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Some reasons for generalising domain theory

Published online by Cambridge University Press:  25 March 2010

MARTIN HYLAND
MARTIN HYLAND*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CMS, Wilberforce Road, Cambridge CB3 OWB, UK Email: [email protected]
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Abstract

One natural way to generalise domain theory is to replace partially ordered sets by categories. This kind of generalisation has recently found application in the study of concurrency. An outline is given of the elegant mathematical foundations that have been developed. This is specialised to give a construction of cartesian closed categories of domains, which throws light on standard presentations of domain theory.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 2: Domains , April 2010 , pp. 239 - 265
Copyright
Copyright © Cambridge University Press 2010

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References

Abramsky, S. (1983) Semantic Foundations for Applicative Multiprogramming. In: Diaz, J. (ed.) Automata, Languages and Programming, Procedings of ICALP’83. Springer-Verlag Lecture Notes in Computer Science 154 1–14.CrossRefGoogle Scholar
Adámek, J. and Rosický, J. (1994) Locally Presentable and Accessible Categories. LMS Lecture Notes Series 189.Google Scholar
Awodey, S. (2006) Category Theory, Oxford Logic Guides 49, Clarendon Press.CrossRefGoogle Scholar
Barendregt, H., Coppo, M. and Dezani-Ciancaglini, M. (1983) A Filter Lambda Model and the Completeness of Type Assignment. Journal of Symbolic Logic 48 931–940.CrossRefGoogle Scholar
Barr, M. and Wells, C. (1984) Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 278, Springer-Verlag.Google Scholar
Bauer, A. and Taylor, P. (2009) The Dedekind Reals in Abstract Stone Duality. Mathematical Structures in Computer Science 19 757–838.CrossRefGoogle Scholar
Beck, J. M. (1967) Triples, Algebras and Cohomology, Ph.D. Dissertation, Columbia University.Google Scholar
Benton, N., Bierman, G., de Paiva, V. and Hyland, M. (1993a) A Term Calculus for Intuitionistic Linear Logic. In: Bezem, M. and Groote, J. F. (eds.) Typed Lambda Calculi and Applications. Springer-Verlag Lecture Notes in Computer Science 664 75–90.CrossRefGoogle Scholar
Benton, N., Bierman, G., de Paiva, V. and Hyland, M. (1993b) Linear lambda-calculus and categorical models revisited. In: Börger, E., Jäger, G., Büning, H. K., Martini, S. and Richter, M. M. (eds.) Proceedings of Computer Science Logic Conference in San Miniato (September 1992). Springer-Verlag Lecture Notes in Computer Science 702 61–84.CrossRefGoogle Scholar
Berline, C. (2000) From Computation to Foundations via Functions and Application: the Lambda-calculus and its Webbed Models. Theoretical Computer Science 249 81–161.CrossRefGoogle Scholar
Berry, G. (1979) Modèles completement adéquats et stables des lambda-calculs typés, Thèse de Doctorat d'Etat, Université de Paris VII.Google Scholar
Cattani, G. L. and Winskel, G. (2005) Profunctors, open maps and bisimulation. Mathematical Structures in Computer Science 15 553–614.CrossRefGoogle Scholar
Coppo, M., Dezani-Ciancaglini, M., Honsell, F. and Longo, G. (1984) Extended Type Structures and Filter Lambda Models. In: Lolli, G., Longo, G. and Marcja, A. (eds.) Logic Colloquium 82, North-Holland241–262.CrossRefGoogle Scholar
Coquand, T., Gunter, C. A. and Winskel, G. (1989) Domain Theoretic Models of Polymorphism. Information and Computation 81 123–167.CrossRefGoogle Scholar
Curien, P.-L., Plotkin, G. D. and Winskel, G. (2000) Bistructures, Bidomains and Linear Logic. In: Plotkin, G., Stirling, C. and Tofte, M. (eds.) Proof, Language, and Interaction. Essays in Honour of Robin Milner, MIT Press 21–54.CrossRefGoogle Scholar
Diers, Y. (1980) Multimonads and Multimonadic Categories. Journal of Pure and Applied Algebra 17 153–170.CrossRefGoogle Scholar
