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A New Foundation for Finitary Corecursion

The Locally Finite Fixpoint and Its Properties
  • Stefan Milius
  • Dirk Pattinson
  • Thorsten Wißmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9634)

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Stefan Milius
    • 1
  • Dirk Pattinson
    • 2
  • Thorsten Wißmann
    • 1
  1. 1.Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  2. 2.The Australian National UniversityCanberraAustralia

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