Context-Free Languages via Coalgebraic Trace Semantics

  • Ichiro Hasuo
  • Bart Jacobs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3629)


We show that, for functors with suitable mild restrictions, the initial algebra in the category of sets and functions gives rise to the final coalgebra in the (Kleisli) category of sets and relations. The finality principle thus obtained leads to the finite trace semantics of non-deterministic systems, which extends the trace semantics for coalgebras previously introduced by the second author. We demonstrate the use of our technical result by giving the first coalgebraic account on context-free grammars, where we obtain generated context-free languages via the finite trace semantics. Additionally, the constructions of both finite and possibly infinite parse trees are shown to be monads. Hence our extension of the application domain of coalgebras identifies several new mathematical constructions and structures.


Parse Tree Shapely Functor Main Technical Result Initial Algebra Trace Semantic 
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  1. [AAMV03]
    Aczel, P., Adáamek, J., Milius, S., Velebil, J.: Infinite trees and completely iterative theories: a coalgebraic view. Theor. Comp. Sci. 300, 1–45 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AK95]
    Adámek, J., Koubek, V.: On the greatest fixed point of a set functor. Theor. Comp. Sci. 150(1), 57–75 (1995)zbMATHCrossRefGoogle Scholar
  3. [ASU86]
    Aho, A.V., Sethi, R., Ullman, J.D.: Compilers: Principles, Techniques, and Tools. Addison-Wesley series in Computer Science. Addison-Wesley, Reading (1986)Google Scholar
  4. [Bar93]
    Barr, M.: Terminal coalgebras in well-founded set theory. Theor. Comp. Sci. 114, 299–315 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Bar04]
    Bartels, F.: On Generalized Coinduction and Probabilistic Specification Formats: Distributive Laws in Coalgebraic Modelling. PhD thesis, Free Univ. Amsterdam (2004)Google Scholar
  6. [BW83]
    Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Heidelberg (1983), Available free for downloading at
  7. [Cho56]
    Chomsky, N.: Three models for the description of language. IRE Transactions on Information Theory 2, 113–124 (1956)CrossRefGoogle Scholar
  8. [CKW91]
    Carboni, A., Kelly, G., Wood, R.: A 2-categorical approach to change of base and geometric morphisms I. Cah. de Top. et Géom. Diff. 32(1), 47–95 (1991)zbMATHMathSciNetGoogle Scholar
  9. [Jac04a]
    Jacobs, B.: Relating two approaches to coinductive solution of recurisve equations. In: Coalgebraic Methods in Computer Science (CMCS 2004). Elect. Notes in Theor. Comp. Sci, vol. 106. Elsevier, Amsterdam (2004)Google Scholar
  10. [Jac04b]
    Jacobs, B.: Trace semantics for coalgebras. In: Coalgebraic Methods in Computer Science (CMCS 2004). Elect. Notes in Theor. Comp. Sci, vol. 106. Elsevier, Amsterdam (2004)Google Scholar
  11. [Jac05a]
    Jacobs, B.: A bialgebraic review of regular expressions, deterministic automata and languages. Techn. Rep. NIII-R05003, Inst. for Computing and Information Sciences, Radboud Univ. Nijmegen (2005)Google Scholar
  12. [Jac05b]
    Jacobs, B.: Introduction to coalgebra. Towards mathematics of states and observations. Draft of a book (2005),
  13. [Jay95]
    Jay, C.: A semantics for shape. Science of Comput. Progr. 25, 251–283 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [LP81]
    Lewis, H.R., Papadimitriou, C.H.: Elements of the Theory of Computation. Prentice-Hall, Englewood Cliffs (1981)zbMATHGoogle Scholar
  15. [Rut00]
    Rutten, J.: Universal coalgebra: a theory of systems. Theor. Comp. Sci. 249, 3–80 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Rut03]
    Rutten, J.: Behavioural differential equations: a coinductive calculus of streams, automata, and power series. Theor. Comp. Sci. 308, 1–53 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Wor]
    A. (Sokolova) Woracek. Personal communicationGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ichiro Hasuo
    • 1
  • Bart Jacobs
    • 1
  1. 1.Institute for Computing and Information SciencesRadboud University NijmegenNijmegenThe Netherlands

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