrary, since there are plenty of biological constraints.