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Domains via graphs

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Abstract

This paper provides a concrete and simple introduction to two pillars of domain theory: (1) solving recursive domain equations, and (2) universal and saturated domains. Our exposition combines Larsen and Winskel’s idea on solving domain equations using information systems with Girard’s idea of stable domain theory in the form of coherence spaces, or graphs. Detailed constructions are given for universal and even homogeneous objects in two categories of graphs: one representing binary complete, prime algebraic domains with complete primes covering the bottom; the other representingw-algebraic, prime algebraic lattices. The back-and-forth argument in model theory helps to enlighten the constructions.

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References

  1. Scott D S. Domains for denotational semantics.Lecture Notes in Computer Science 140, 1982, pp.577–613.

    Article  Google Scholar 

  2. Girard Jean-Yves. Linear logic.Theoretical Computer Science, 1987, 50: 1–102.

    Article  MATH  Google Scholar 

  3. Larsen K, Winskel G. Using information systems to solve recursive domain equations effectively.Lecture Notes in Computer Science 173, 1984, pp.109–130.

    MathSciNet  Google Scholar 

  4. Droste M, Göbel R. Universal domains in the theory of denotational semantics of programming languages. InProc. the IEEE 5th Annual Symposium on Logic in Computer Science Philadelphia, 1990, pp.19–34.

  5. Barwise Jon. Back and forth through infinitary logic.Studies in Model Theory, MAA Studies in Math., 1973, 8: 5–34.

    Google Scholar 

  6. Winskel G. An introduction to event structures.Lecture Notes in Computer Science 354, 1988, pp.364–399.

    Article  MathSciNet  Google Scholar 

  7. Berry G, Curien P-L. Sequential algorithms on concrete data structures.Theoretical Computer Science, 1982 20: 265–321.

    MATH  MathSciNet  Google Scholar 

  8. Chen Y-X. Stone duality and representation of stable domain.International Journal of Computer and Mathematics with Application, 1997, 34: 27–34.

    Article  MATH  Google Scholar 

  9. Zhang G-Q. DI-domains as prime information systems.Information and Computation, 1992, 100(2): 151–177.

    Article  MATH  MathSciNet  Google Scholar 

  10. Zhang G-Q. Some monoidal closed categories of domains and event structures.Mathematical Structures in Computer Science, 1993, 3: 259–276.

    Article  MATH  Google Scholar 

  11. Zhang G-Q. Logic of Domains. Birkhauser, Boston, 1991.

    MATH  Google Scholar 

  12. Zhang G-Q. Universal quasi-prime algebraic domains. InProceedings of the 9th International Conference on the Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science 802, 1994, pp.454–473.

  13. Zhang G-Q. The largest Cartesian closed category of stable domain.Theoretical Computer Science, 1996, 166: 203–219.

    Article  MATH  MathSciNet  Google Scholar 

  14. Bacsich P, Hughs D R. Syntactic characterisations of amalgamation, convexity and related properties.The Journal of Symbolic Logic, 1974, 39(3): pp.433–451.

    Article  MATH  MathSciNet  Google Scholar 

  15. Asperti A, Longo G. Categories, Types and Structures. MIT Press, 1991.

  16. Gunter C. Universal profinite domains.Information and Computation, 1987, 72: 1–30.

    Article  MATH  MathSciNet  Google Scholar 

  17. Gunter C, Jung A. Coherence and consistency in domains.Journal of Pure and Applied Algebra., 1990, 63: 49–66.

    Article  MATH  MathSciNet  Google Scholar 

  18. Plotkin G.T w as a universal domain.Journal of Computer and System Sciences, 1978, 17: 209–236.

    Article  MATH  MathSciNet  Google Scholar 

  19. Scott D S. Data types as lattices.SIAM J. Computing, 1976, 5: 522–587.

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Case Western Reserve University, 44106, Cleveland, OH, USA

    Zhang Guoqiang

  2. College of Mathematical Science, Shanghai Teachers University, 200234, Shanghai, P.R. China

    Chen Yixiang

Authors
  1. Zhang Guoqiang
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  2. Chen Yixiang
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Corresponding author

Correspondence to Zhang Guoqiang.

Additional information

This work is supported by the National Natural Science Foundation of China (No.69873034), the Foundation for University Key Teachers by the Ministry of Education of China, and the Shu Guang Project by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (99SG46).

ZHANG Guoqiang received his Ph.D. degree in CS from Cambridge University, England in 1990. He is currently an associate professor at the EECS Department of CWRU, after serving as a faculty member at the University of Georgia. He held visiting positions at CWRU (98–99), the University of Michigan (93–95), and Aarhus University (88–89). He is the author of the research monograph “Logic of Domains” (Birkhauser, 1991), the editor-in-chief of the Kluwer book series “Semantic Structures in Computation”, and program committee co-chair of the International Domain Theory Conference series. He has published extensively in leading computer science journals such as Theoretical Computer Science, Information and Computation, and conferences such as ICALP, MFPS, LICS, and AAAI.

CHEN Yixiang received the Ph.D. degree in mathematics from Sichuan University in 1995. He worked as a post-doctor at the College of Mathematical Science of Suzhou University from Dec. 1995 to Sept. 1997. Now he is a professor at the College of Mathematical Science of Shanghai Teachers University. His research interests include formal semantics of programming, domain theory, and fuzzy systems.

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Zhang, G., Chen, Y. Domains via graphs. J. Comput. Sci. & Technol. 16, 505–521 (2001). https://doi.org/10.1007/BF02943235

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  • DOI: https://doi.org/10.1007/BF02943235

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