Advertisement

A Logic of Coequations

  • Jiri Adámek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)

Abstract

By Rutten’s dualization of the Birkhoff Variety Theorem, a collection of coalgebras is a covariety (i.e., is closed under coproducts, subcoalgebras, and quotients) iff it can be presented by a subset of a cofree coalgebra. We introduce inference rules for these subsets, and prove that they are sound and complete. For example, given a polynomial endofunctor of a signature Σ, the cofree coalgebra consists of colored Σ-trees, and we prove that a set T of colored trees is a logical consequence of a set S iff T contains every tree such that all recolorings of all its subtrees lie in S. Finally, we characterize covarieties whose presentation needs only n colors.

Keywords

Logical Consequence Inference Rule Free Algebra Small Cardinal Unique Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adámek, J.: Free algebras and automata realizations in the language of categories. Commentationes Mathematicae Universitatis Carolinae 15, 589–602 (1974)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Adámek, J.: On a description of terminal coalgebras and iterative theories. Electronic Notes in Theoretical Computer Science, vol. 82.1 (2003); Full version in Information and Computation (to appear)Google Scholar
  3. 3.
    Adámek, J.: Birkhoff’s covariety theorem without limitation. Commentationes Mathematicae Universitatis Carolinae 46, 197–215 (2005)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Adámek, J., Milius, S., Velebil, J.: On coalgebra based on classes. Theoretical Computer Science 316, 3–23 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Adámek, J., Porst, H.-E.: On tree coalgebras and coalgebra presentations. Theoretical Computer Science 311, 257–283 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Arbib, M.A., Manes, E.G.: Parametrized data types do not need highly constrained parameters. Information and Control 52, 130–158 (1982)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Awodey, S., Hughes, J.: Modal operators and the formal dual of Birkoff’s completeness theorem. Mathematical Structures in Computer Science 13, 233–258 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Barr, M.: Terminal coalgebras in well-founded set theory. Theoretical Computer Science 124, 182–192 (1984)MathSciNetGoogle Scholar
  9. 9.
    Birkhoff, G.: On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society 31, 433–454 (1935)CrossRefGoogle Scholar
  10. 10.
    Gumm, H.P.: Elements of the general theory of coalgebras (preprint 1999)Google Scholar
  11. 11.
    Gumm, H.P.: Birkoff’s variety theorem for coalgebras. Contributions to General Algebra 13, 159–173 (2000)Google Scholar
  12. 12.
    Gumm, H.P., Schröder, T.: Covarieties and complete covarieties. Electronic Notes in Theoretical Computer Science 11 (1998)Google Scholar
  13. 13.
    Jacobs, B.: The temporal logic of coalgebras via Galois algebras. Mathematical Structures in Computer Science 12, 875–903 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rutten, J.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 3–80 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Worrell, J.: On the final sequence of a finitary set functor. Theoretical Computer Science 338, 184–199 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jiri Adámek
    • 1
  1. 1.Institute of Theoretical Computer ScienceTechnical UniversityBraunschweigGermany

Personalised recommendations