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On the ubiquity of certain total type structures

Published online by Cambridge University Press:  01 October 2007

JOHN LONGLEY*
Affiliation:
Laboratory for Foundations of Computer Science, School of Informatics, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, U.K.

Abstract

It is an empirical observation arising from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene–Kreisel continuous functionals, its effective substructure Ceff and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously.

In this paper we present some new results that go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, Ceff or HEO (as appropriate). We obtain versions of our results for both the ‘standard’ and ‘modified’ extensional collapse constructions. The proofs make essential use of a technique due to Normann.

Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the three type structures under consideration are highly canonical mathematical objects.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

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References

Abramsky, S. and McCusker, G. (1999) Game semantics. In: Schwichtenberg, H. and Berger, U. (eds.) Computational Logic: Proceedings of the 1997 Marktoberdorf Summer School, Springer 156.Google Scholar
Amadio, R. and Curien, P.-L. (1998) Domains and Lambda Calculi, Cambridge Tracts in Theoretical Computer Science 46, Cambridge University Press.Google Scholar
Awodey, S., Birkedal, L. and Scott, D. S. (2002) Local realizability toposes and a modal logic for computability. Mathematical Structures in Computer Science 12 319334.Google Scholar
Barendregt, H. P. (1984) The Lambda Calculus: Its syntax and semantics (revised), Studies in Logic and the Foundations of Mathematics 103, North-Holland.Google Scholar
Bauer, A. (2000) The Realizability Approach to Computable Analysis and Topology, Ph.D. thesis, Carnegie Mellon University. (Available as technical report CMU-CS-00-164.)Google Scholar
Bauer, A. (2002) A relationship between equilogical spaces and Type Two Effectivity. Mathematical Logic Quarterly 48 (1)115.Google Scholar
Bauer, A., Escardó, M. and Simpson, A. (2002) Comparing functional paradigms for exact real-number computation. In: Proceedings of ICALP 2002. Springer-Verlag Lecture Notes in Computer Science 2380 488500.Google Scholar
Beeson, M. (1985) Foundations of Constructive Mathematics, Springer-Verlag.Google Scholar
Berger, U. (1993) Total sets and objects in domain theory. Annals of Pure and Applied Logic 60 91117.Google Scholar
Berger, U. (2002) Computability and totality in domains. Mathematical Structures in Computer Science 12 281294.CrossRefGoogle Scholar
Bergstra, J. A. (1978) The continuous functionals and 2 E. In: Fenstad, J. E., Gandy, R. O. and Sacks, G. E. (eds.) Generalized Recursion Theory II, North-Holland 3953.Google Scholar
Berry, G. (1978) Stable models of typed lambda-calculi. In: Proceedings of 5th International Colloquium on Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 62 7289.Google Scholar
Berry, G. and Curien, P.-L. (1982) Sequential algorithms on concrete data structures. Theoretical Computer Science 20 (3)265321.Google Scholar
Bezem, M. (1985) Isomorphisms between HEO and HROE, ECF and ICFE. Journal of Symbolic Logic 50 359371.CrossRefGoogle Scholar
Bucciarelli, A. and Ehrhard, T. (1991) Sequentiality and strong stability. In: Proceedings of 6th Annual Symposium on Logic in Computer Science, IEEE 138145.Google Scholar
Ershov, Yu. L. (1972a) Computable functionals of finite types. Algebra i Logika 11 (4)367–437. (English translation: Algebra and Logic, AMS 11 (4) 203–242.)Google Scholar
Ershov, Yu. L. (1972b) Everywhere-defined continuous functionals. Algebra i Logika 11 (6) 656–665. (English translation: Algebra and Logic, AMS 11 (6) 363368.)CrossRefGoogle Scholar