Edalat, A. (1997) Domains for Computation in Mathematics, Physics and Exact Real Arithmetic. Bulletin of Symbolic Logic 3 401–452.CrossRefGoogle Scholar
Eilenberg, S. and Kelly, G. M. (1966) Closed Categories. In: Proceedings of the Conference on Categorical Algebra, La Jolla 1965 421–562.CrossRefGoogle Scholar
Eilenberg, S. and Wright, J. B. (1967) Automata in general algebras. Information and Control 11 452–470.CrossRefGoogle Scholar
Engeler, E. (1981) Algebras and Combinators. Algebra Universalis 13 389–392.CrossRefGoogle Scholar
Fiore, M. P., Plotkin, G. D. and Power, J. (1997) Complete cuboidal sets in axiomatic domain theory. In: Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press 268–279.Google Scholar
Fiore, M. P. and Plotkin, G. D. (1997) An extension of models of axiomatic domain theory to models of synthetic domain theory. In: van Dalen, D. and Bezem, M. (eds.) Computer Science Logic, Proceedings of 10th International Workshop, CSL'96. Springer-Verlag Lecture Notes in Computer Science 1258 129–149.CrossRefGoogle Scholar
Fiore, M., Gambino, N., Hyland, M. and Winskel, G. (2008) The cartesian closed bicategory of generalised species of structures. Journal of the London Mathematical Society 77 203–220.CrossRefGoogle Scholar
Fiore, M., Gambino, N., Hyland, M. and Winskel, G. (in preparation) Kleisli bicategories.Google Scholar
Girard, J.-Y. (1987) Linear Logic. Theoretical Computer Science 50 1–102.CrossRefGoogle Scholar
Artin, A. G. M. and Verdier, J. L. (1972) Théorie des Topos et Cohomologie Étale des Schemas. SGA 4, vol. 1. Springer-Verlag Lecture Notes in Mathematics 269.Google Scholar
Hasegawa, R. (2002) Two applications of analytic functors. Theoretical Computer Science 272 113–175.CrossRefGoogle Scholar
Houston, R. (2008) Finite products are biproducts in a compact closed category. Journal of Pure and Applied Algebra 212 394–400.CrossRefGoogle Scholar
Hyland, J. M. E. (1976) A syntactic characterization of the Equality in some Models for the Lambda Calculus. Journal of the London Mathematical Society 12 361–370.CrossRefGoogle Scholar
Hyland, J. M. E. (1982) The effective topos. In: Troelstra, A. S. and van Dalen, D. (eds.) The L.E.J. Brouwer Centenary Symposium, North-Holland165–216.Google Scholar
Hyland, J. M. E. (1991) First steps in synthetic domain theory. In: Carboni, I. A., Pedicchio, M.-C. and Rosolini, G. (eds.) Category Theory. Springer-Verlag Lecture Notes in Mathematics 1488 280–301.Google Scholar
Hyland, J. M. E. (2002) Proof Theory in the Abstract. Annals of Pure and Applied Logic 114 43–78.CrossRefGoogle Scholar
Hyland, J. M. E. and Pitts, A. M. (1989) The Theory of Constructions: Categorical Semantics and Topos Theoretic Models. In: Gray, J. W. and Scedrov, A. (eds.) Categories in Computer Science and Logic. Contemporary Mathematics 92 137–199.CrossRefGoogle Scholar
Hyland, M. and Power, J. (2002) Pseudo-commutative monads and pseudo-closed 2-categories. Journal of Pure and Applied Algebra 175 141–185.CrossRefGoogle Scholar
Hyland, M. and Schalk, A. (2003) Glueing and Orthogonality for Models of Linear Logic. Theoretical Computer Science 294 183–231.CrossRefGoogle Scholar
Hyland, M., Nagayama, M., Power, J. and Rosolini, G. (2006) A Category-Theoretic Formulation for Engeler-style Models of the Untyped lambda-Calculus. Proc MFCSIT 2004. Electronic Notes in Theoretical Computer Science 161 43–57.CrossRefGoogle Scholar
Im, G. B. and Kelly, G. M. (1986) A universal property of the convolution monoidal structure. Journal of Pure and Applied Algebra 43 75–88.CrossRefGoogle Scholar
Johnstone, P. T. (1982) Stone Spaces. Cambridge Studies in Advanced Mathematics 3, Cambridge University Press.Google Scholar
Joyal, A. (1981) Une théorie combinatoire des séries formelles. Advances in Mathematics 42 1–82.CrossRefGoogle Scholar
Kelly, G. M. (1980) Basic Concepts of Enriched Category Theory. LMS Lecture Note Series 64, Cambridge University Press.Google Scholar
