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Proper Functors and their Rational Fixed Point

Author Stefan Milius

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Stefan Milius

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Stefan Milius. Proper Functors and their Rational Fixed Point. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CALCO.2017.18

Abstract

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be a subcoalgebra of the final coalgebra. Inspired by Ésik and Maletti's notion of proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata.
Keywords
  • proper functor
  • proper semiring
  • coalgebra
  • rational fixed point

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References

  1. Jiří Adámek, Stefan Milius, and Jiří Velebil. Iterative algebras at work. Math. Structures Comput. Sci., 16(6):1085-1131, 2006. Google Scholar
  2. Jiří Adámek, Stefan Milius, and Jiří Velebil. On second-order iterative monads. Theoret. Comput. Sci., 412:4969-4988, 2011. Google Scholar
  3. Jiří Adámek, Stefan Milius, and Jiří Velebil. Semantics of higher-order recursion schemes. Log. Methods Comput. Sci., 7(1:15):43 pp., 2011. Google Scholar
  4. Jiří Adámek and Jiří Rosický. Locally presentable and accessible categories. Cambridge University Press, 1994. Google Scholar
  5. Jiří Adámek, Jiří Rosický, and Enrico Vitale. Algebraic Theories. Cambridge University Press, 2011. Google Scholar
  6. Jean Berstel and Christophe Reutenauer. Rational Series and Their Languages. Springer-Verlag, 1988. Google Scholar
  7. Marcello Bonsangue, Stefan Milius, and Alexandra Silva. Sound and complete axiomatizations of coalgebraic language equivalence. ACM Trans. Comput. Log., 14(1:7), 2013. Google Scholar
  8. Bruno Courcelle. Fundamental properties of infinite trees. Theoret. Comput. Sci., 25:95-169, 1983. Google Scholar
  9. Zoltán Ésik and Werner Kuich. Free iterative and iteration k-semialgebras. Algebra Univ., 67(2):141-162, 2012. Google Scholar
  10. Zoltán Ésik and Andreas Maletti. Simulation vs. equivalence. In Proc. 6th Int. Conf. Foundations of Computer Science, pages 119-122. CSREA Press, 2010. Google Scholar
  11. Zoltán Ésik and Andreas Maletti. Simulations of weighted tree automata. In Proc. CIAA'11, volume 6482 of Lecture Notes Comput. Sci., pages 321-330. Springer, 2011. Google Scholar
  12. Marcelo Fiore, Gordon D. Plotkin, and Daniele Turi. Abstract syntax and variable binding. In Proc. LICS'99, pages 193-202. IEEE Press, 1999. Google Scholar
  13. Peter Freyd. Rédei’s finiteness theorem for commutative semigroups. Proc. Amer. Math. Soc., 19(4), 1968. Google Scholar
  14. Susanna Ginali. Regular trees and the free iterative theory. J. Comput. System Sci., 18:228-242, 1979. Google Scholar
  15. Sergey Goncharov, Stefan Milius, and Alexandra Silva. Towards a coalgebraic Chomsky hierarchy (extended abstract). In Proc. TCS'14, volume 8705 of Lecture Notes Comput. Sci., pages 265-280. Springer, 2014. Google Scholar
  16. Bart Jacobs, Alexandra Silva, and Ana Sokolova. Trace semantics via determinization. J. Comput. System Sci., 2015. Google Scholar
  17. Stefan Milius. A sound and complete calculus for finite stream circuits. In Proc. LICS'10, pages 449-458. IEEE Computer Society, 2010. Google Scholar
  18. Stefan Milius. Proper functors and their rational fixed point. Full version; available at https://arxiv.org/abs/1705.09198, 2017.
  19. Stefan Milius, Dirk Pattinson, and Thorsten Wißmann. A new foundation for finitary corecursion: The locally finite fixpoint and its properties. In Proc. FoSSaCS'16, volume 9634 of Lecture Notes Comput. Sci. (ARCoSS), pages 107-125. Springer, 2016. Google Scholar
  20. Stefan Milius, Lutz Schröder, and Thorsten Wißmann. Regular behaviours with names: On rational fixpoints of endofunctors on nominal sets. Appl. Categ. Structures, 24(5):663-701, 2016. Google Scholar
  21. Stefan Milius and Thorsten Wißmann. Finitary corecursion for the infinitary lambda calculus. In Proc. CALCO'15, volume 35 of LIPIcs, pages 336-351, 2015. Google Scholar
  22. Ion Petre and Arto Salomaa. Algebraic systems and pushdown automata. In Handbook of Weighted Automata, pages 257-289. Springer, 2009. Google Scholar
  23. Lásló Rédei. The Theory of Finitely Generated Commutative Semigroups. Pergamon, Oxford-Edinburgh-New York, 1965. Google Scholar
  24. Jan Rutten. Automata and coinduction (an exercise in coalgebra). In Proc. CONCUR 1998, volume 1466 of Lecture Notes Comput. Sci., pages 194-218. Springer, 1998. Google Scholar
  25. Jan Rutten. Universal coalgebra: a theory of systems. Theoret. Comput. Sci., 249(1):3-80, 2000. Google Scholar
  26. Jan Rutten. Rational streams coalgebraically. Log. Methods Comput. Sci., 4(3:9):22 pp., 2008. Google Scholar
  27. Marcel Paul Schützenberger. On the definition of a family of automata. Inform. and Control, 4(2-3):275-270, 1961. Google Scholar
  28. Alexandra Silva, Filippo Bonchi, Marcello Bonsangue, and Jan Rutten. Generalizing determinization from automata to coalgebras. Log. Methods Comput. Sci, 9(1:9), 2013. Google Scholar
  29. Alexandra Silva, Marcello Bonsangue, and Jan Rutten. Non-deterministic Kleene coalgebras. Log. Methods Comput. Sci., 6(3:23):39 pp., 2010. Google Scholar
  30. Ana Sokolova and Harald Woracek. Congruences of convex algebras. J. Pure Appl. Algebra, 219(8):3110-3148, 2015. Google Scholar
  31. Henning Urbat. Finite behaviours and finitary corecursion. In Proc. CALCO'17, volume 72 of LIPIcs, pages 24:1-24:15, 2017. Google Scholar
  32. Joost Winter. A completeness result for finite λ-bisimulations. In Proc. FoSSaCS'15, volume 9034 of Lecture Notes Comput. Sci., pages 117-132, 2015. Google Scholar
  33. Joost Winter, Marcello Bonsangue, and Jan Rutten. Coalgebraic characterizations of context-free languages. Log. Methods Comput. Sci., 9(3), September 2013. Google Scholar
  34. Joost Winter, Marcello Bonsangue, and Jan Rutten. Context-free coalgebras. J. Comput. System Sci., 81(5):911-939, 2015. Google Scholar
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