A New Foundation for Finitary Corecursion

Abstract

Abstract. This paper contributes to a theory of the behaviour of "finite-state" systems that is generic in the system type. We propose that such systems are modeled as coalgebras with a finitely generated carrier for an endofunctor on a locally finitely presentable category. Their behaviour gives rise to a new fixpoint of the coalgebraic type functor called locally finite fixpoint (LFF). We prove that if the given endofunctor preserves monomorphisms then the LFF always exists and is a subcoalgebra of the final coalgebra (unlike the rational fixpoint previously studied by Adámek, Milius and Velebil). Moreover, we show that the LFF is characterized by two universal properties: 1. as the final locally finitely generated coalgebra, and 2. as the initial fg-iterative algebra. As instances of the LFF we first obtain the known instances of the rational fixpoint, e.g. regular languages, rational streams and formal power-series, regular trees etc. And we obtain a number of new examples, e.g. (realtime deterministic resp. non-deterministic) context-free languages, constructively S-algebraic formal power-series (and any other instance of the generalized powerset construction by Silva, Bonchi, Bonsangue, and Rutten) and the monad of Courcelle's algebraic trees.

Product Rules and Distributive Laws

“…Note that, when instantiating the above construction with the laws λ 0 from Section 2.2, λ brz , and λ 2 from 2.3, the resulting composite distributive law is a law between the monad S − and the cofree copointed endofunctor on K × − A . This law differs from (but is closely related to) the laws obtained in [WBR13] and [MPW16], in which a distributive law is defined between a suitably defined monad on K A + − and the endofunctor K × − A (without copoint). However, the resulting lawλ still has the property, that a language is contextfree (resp.…”