A New Foundation for Finitary Corecursion

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


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.


  1. 1.
    Aczel, P., Adámek, J., Milius, S., Velebil, J.: Infinite trees and completely iterative theories: a coalgebraic view. Theoret. Comput. Sci. 300, 1–45 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adámek, J., Milius, S., Velebil, J.: Semantics of higher-order recursion schemes. Log. Methods Comput. Sci. 7(1:15), 43 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Adámek, J., Milius, S., Velebil, J.: Equational properties of iterative monads. Inf. Comput. 208, 1306–1348 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Adámek, J., Milius, S., Velebil, J.: Elgot theories: a new perspective of the equational properties of iteration. Math. Struct. Comput. Sci. 21(2), 417–480 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Adámek, J.: Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolin. 015(4), 589–602 (1974)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Adámek, J., Milius, S., Velebil, J.: Iterative algebras at work. Math. Struct. Comput. Sci. 16(6), 1085–1131 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Adámek, J., Milius, S., Velebil, J.: On second-order iterative monads. Theoret. Comput. Sci. 412(38), 4969–4988 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Cambridge University Press, New York (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Barr, M.: Coequalizers and free triples. Math. Z. 116, 307–322 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bartels, F.: On generalized coinduction and probabilistic specification formats: Distributive laws in coalgebraic modelling. Ph.D. thesis, Vrije Universiteit Amsterdam.(2004)Google Scholar
  11. 11.
    Bloom, S.L., Ésik, Z.: Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bonsangue, M., Milius, S., Silva, A.: Sound and complete axiomatizations of coalgebraic language equivalence. ACM Trans. Comput. Log. 14((1: 7)), 52.(2013)Google Scholar
  13. 13.
    Bonsangue, M.M., Milius, S., Rot, J.: On the specification of operations on the rational behaviour of systems. In: Luttik, B., Reniers, M.A. (eds.) Proceedings of Combined Workshop on Expressiveness in Concurrency and Structural Operational Semantics (EXPRESS/SOS’12), Electronic Proceedings of Theoretical Computer Science, vol. 89, pp. 3–18.(2012)Google Scholar
  14. 14.
    Borceux, F.: Handbook of Categorical Algebra: Volume 1, Basic Category Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  15. 15.
    Cenciarelli, P., Moggi, E.: A syntactic approach to modularity in denotational semantic. In: Proceedings of 5th CTCS. CWI Technical report.(1993)Google Scholar
  16. 16.
    Courcelle, B.: Fundamental properties of infinite trees. Theoret. Comput. Sci. 25, 95–169 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata, 1st edn. Springer Publishing Company, Berlin (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Elgot, C.: Monadic computation and iterative algebraic theories. In: Rose, H.E., Sheperdson, J.C. (eds.) Logic Colloquium 1973, vol. 80, pp. 175–230. North-Holland Publishers, Amsterdam (1975)Google Scholar
  19. 19.
    Fliess, M.: Sur divers produits de séries formelles. Bulletin de la Société Mathématique de France 102, 181–191 (1974)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien. Lecture Notes in Mathematics, vol. 221. Springer, Heidelberg (1971)Google Scholar
  21. 21.
    Ghani, N., Lüth, C., Marchi, F.D.: Monads of coalgebras: rational terms and term graphs. Math. Struct. Comput. Sci. 15, 433–451 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Goncharov, Sergey: Trace Semantics via Generic Observations. In: Heckel, Reiko, Milius, Stefan (eds.) CALCO 2013. LNCS, vol. 8089, pp. 158–174. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  23. 23.
    Goncharov, Sergey, Milius, Stefan, Silva, Alexandra: Towards a Coalgebraic Chomsky Hierarchy. In: Diaz, Josep, Lanese, Ivan, Sangiorgi, Davide (eds.) TCS 2014. LNCS, vol. 8705, pp. 265–280. Springer, Heidelberg (2014)Google Scholar
