On the semantic equivalence of language syntax formalisms

Several formalisms for language syntax specification exist in literature. In this paper, we prove that longstanding syntactical transformations between context-free grammars and algebraic signatures are adjoint functors and/or adjoint equivalences that preserve the abstract syntax of the generated terms. The main result is a categorical equivalence between the categories of algebras (i.e., all the possible semantics) over the objects in these formalisms up to the provided syntactical transformations, namely that all these language specification frameworks are essentially the same from a semantic perspective.