Skip to main content
SpringerLink
Log in

Sequents, Frames, and Completeness

  • Conference paper
  • First Online:
Book cover Computer Science Logic (CSL 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1862))

Included in the following conference series:

Abstract

Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the unifying principle of the spatiality of coherence logic. In particular, the domain of disjunctive states, derived from the hyperresolution rule as used in disjunctive logic programs, can be seen as the frame freely generated from the opposite of a sequent structure. At the categorical level, we present equivalences among the categories of sequent structures, distributive lattices, and spectral locales using appropriate morphisms.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. S. Abramsky and A. Jung, Domain theory, in: Handbook of Logic in Computer Science, Vol 3, (Clarendon Press, 1995).

    Google Scholar 

  2. G. Birkho.. On the combination of subalgebras. Proc. Camb. Philos. Soc. 29, 441–464, 1933.

    Article  Google Scholar 

  3. J. Cederquist and Th. Coquand. Entailment relations and distributive lattices. To appear in the Proceedings of Logic Colloquium 98.

    Google Scholar 

  4. Th. Coquand and H. Persson. Valuations and Dedekind Prague Theorem. To appear in the Journal of Pure and Applied Logic.

    Google Scholar 

  5. M. Droste and R. Göbel. Non-deterministic information systems and their domains. Theoretical Computer Science 75, 289–309, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Fourman, R. Grayson. Formal Spaces. in L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), 107–122, North-Holland, Amsterdam-New York, 1982.

    Google Scholar 

  7. P. Johnstone. Stone Spaces. Cambridge University Press, 1982.

    Google Scholar 

  8. A. Jung, M.A.M. Moshier and M. Kegelmann. Multi lingual sequent calculus and coherent spaces. Fundamenta Informaticae, vol 37, 1999, pages 369–412.

    MATH  MathSciNet  Google Scholar 

  9. G. Gentzen Collected Works. Edited by Szabo, Not-Holland, 1969.

    Google Scholar 

  10. V. Lifschitz. Semantical completeness theorems in logic and algebra. Proc. Am. Math. Soc., vol. 79, 1980, p. 89–96.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. MacLane, Categories for the working mathematician. Springer-Verlag, 1971.

    Google Scholar 

  12. J.A. Robinson. The generalised resolution principle. Machine Intelligence, vol. 3, p. 77–93. 1968.

    MATH  Google Scholar 

  13. W. Rounds and G.-Q. Zhang. Clausal logic and logic programming in algebraic domains. Submitted. Copy at: http://www.cs.uga.edu/~gqz

  14. G. Sambin. Pretopologies and completeness proofs. J. Symbolic Logic 60 (1995), no. 3, 861–878.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Scott. Completeness and axiomatizability. Proceedings of the Tarski Symposium, 1974, p. 411–435.

    Google Scholar 

  16. D. Scott. Background to formalisation. in Truth, Syntax and Modality, H. Leblanc, ed., p. 411–435, 1973.

    Google Scholar 

  17. D. Scott. On engendring an illusion of understanding. Journal of Philosophy, p. 787–807, 1971.

    Google Scholar 

  18. D. Scott. Domains for denotational semantics. in: Lecture Notes in Computer Science 140, 577–613, 1982.

    Chapter  Google Scholar 

  19. M. H. Stone. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111, 1936.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Tarski. Logic, semantics, metamathematics. Oxford, 1956.

    Google Scholar 

  21. G.-Q. Zhang. Logic of Domains. Birkhauser Boston, Inc., Boston, MA, 1991.

    MATH  Google Scholar 

  22. G.-Q. Zhang. Disjunctive systems and L-domains. 19th International Colloquium on Automata, Languages, and Programming (ICALP’92), Lecture Notes in Computer Science 623, 1992, pp. 284–295.

    Google Scholar 

  23. G.-Q. Zhang and W. C. Rounds. An information-system representation of the Smyth powerdomain. International Symposium on Domain Theory. Shanghai, China, October 1999. Copy at: http://www.cs.uga.edu/~gqz

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Göteborg, S 412 96, Göteborg, Sweden

    Thierry Coquand

  2. Department of Computer Science, University of Georgia, Athens, GA, 30602, USA

    Guo-Qiang Zhang

Authors
  1. Thierry Coquand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guo-Qiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-Qiang Zhang .

Editor information

Editors and Affiliations

  1. Department of Computer Science and Department of Biology, Boston College, Fulton Hall 410B, Chestnut Hill, MA, 02467, USA

    Peter G. Clote

  2. Mathematisches Institut, Ludwig-Maximilian-Universität München, Theresienstr. 39, 80333, München, Germany

    Helmut Schwichtenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Coquand, T., Zhang, GQ. (2000). Sequents, Frames, and Completeness. In: Clote, P.G., Schwichtenberg, H. (eds) Computer Science Logic. CSL 2000. Lecture Notes in Computer Science, vol 1862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44622-2_18

Download citation

  • DOI: https://doi.org/10.1007/3-540-44622-2_18

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67895-3

  • Online ISBN: 978-3-540-44622-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation