Skip to main content
SpringerLink
Log in

Determinisation of Relational Substitutions in Ordered Categories with Domain

  • Conference paper
Relations and Kleene Algebra in Computer Science (RelMiCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4988))

Included in the following conference series:

  • 395 Accesses

Abstract

We present two different relational generalisations of substitutions, show that they both produce locally ordered categories with domain, and then develop the single-morphism “determiniser” concept that relies only on this framework, while still corresponding to conventional two-morphism unification in both examples. Central to this development is the determinacy concept of “characterisation by domain” introduced by Desharnais and Möller for Kleene algebras with domain; this is here applied in the weakest possible setting.

This research is supported by NSERC (National Science and Engineering Research Council of Canada).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Backhouse, R.C.: Constructive Lattice Theory (1993), http://www.cs.nott.ac.uk/~rcb/papers/abstract.html#isos

  2. Beck, J.: Distributive laws. In: Appelgate, H., Eckmann, B. (eds.) Seminar on triples and categorical homology theory, ETH, 1966-67. Lect. Notes in Math., vol. 80, pp. 119–140. Springer, Heidelberg (1969)

    Chapter  Google Scholar 

  3. Desharnais, J., Möller, B.: Characterizing Determinacy in Kleene Algebras. Information Sciences 139, 253–273 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Desharnais, J., Möller, B., Struth, G.: Kleene Algebra with Domain. ACM Transactions on Computational Logic 7(4), 798–833 (2006)

    MathSciNet  Google Scholar 

  5. Eklund, P., Galán, M.A., Ojeda-Aciego, M., Valverde, A.: Set functors and generalised terms. In: IPMU 2000, 8th Information Proc. and Management of Uncertainty in Knowledge-Based Systems Conference, vol. III, pp. 1595–1599 (2000)

    Google Scholar 

  6. Eklund, P., Galán, M.A., Medina, J., Ojeda Aciego, M., Valverde, A.: A categorical approach to unification of generalised terms. Electronic Notes in Computer Science, 66(5), 41–51 (2002), Special Issue: UNCL 2002, Unification in Non-Classical Logics (ICALP 2002 Satellite Workshop)

    Google Scholar 

  7. Eklund, P., Galán, M.A., Medina, J., Ojeda Aciego, M., Valverde, A.: Set functors, L-Fuzzy Set Categories, and Generalized Terms. Computers and Mathematics with Applications 43(6–7), 693–705 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Eklund, P., Gähler, W.: Partially ordered monads and powerset Kleene algebras. In: Proc. 10th Information Processing and Management of Uncertainty in Knowledge Based Systems Conference (IPMU 2004) (2004)

    Google Scholar 

  9. Goguen, J.A.: What is Unification? In: Aït-Kaci, H., Nivat, M. (eds.) Resolution of Equations in Algebraic Structures, Algebraic Techniques, vol. 1, pp. 217–261. Academic Press, Boston (1989)

    Google Scholar 

  10. Jones, M.P., Duponcheel, L.: Composing Monads. Research Report YALEU/DCS/RR-1004, Yale University, New Haven, Connecticut, USA (1993)

    Google Scholar 

  11. Kahl, W.: Refactoring Heterogeneous Relation Algebras around Ordered Categories and Converse. J. Relational Methods in Comp. Sci. 1, 277–313 (2004)

    Google Scholar 

  12. Kawahara, Y.: Notes on the Universality of Relational Functors. Mem. Fac. Sci. Kyushu Univ. Ser. A 27(2), 275–289 (1973)

    MATH  MathSciNet  Google Scholar 

  13. Kozen, D.: Natural Transformations as Rewrite Rules and Monad Composition. Techn. Rep. TR2004-1942, Computer Science Dept., Cornell University (2004)

    Google Scholar 

  14. Lawvere, F.W.: Functorial Semantics of Algebraic Theories. Proc. Nat. Acad. Sci. USA 50, 869–872 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  15. Marciniec, J.: Infinite Set Unification with Application to Categorial Grammar. Studia Logica 58, 339–355 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rydeheard, D., Burstall, R.: A categorical unification algorithm. In: Poigné, A., Pitt, D.H., Rydeheard, D.E., Abramsky, S. (eds.) Category Theory and Computer Programming. LNCS, vol. 240, pp. 493–505. Springer, Heidelberg (1986)

    Google Scholar 

  17. Spivey, J.M.: The Z Notation: A Reference Manual, Prentice Hall International Series in Computer Science, 2nd edn. Prentice Hall, Englewood Cliffs (1992), http://spivey.oriel.ox.ac.uk/~mike/zrm/

    Google Scholar 

  18. Winter, M.: Goguen Categories: A Categorical Approach to L-fuzzy Relations. In: Trends in Logic, vol. 25 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. McMaster University, Hamilton, Ontario, Canada

    Wolfram Kahl

Authors
  1. Wolfram Kahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rudolf Berghammer Bernhard Möller Georg Struth

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kahl, W. (2008). Determinisation of Relational Substitutions in Ordered Categories with Domain . In: Berghammer, R., Möller, B., Struth, G. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2008. Lecture Notes in Computer Science, vol 4988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78913-0_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-78913-0_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78912-3

  • Online ISBN: 978-3-540-78913-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation