Abstract
In our previous work [17] we have shown that for any ω-algebraic meet-cpo D, if all higher-order stable function spaces built from D are ω-algebraic, then D is finitary. This accomplishes the first of a possible, two-step process in solving the problem raised in [1,2]: whether the category of stable bifinite domains of Amadio-Droste-Göbel [1,6] is the largest cartesian closed full sub-category within the category of ω-algebraic meet-cpos with stable functions. This paper presents results on the second step, which is to show that for any ω-algebraic meet-cpo D satisfying axioms \({\sf M}\) and \({\sf I}\) to be contained in a cartesian closed full sub-category using ω-algebraic meet-cpos with stable functions, it must not violate \({\sf MI}^{\infty}\;\). We introduce a new class of domains called weakly distributive domains and show that for these domains to be in a cartesian closed category using ω-algebraic meet-cpos, property \({\sf MI}^{\infty}\;\) must not be violated. We further demonstrate that principally distributive domains (those for which each principle ideal is distributive) form a proper subclass of weakly distributive domains, and Birkhoff’s M 3 and N 5 [5] are weakly distributive (but non-distributive). We introduce also the notion of meet-generators in constructing stable functions and show that if an ω-algebraic meet-cpo D contains an infinite number of meet-generators, then [D →D] fails \({\sf I}\). However, the original problem of Amadio and Curien remains open.
This work is partially supported by NSFC 60673045, NSFC 60373050, NSFC major research program 60496321 and NSFC 60421001.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amadio, R.-M.: Bifinite domains: stable case. LNCS, vol. 530, pp. 16–33. Springer, Heidelberg (1991)
Amadio, R.-M., Curien, P.-L.: Domains and Lambda-Calculi. Cambridge Tracts in Theeoretical Computer Science, vol. 46. Cambridge University Press, Cambridge (1998)
Berry, G.: Modèles complètement adéquats et stables des lambda-calculs typés. Thèse de Doctorat d’Etat, Université Paris VII (1979)
Coquand, T., Gunter, C.A., Winskel, G.: DI-domains as a model of polymorphism. LNCS, vol. 298, pp. 344–363. Springer, Heidelberg (1987)
Davey, B.-A., Priestley, H.-A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)
Droste, M.: On stable domains. Theoretical Computer Science 111, 89–101 (1993)
Droste, M.: Cartesian closed categories of stable domains for polymorphism. Preprint, Universität GHS Essen
Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)
Milner, R.: Fully abstract models of typed λ-calculi. Theoretical Computer Science 4, 1–22 (1977)
Plotkin, G.: A powerdomain construction. SIAM J. Computing 5, 452–487 (1976)
Reddy, U.: Global state considered unnecessary: An introduction to object-based semantics. J. Lisp and Symbolic Computation 9, 7–76 (1996)
Smyth, M.B.: The largest cartesian closed category of domains. Theoretical Computer Science 27, 109–120 (1983)
Winskel, G.: Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. LNCS, vol. 354, pp. 364–399. Springer, Heidelberg (1988)
Zhang, G.-Q.: dI-domains as prime information systems. Information and Computation 100, 151–177 (1992)
Zhang, G.-Q.: The largest cartesian closed category of stable domains. Theoretical Computer Science 166, 203–219 (1996)
Zhang, G.-Q.: Logic of Domains. Birkhauser (1991)
Zhang, G.-Q., Jiang, Y.: On a problem of Amadio and Curien: The finite antichain condition. Information and Computation 202, 87–103 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Jiang, Y., Zhang, GQ. (2007). Weakly Distributive Domains. In: Della Rocca, S.R. (eds) Typed Lambda Calculi and Applications. TLCA 2007. Lecture Notes in Computer Science, vol 4583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73228-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-73228-0_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73227-3
Online ISBN: 978-3-540-73228-0
eBook Packages: Computer ScienceComputer Science (R0)