Ershov, Yu. L. (1974) Maximal and everywhere defined functionals. Algebra i Logika 13 (4)374397. (English translation: Algebra and Logic, AMS 13 (4) 210–225.)Google Scholar
Ershov, Yu. L. (1976a) Hereditarily effective operations. Algebra i Logika 15 (6)642654. (English translation: Algebra and Logic, AMS 15 (6) 400–409.)Google Scholar
Ershov, Yu. L. (1976b) Model C of the partial continuous functionals. In: Gandy, R. O. and Hyland, J. M. E. (eds.) Logic Colloquium 1976, North-Holland 455467.Google Scholar
Escardó, M., Lawson, J. and Simpson, A. (2004) Comparing cartesian closed categories of (core) compactly generated Spaces. Topology and its Applications 143 105145.Google Scholar
Gandy, R. O. (1962) Effective operations and recursive functionals (abstract). Journal of Symbolic Logic 27 378379.Google Scholar
Gandy, R. O. and Hyland, J. M. E. (1977) Computable and recursively countable functions of higher type. In: Gandy, R. O. and Hyland, J. M. E. (eds.) Logic Colloquium 1976, North-Holland 407438.Google Scholar
Hinata, S. and Tugué, S. (1969) A note on continuous functionals. Annals of the Japan Association for Philosophy of Science 3 138145.Google Scholar
Honsell, F. and Sannella, D. (1999) Pre-logical relations. In: Proceedings Computer Science Logic 1999. Springer-Verlag Lecture Notes in Computer Science 1683 546561.Google Scholar
Hyland, J. M. E. (1975) Recursion theory on the countable functionals, Ph.D. thesis, University of Oxford.Google Scholar
Hyland, J. M. E. (1979) Filter spaces and continuous functionals. Annals of Mathematical Logic 16 101143.CrossRefGoogle Scholar
Hyland, J. M. E. (1982) The effective topos. In: Troelstra, A. S. and vanDalen, D. Dalen, D. (eds.) The L. E. J. Brouwer Centenary Symposium, North-Holland 165216.Google Scholar
Hyland, J. M. E. (1997) Game semantics. In: Pitts, A. M. and Dybjer, P. (eds.) Semantics and Logics of Computation, Cambridge University Press 131194.Google Scholar
Kleene, S. C. (1952) Introduction to Metamathematics, Wolter-Noordhoff and North-Holland.Google Scholar
Kleene, S. C. (1959a) Recursive functionals and quantifiers of finite types I. Transactions of the American Mathematical Society 91 152.Google Scholar
Kleene, S. C. (1959b) Countable functionals. In: Heyting, A. (ed.) Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957, North-Holland 81100.Google Scholar
Kleene, S. C. and Vesley, R. E. (1965) The Foundations of Intuitionistic Mathematics, North-Holland (1965).Google Scholar
Kreisel, G. (1959) Interpretation of analysis by means of functionals of finite type. In: Heyting, A. (ed.) Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957, North-Holland 101128.Google Scholar
Kreisel, G., Lacombe, D. and Shoenfield, J. R. (1959) Partial recursive functionals and effective operations. In: Heyting, A. (ed.) Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957, North-Holland 290297.Google Scholar
Lietz, P. and Streicher, T. (2002) Impredicativity entails untypedness, Mathematical Structures in Computer Science 12 335347.CrossRefGoogle Scholar
Longley, J. R. (1995) Realizability Toposes and Language Semantics, Ph.D. thesis, University of Edinburgh. (Available as technical report ECS-LFCS-95-332.)Google Scholar
Longley, J. R. (1999a) Matching typed and untyped realizability. Electronic Notes in Theoretical Computer Science 23 (1).CrossRefGoogle Scholar
Longley, J. R. (1999b) Unifying typed and untyped realizability. (Unpublished note available at http://www.inf.ed.ac.uk/home/jrl/unifying.txt.)Google Scholar
Longley, J. R. (2002) The sequentially realizable functionals. Annals of Pure and Applied Logic 117 (1)193.Google Scholar
Longley, J. R. (2003) Universal types and what they are good for. In: Zhang, G.-Q. et al. . (eds.) Domain theory, Logic and Computation: Proceedings of 2nd International Symposium on Domain Theory, Sichuan, China, 2001, Kluwer 2563.Google Scholar
