Regular Behaviours with Names

Nominal sets provide a framework to study key notions of syntax and semantics such as fresh names, variable binding and α-equivalence on a conveniently abstract categorical level. Coalgebras for endofunctors on nominal sets model, e.g., various forms of automata with names as well as infinite terms with variable binding operators (such as λ-abstraction). Here, we first study the behaviour of orbit-finite coalgebras for functors F̄ on nominal sets that lift some finitary set functor F . We provide sufficient conditions under which the rational fixpoint of F̄ , i.e. the collection of all behaviours of orbit-finite F̄ -coalgebras, is the lifting of the rational fixpoint of F . Second, we describe the rational fixpoint of the quotient functors: we introduce the notion of a sub-strength of an endofunctor on nominal sets, and we prove that for a functor G with a sub-strength the rational fixpoint of each quotient of G is a canonical quotient of the rational fixpoint of G. As applications, we obtain a concrete description of the rational fixpoint for functors arising from so-called binding signatures with exponentiation, such as those arising in coalgebraic models of infinitary λ-terms and various flavours of automata.