Context-Free Languages via Coalgebraic Trace Semantics

In this paper we identify context-free grammars as coalgebras. To obtain the associated context-free languages (consisting of only finite-length strings) we introduce a general and novel technique of finite trace semantics for coalgebras. It builds on top of the (possibly infinite) trace semantics introduced earlier by the second author, but extracts only finite behavior. Interestingly the finite trace is uniquely characterized corecursively and hence it yields a final coalgebra in a suitable Kleisli category, while the ordinary trace is not unique and yields a weakly final coalgebra. Additionally, the constructions of both finite and possibly infinite parse trees are shown to be monads. Hence our extension of the application domain of coalgebras identifies several new mathematical constructions and structures.