A New Foundation for Finitary Corecursion and Iterative Algebras
Abstract
This paper contributes to a theory of the behaviour of "finitestate" systems that is generic in the system type. We propose that such systems are modelled 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 is finitary and 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 fgiterative algebra. As instances of the LFF we first obtain the known instances of the rational fixpoint, e.g. regular languages, rational streams and formal powerseries, regular trees etc. Moreover, we obtain a number of new examples, e.g. (realtime deterministic resp. nondeterministic) contextfree languages, constructively Salgebraic formal powerseries (in general, the behaviour of finite coalgebras under the coalgebraic language semantics arising from the generalized powerset construction by Silva, Bonchi, Bonsangue, and Rutten), and the monad of Courcelle's algebraic trees.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.08070
 Bibcode:
 2018arXiv180208070M
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 Extended journal version of arXiv:1601.01532