Advertisement

Semantics of Higher-Order Recursion Schemes

  • Jiří Adámek
  • Stefan Milius
  • Jiří Velebil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)

Abstract

Higher-order recursion schemes are equations defining recursively new operations from given ones called “terminals”. Every such recursion scheme is proved to have a least interpreted semantics in every Scott’s model of λ-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite λ-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Whereas Fiore et al proved that the presheaf F λ of λ-terms is an initial H λ -monoid, we work with the presheaf R λ of rational infinite λ-terms and prove that this is an initial iterative H λ -monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in R λ .

Keywords

Higher-order recursion schemes infinite λ-terms sets in context rational tree 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adámek, J., Milius, S., Velebil, J.: Iterative algebras at work. Math. Structures Comput. Sci. 16, 1085–1131 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adámek, J., Trnková, V.: Automata and algebras in a category. Kluwer Academic Publishers, Dordrecht (1990)MATHGoogle Scholar
  3. 3.
    Aehlig, K.: A finite semantics of simply-typed lambda terms for infinite runs of automata. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 104–118. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Damm, W.: Higher-order program schemes and their languages. LNCS, vol. 48, pp. 51–72. Springer, Heidelberg (1979)Google Scholar
  5. 5.
    Fiore, M.: Second order dependently sorted abstract syntax. In: Proc. Logic in Computer Science 2008, pp. 57–68. IEEE Press, Los Alamitos (2008)Google Scholar
  6. 6.
    Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proc. Logic in Computer Science 1999, pp. 193–202. IEEE Press, Los Alamitos (1999)Google Scholar
  7. 7.
    Garland, S.J., Luckham, D.C.: Program schemes, recursion schemes and formal languages. J. Comput. Syst. Sci. 7, 119–160 (1973)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Guessarian, I.: Algebraic semantics. LNCS, vol. 99. Springer, Heidelberg (1981)MATHGoogle Scholar
  9. 9.
    Janelidze, G., Kelly, G.M.: A note on actions of a monoidal category. Theory Appl. Categ. 9, 61–91 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Lack, S.: On the monadicity of finitary monads. J. Pure Appl. Algebra 140, 65–73 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Matthes, R., Uustalu, T.: Substitution in non-wellfounded syntax with variable binding. Theoret. Comput. Sci. 327, 155–174 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Milius, S.: Completely iterative algebras and completely iterative monads. Inform. and Comput. 196, 1–41 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Milius, S., Moss, L.: The category theoretic solution of recursive program schemes. Theoret. Comput. Sci. 366, 3–59 (2006); corrigendum 403, 409–415 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Miranda, G.: Structures generated by higher-order grammars and the safety constraint. Ph.D. Thesis, Merton College, Oxford (2006)Google Scholar
  15. 15.
    Power, J.: A unified category theoretical approach to variable binding. In: Proc. MERLIN 2003 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jiří Adámek
    • 1
  • Stefan Milius
    • 1
  • Jiří Velebil
    • 2
  1. 1.Institut für Theoretische InformatikTechnische Universität BraunschweigGermany
  2. 2.Faculty of Electrical EngineeringCzech Technical University of PragueCzech Republic

Personalised recommendations