Abstract
Issues in the mathematical semantics of two restrictions of the λ-calculus, i.e. λI-calculus and λv-calculus, are discussed. A fully abstract model for the natural evaluation of the former is defined using complete partial orders and strict Scott-continuous functions. A correct, albeit non-fully abstract, model for the SECD evaluation of the latter is denned using Girard's coherence spaces and stable functions. These results are used to illustrate the interest of the analysis of the fine structure of mathematical models of programming languages.
In conclusion we have that indeed the “coherent” model succeeds in equating observationally equivalent terms that the “continuous” model tells apart, as our original intuition suggested. And this is because the Scott-continuous functions which separate those terms are indeed parallel and hence not stable. But some-what surprisingly Berry's order introduces perverse stable functions which tell apart observationally equivalent terms which are instead equated in the continuous model. Although neither the coherent model nor the continuous one are fully abstract, nevertheless the investigation carried out was shown to be rewarding.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Barendregt, H.: The Lambda, Calculus, its Syntax and Semantics. North Holland. Amsterdam (1984)
Böhm, C.: Lambda Calculus and Computer Science Theory. LNCS 37 Springer-Verlag (1975)
Church, A.: The Calculi of Lambda Conversion. Princeton University Press Princeton (1941)
Coppo, M., Dezani-Ciancaglini, M., Zacchi, M.: Type Theories, Normal Forms and D ∞-Lambda-Models. Information and Computation 72(2) (1987) 85–116
Dezani-Ciancaglini, M., Honsell, F., Ronchi Della Rocca, S.: Models for theories of functions depending on all their arguments (abstract). Journal of Symbolic Logic 51(3) (1986) 399–402
Egidi, L., Honsell, F., Ronchi Della Rocca, S.: Operational, Denotational and Logical descriptions: a case study. Fundamenta Informaticae 16(2) (1992) 149–169
Girard, J.Y.: The system F of variable types, fifteen years later. TCS 45 (1986) 159–192
Girard, J. Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press. Cambridge (1989)
Hindley, J. R., Longo, G.: Lambda Calculus Models and Extensionality. Z. Math. Logik Grundlag. Math. 26 (1980) 289–310
Landin, P. J.: The mechanical evaluation of expressions. Computer J. 6(4) (1964) 308–320
Meyer, A.: What is a, Model of the Lambda Calculus? Information and Control 52 (1982) 87–122
Plotkin, G. D.: Call-by-name, call-by-value and the λ-calculus. TCS 1 (1975) 125–159
Smith, M. B., Plotkin, G. D.: The category-theoretic solution of recursive domain equations. SIAM J. of Computing 11(5) (1982) 761–783
Author information
Authors and Affiliations
Editor information
Additional information
dedicated to Corrado Böhm on the occasion of his 70th birthday
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Honsell, F., Lenisa, M. (1993). Some results on the full abstraction problem for restricted lambda calculi. In: Borzyszkowski, A.M., Sokołowski, S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57182-5_6
Download citation
DOI: https://doi.org/10.1007/3-540-57182-5_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57182-7
Online ISBN: 978-3-540-47927-7
eBook Packages: Springer Book Archive