  24. 24.
    Harrison, M.A., Havel, I.M.: Real-time strict deterministic languages. SIAM J. Comput. 1(4), 333–349 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hyland, M., Plotkin, G., Power, J.: Combining effects: sum and tensor. Theoret. Comput. Sci. 357(1–3), 70–99 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jacobs, Bart: A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages. In: Futatsugi, Kokichi, Jouannaud, Jean-Pierre, Meseguer, José (eds.) Algebra, Meaning, and Computation. LNCS, vol. 4060, pp. 375–404. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. 27.
    Klin, B.: Bialgebras for structural operational semantics: an introduction. Theoret. Comput. Sci. 412(38), 5043–5069 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lambek, J.: A fixpoint theorem for complete categories. Math. Z. 103, 151–161 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    MacLane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, 2nd edn. Springer, New York.(1998)Google Scholar
  30. 30.
    Milius, S.: A sound and complete calculus for finite stream circuits. In: Proceedings of 25th Annual Symposium on Logic in Computer Science (LICS 2010), pp. 449–458.(2010)Google Scholar
  31. 31.
    Milius, S., Bonsangue, M.M., Myers, R.S., Rot, J.: Rational operation models. Electron. Notes Theoret. Comput. Sci. 298, 257–282. In: Mislove, M. (ed.) Proceedings of 29th conference on Mathematical Foundations of Programming Science (MFPS XXIX).(2013)Google Scholar
  32. 32.
    Milius, S., Moss, L.S.: The category theoretic solution of recursive program schemes. Theoret. Comput. Sci. 366, 3–59 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Milius, S., Pattinson, D., Wißmann, T.: A new foundation for finitary corecursion: the locally finite fixpoint and its properties (2015)..
  34. 34.
    Milius, S., Wißmann, T.: Finitary corecursion for the infinitary lambda calculus. In: Proceedings of 6th Conference on Algebra and Coalgebra in Computer Science, CALCO 2015. Leibniz International Proceedings in Informatics.(2015)Google Scholar
  35. 35.
    Myers, R.: Rational coalgebraic machines in varieties: Languages, completeness and automatic proofs. Ph.D. thesis, Imperial College London, Department of Computing.(2011)Google Scholar
  36. 36.
    Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series, pp. 257–289.. Springer, Heidelberg (2009)Google Scholar
  37. 37.
    Plotkin, G., Turi, D.: Towards a mathematical operational semantics. In: Proceedings of 12th LICS Conference, pp. 280–291. IEEE Computer Society Press.(1997)Google Scholar
  38. 38.
    Rutten, J.J.M.M.: Automata and Coinduction. In: Sangiorgi, Davide, de Simone, Robert (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  39. 39.
    Rutten, J.: Universal coalgebra: a theory of systems. Theoret. Comput. Sci. 249(1), 3–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Silva, A., Bonchi, F., Bonsangue, M.M., Rutten, J.J.M.M.: Quantitative kleene coalgebras. Inf. Comput. 209(5), 822–849 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Silva, A., Bonchi, F., Bonsangue, M.M., Rutten, J.: Generalizing determinization from automata to coalgebras. Log. Meth. Comput. Sci. 9(1), 1–27 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Silva, A., Bonsangue, M.M., Rutten, J.: Non-deterministic Kleene coalgebras. Log. Meth. Comput. Sci. 6(3: 23), 39.(2010)Google Scholar
  43. 43.
    Winter, J., Bonsangue, M., Rutten, J.: Coalgebraic characterizations of context-free languages. Log. Meth. Comput. Sci. 9(3), 1–39 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Winter, J., Bonsangue, M.M., Rutten, J.J.: Context-free coalgebras. J. Comput. Syst. Sci. 81(5), 911–939 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Wißmann, T.: The locally finite fixpoint and its properties. Master’s thesis, Friedrich-Alexander Universität Erlangen-Nürnberg (April 2015).

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

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