Axioms and (counter)examples in synthetic domain theory

Annals of Pure and Applied Logic

Volumn 104, Issue 1-3, 2000, Pages 233-278

  Van Oosten, Jaap ^a   Simpson, Alex K ^b  

^a UTRECHT UNIVERSITY   (Netherlands)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Realizability; Synthetic domain theory; Topos theory

Indexed keywords

EID: 0005500272     PISSN: 01680072     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-0072(00)00014-2     Document Type: Article

References (34)

1. R.L. Croie, A.M. Pitts, New foundations for fixpoint computations: FIX-Hyperdoctrines and the FIX-Logic, Inform. Comput. 98 (1992) 171-210.
2. M.P. Fiore, G.D. Plotkin, An extension of models of axiomatic domain theory to models of synthetic domain theory, in: Proc. CSL 96, Lecture Notes in Computer Science, Vol. 1258, Springer, Berlin, 1997, pp. 129-149.
3. M.P. Fiore, G. Rosolini, Two models of synthetic domain theory, J. Pure Appl. Algebra 116 (1997) 151-162.
4. M.P. Fiore, G. Rosolini. Domains in H, Theoret. Comput. Sci. (2000), to appear.
5. P.J. Freyd, Algebraically complete categories, in: Category Theory, Lecture Notes in Mathematics, Vol. 1488, Springer, Berlin, 1991.
6. P.J. Freyd, P. Mulry, G. Rosolini, D.S. Scott, Extensional PERs, Proc. 5th Annu. Symp. on Logic in Computer Science (1990).
7. J.M.E. Hyland, The Effective Topos, in: A.S. Troelstra, D. van Dalen (Eds.), The L.E.J. Brouwer Centenary Symposium, North-Holland, Amsterdam, 1982.
8. J.M.E. Hyland, First steps in synthetic domain theory, in: Category Theory, Lecture Notes in Mathematics, Vol. 1488, Springer, Berlin, 1991.
9. J.M.E. Hyland, L. Ong, in: Modified realizability toposes and strong normalization proofs, Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, Vol. 664, Springer, Berlin, 1993.
10. J.M.E. Hyland, E. Robinson, G. Rosolini, The discrete objects in the effective topos, Proc. London Math. Soc. 3 (60) (1990) 1-60.
11. M. Jibladze, A presentation of the initial lift algebra, J. Pure Appl. Algebra 116 (1997) 185-198.
12. P.T. Johnstone, On a topological topos, Proc. London Math. Soc. 38 (1977) 237-271.
13. A. Joyal, I. Moerdijk, Algebraic Set Theory, LMS Lecture Notes, Vol. 220, Cambridge University Press, Cambridge, 1995.
14. F.W. Lawvere, Diagonal arguments and cartesian closed categories, in: Category Theory, Homology Theory and their Applications II, Lecture Notes in Mathematics, Vol. 92, Springer, Berlin, 1968, pp. 134-145.
15. J.R. Longley, The Sequentially Realizable Functionals, 1998, Technical Report, University of Edinburgh, no. ECS-LFCS-98-402 (1998), Ann. Pure Appl. Logic, accepted.
16. J.R. Longley, A.K. Simpson, A uniform approach to domain theory in realizability models, Math. Struct. Comput. Sci. 7 (1997) 469-505.
17. S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Universitext, Springer, Berlin, 1992.
18. M. Menni, A.K. Simpson, The largest topological subcategory of countably-based equilogical spaces, Proc. MFPS '99, Electronical Notes in Computer Science, 1999, to appear.
19. J. van Oosten, in: Models and Computability, Invited Papers from the 1997 Logic Colloquium in Leeds, LMS Lecture Series in Mathematics 259, Cambridge University Press, 1999, pp. 389-406.
20. J. van Oosten, The modified realizability topos, J. Pure Appl. Algebra 116 (1997) 273-289.
21. W.K.-S. Phoa, Domain theory in realizability toposes, Ph.D. Dissertation, Cambridge (1990).
22. W.K.-S. Phoa, Effective domains and intrinsic structure, in: Proc. 5th Ann. Symp. on Logic in Computer Science (1990).
23. B. Reus, Program verification in synthetic domain theory, Ph.D. Dissertation, Munich (1995).
24. B. Reus, Th. Streicher, General synthetic domain theory - a logical approach, in: Category Theory and Computer Science, Proc. CTCS '97 Lecture Notes in Computer Science, Vol. 1290, Springer, Berlin, 1997, pp. 293-313. (Expanded version in: Math. Struct. Comp. Science 9 (1999) 177-223).
25. B. Reus, Th. Streicher, General synthetic domain theory - a logical approach, in: Category Theory and Computer Science, Proc. CTCS '97 Lecture Notes in Computer Science, Vol. 1290, Springer, Berlin, 1997, pp. 293-313. (Expanded version in: Math. Struct. Comp. Science 9 (1999) 177-223).
26. G. Rosolini, Continuity and effectivity in topoi, Ph.D. thesis, University of Oxford (1986).
27. G. Rosolini, Categories and Effective Computations, in: Lecture Notes in Computer Science, Vol. 283, Springer, Berlin, 1987, pp. 1-11.
28. D.S. Scott, Relating theories of the λ-calculus, in: J.P. Seldin, J.R. Hindley (Eds.), To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, New York, 1980.
29. A.K. Simpson, Computational adequacy in an elementary topos, in: Computer Science Logic, Proc. CSL '98, Brno, Lecture Notes in Computer Science, Vol. 1584, Springer, Berlin, 1998, pp. 323-342.
30. A.K. Simpson, Algebraic Compactness in Intuitionistic Set Theory, Talk at PSSL, Edinburgh, 1995, 1999, in preparation.
31. T. Streicher, Investigations into intensional type theory, Habilitation Thesis, Darmstadt (1993).
32. Th. Streicher, Inductive construction of repletion, Appl. Categorical Struct. 7 (1999) 185-207.
33. P. Taylor, The fixed point property in synthetic domain theory, Proc. 6th Annu. Symp. on Logic in Computer Science (1991).
34. A.S. Troelstra, Metamathematical Investigations in Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, Vol. 344, Springer, Berlin, 1973.