Kelly, G. M. and Laplaza, M. (1990) Coherence for compact closed categories. Journal of Pure and Applied Algebra 19 193–213.CrossRefGoogle Scholar
Kock, A. (1970) Monads on Symmetric Monoidal Closed Categories. Archiv der Mathematik 21 1–10.CrossRefGoogle Scholar
Kock, A. (1971a) Closed categories generated by Commutative Monads. Journal of the Australian Mathematical Society 12 405–424.CrossRefGoogle Scholar
Kock, A. (1971b) Bilinearity and Cartesian Closed Monads. Mathematica Scandinavica 29 161–174.CrossRefGoogle Scholar
Kock, A. (1972) Strong Functors and Monoidal Monads. Archiv der Mathematik 23 113–120.CrossRefGoogle Scholar
Kreisel, G. (1971) Some reasons for generalizing recusion theory. In: Gandy, R. O. and Yates, C. E. M. (eds.) Logic Colloquium '69, Proceedings of the summer school and colloquium in Mathematical Logic, North-Holland139–198.Google Scholar
Lambek, J. and Scott, P. J. (1986) Introduction to higher order categorical logic, Cambridge Studies in Advanced Mathematics 7, Cambridge University Press.Google Scholar
Lehmann, D. J. (1976) Categories for fixed point semantics. In: Proceedings of the 17th IEEE Annual Symposium on Foundations of Computer Science 122–26.Google Scholar
Longley, J. R. (2007) On the ubiquity of certain total type structures. Mathematical Structures in Computer Science 17 841–953.CrossRefGoogle Scholar
LopezFranco, I. Franco, I. (2008) Autonomous pseudo-monoids, Ph.D. dissertation, University of Cambridge.Google Scholar
MacLane, S. Lane, S. (1971) Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag.Google Scholar
Platek, R. A. (1966) Foundations of Recursion Theory, Ph.D. dissertation, Stanford University.Google Scholar
Plotkin, G. D. (1976) A Power Domain Construction. SIAM Journal on Computing 5 452–487.CrossRefGoogle Scholar
Plotkin, G. D. and Winskel, G. (1994) Bistructures, Bidomains and Linear Logic. In: Proceedings of ICALP 352–363.CrossRefGoogle Scholar
Reus, B. and Streicher, T. (1999) General synthetic domain theory – a logical approach. Mathematical Structures in Computer Science 9 177–223.CrossRefGoogle Scholar
Scott, D. S. (1976) Continuous lattices. In: Lawvere, F. W. (ed.) Toposes, Algebraic Geometry and Logic. Springer-Verlag Lecture Notes in Mathematics 274 97–136.CrossRefGoogle Scholar
Taylor, P. (1986) Recursive Domains, Indexed Category Theory and Polymorphism, Ph.D. dissertation, University of Cambridge.Google Scholar
Taylor, P. (1990) An Algebraic Approach to Stable Domains. Journal of Pure and Applied Algebra 64 171–203.CrossRefGoogle Scholar
Taylor, P. (1991) The fixed point property in synthetic domain theory. In: Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press 152–160.Google Scholar
Taylor, P. (2000) Geometric and Higher Order Logic in terms of ASD. Theory and Applications of Categories 7 284–338.Google Scholar
Taylor, P. (2002a) Sober Spaces and Continuations. Theory and Applications of Categories 10 248–299.Google Scholar
Taylor, P. (2002b) Subspaces in ASD. Theory and Applications of Categories 10 300–366.Google Scholar
van Oosten, J. (2008) Realizability: An Introduction to its Categorical Side, Studies in Logic and The Foundations of Mathematics 152, Elsevier.Google Scholar
Winskel, G. (1980) Events in Computation, Ph.D. dissertation, University of Edinburgh.Google Scholar
Winskel, G. (1993) The formal semantics of programming languages, an introduction, MIT Press.CrossRefGoogle Scholar
Winskel, G. (1994) Stable Bistructure Models of PCF. In: Proceedings of the 19th International Symposium on Mathematical Foundations of Computer Science. Springer-Verlag Lecture Notes in Computer Science 841 177–197.CrossRefGoogle Scholar
Winskel, G. and Zappa Nardelli, F. (2004) NEW-HOPLA a higher-order process language with name generation. In: Proceedings of TCS 2004, Third IFIP International Conference on Theoretical Computer Science 521–534.CrossRefGoogle Scholar