Longley, J. R. (2005a) Notions of computability at higher types I. In: Cori, R., Razborov, A., Todorčević, S. and Wood, C. (eds.) Logic Colloquium 2000: Proceedings of the ASL meeting held in Paris. Lecture Notes in Logic, ASL 200 32142.Google Scholar
Longley, J. R. (2005b) On the ubiquity of certain total type structures – extended abstract. In: Proceedings of the Workshop on Domains VI, Birmingham, United Kingdom. Electronic Notes in Theoretical Computer Science 73 87109.Google Scholar
Longley, J. R. (2007) Notions of computability at higher types II. (In preparation.)Google Scholar
Milner, R. (1977) Fully abstract models of typed λ-calculi. Theoretical Computer Science 4 122.Google Scholar
Milner, R., Tofte, M., Harper, R. and MacQueen,, D. (1997) The Definition of Standard ML (revised), MIT Press.CrossRefGoogle Scholar
Moggi, E. (1991) Computational lambda-calculus and monads. Information and Computation 93 (1).Google Scholar
Normann, D. (1980) Recursion on the countable functionals. Springer-Verlag Lecture Notes in Mathematics 811.Google Scholar
Normann, D. (1981) The continuous functionals: computations, recursions and degrees. Annals of Pure and Applied Logic 21 126.Google Scholar
Normann, D. (1999) The continuous functionals. In: Griffor, E. R. (ed.) Handbook of Computability Theory, North-Holland 251275.Google Scholar
Normann, D. (2000) Computability over the partial continuous functionals. Journal of Symbolic Logic 65 11331142.Google Scholar
Normann, D. (2005) Comparing hierarchies of total functionals. Logical Methods in Computer Science 1 (2).Google Scholar
Normann, D., Palmgren, E. and Stoltenberg-Hansen,, V. (1999) Hyperfinite type structures. Journal of Symbolic Logic 64 12161242.CrossRefGoogle Scholar
van Oosten, J. (1997) The modified realizability topos. Journal of Pure and Applied Algebra 116 273289.Google Scholar
vanOosten, J. Oosten, J. (1999) A combinatory algebra for sequential functionals of finite type. In: Cooper, S. B. and Truss, J. K. (eds.) Models and Computability, Cambridge University Press 389406.Google Scholar
Platek, R. (1966) Foundations of recursion theory, Ph.D. thesis, Stanford University.Google Scholar
Plotkin, G. D. (1977) LCF considered as a programming language. Theoretical Computer Science 5 223255.Google Scholar
Plotkin, G. D. (1978) as a universal domain. Journal of Computer and System Sciences 17 209236.Google Scholar
Plotkin, G. (1993) Set-theoretical and other elementary models of the λ-calculus. Theoretical Computer Science 121 351409. (First written in 1972 and circulated in unpublished form.)Google Scholar
Plotkin, G. D. (1997) Full abstraction, totality and PCF. Mathematical Structures in Computer Science 9 (1)120.Google Scholar
Schwichtenberg, H. (1996) Density and choice for total continuous functionals. In: Odifreddi, P. (ed.) Kreiseliana: About and around Georg Kreisel, A. K. Peters 335362.Google Scholar
Scott, D. S. (1972) Continuous lattices. In: Lawvere, F. W. (ed.) Toposes, Algebraic Geometry and Logic, Springer 97136.Google Scholar
Scott, D. S. (1976) Data types as lattices. SIAM Journal of Computing 5 (3)522587.Google Scholar
Scott, D. S. (1993) A type-theoretical alternative to ISWIM, CUCH, OWHY. Theoretical Computer Science 121 411440. (First written in 1969 and circulated in unpublished form since then.)Google Scholar
Soare, R. I. (1999) The history and concept of computability. In: Griffor, E. R. (ed.) Handbook of Computability Theory, North-Holland 336.Google Scholar
Stoltenberg-Hansen, V., Lindström, I. and Griffor, E. R. (1994) Mathematical Theory of Domains, Cambridge Tracts in Theoretical Computer Science 22, Cambridge University Press.Google Scholar
Troelstra, A. S. (1973) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer-Verlag Lecture Notes in Mathematics 344.Google Scholar
Weihrauch, K. (2000) Computability, EATCS Monographs on Theoretical Computer Science 9, Springer-Verlag.Google Scholar