Towards a Coalgebraic Chomsky Hierarchy

The Chomsky hierarchy plays a prominent role in the foundations of theoretical computer science relating classes of formal languages of primary importance as well as the grammars and machine models that represent those language classes. It is well-known that regular languages can be captured by coalgebras. They form precisely the image of all finite coalgebras for the set functor X 7→ 2×X under the unique homomorphisms into the final coalgebra, which is carried by all formal languages in this case. There have also been attempts to model context-free grammars (in Greibach normal form) coalgebraically [2, 5] and hence, to capture context-free languages. But what about other language classes and other machine models in the Chomsky Hierarchy? Here we bring together several ideas in order to treat machine models that are automata extended by extra storage; in particular, we show how to model machines with a stack or a Turing tape coalgebraically, and so we obtain the first coalgebraic account of recursively enumerable languages. Our central notion is that of a T-automaton, i.e. a finite coalgebra m : X → B × (TX), where A is a set of inputs, T = (T, η, μ) is a finitary monad, which we think as modelling extra storage, and B is an (EilenbergMoore) T-algebra a : TB → B of outputs. The semantics of a T-automaton can then be defined via the generalized power-set construction [4]: notice first that the functor LX = B ×X (being essentially given by product) lifts to the category of T-algebras; then one can extend the coalgebra structure m to the free T -algebra on X to obtain an L-coalgebra m : TX → B × (TX). Observe that the final L-coalgebra is carried by the set B ∗ of formal power-series on B. Finally, one defines the semantic map as J--Km = h · ηX : X → B ∗ , where h : TX → BA is the unique L-coalgebra homomorphism from (TX,m) to the final L-coalgebra. In order to model machines with pushdown storage we consider the stack monad: for a finite set Γ of stack symbols this is the submonad T of the store monad (--×Γ ∗)Γ for which elements 〈r, t〉 of TX satisfy the following restriction: there exists a k (depending on r, t) such that for every w ∈ Γ k and u ∈ Γ ∗ we have r(wu) = r and t(wu) = t(w)u. Goncharov [1, Proposition 5] has shown that the stack monad can be presented by operations and equations by the stack theory w.r.t. Γ = {γ1, . . . , γn}: this theory consists of the operations pop : n+ 1→ 1 and pushi : 1→ 1 for i = 1, . . . , n. These operations are subject to the following axioms: