Abstract
Proof principles for reasoning about various semantics of untyped λ-calculus are discussed. The semantics are determined operationally by fixing a particular reduction strategy on λ-terms and a suitable set of values, and by taking the corresponding observational equivalence on terms. These principles arise naturally as co-induction principles, when the observational equivalences are shown to be induced by the unique mapping into a final F-coalgebra, for a suitable functor F. This is achieved either by induction on computation steps or exploiting the properties of some, computationally adequate, inverse limit denotational model. The final F-coalgebras cannot be given, in general, the structure of a “denotational” λ-model. Nevertheless the “final semantics” can count as compositional in that it induces a congruence. We utilize the intuitive categorical setting of hypersets and functions. The importance of the principles introduced in this paper lies in the fact that they often allow to factorize the complexity of proofs (of observational equivalence) by “straight” induction on computation steps, which are usually lengthy and error-prone.
Work supported by EEC Science contract MASK, HCM contract “Lambda Calcul Typé” and MURST 40% and 60% grants.
Preview
Unable to display preview. Download preview PDF.
References
S. Abramsky, L. Ong, Full Abstraction in the Lazy Lambda Calculus, Information and Computation, 105(2):159–267, 1993.
P. Aczel, Non-wellfounded sets, Number 14, Lecture Notes CSLI, 1988.
P. Aczel, N. Mendler, A final coalgebra theorem Category Thepry and Computer Science Proceedings, D. Pitt et al. eds., Springer LNCS n.389:357–365, 1989.
H. Barendregt, The Lambda Calculus, its Syntax and Semantics, North Holland, Amsterdam, 1984.
M. Coppo, M. Dezani-Ciancaglini, M. Zacchi, Type Theories, Normal Forms and D ∞-Lambda-Models, Information and Computation, 72(2):85–116, 1987.
L. Egidi, F. Honsell, S. Ronchi Della Rocca, Operational, denotational and logical Descriptions: a Case Study, Fundamenta Informaticae, 16(2):149–169, 1992.
M. Fiore, A Coinduction Principle for Recursive Data Types Based on Bisimulation, 8th LICS Conference Proceedings, IEEE Computer Society Press:110–119, 1993.
M. Forti, F. Honsell, Set Theory with Free Construction Principles, Annali Scuola Normale Sup. Pisa, Cl. Sci., (IV), 10:493–522, 1983.
M. Forti, F. Honsell, M. Lenisa, Processes and Hyperuniverses, MFCS'94 Conference Proceedings, I. Privara et al. eds., Springer LNCS n.841:352–363, 1994.
F. Honsell, M. Lenisa, Some Results on Restricted λ-calculi, MFCS'93 Conference Proceedings, A. Borzyszkowski et al. eds., Springer LNCS n.711:84–104, 1993.
F. Honsell, S. Ronchi Della Rocca, An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus, J. of Computer and System Sciences (45) 1:49–75, 1992.
R. Milner, Operational and Algebraic Semantics of Concurrent Processes, Handbook of Theoretical Computer Science, Ch.19, 1990.
C.H.L. Ong, The lazy lambda calculus: an investigation into the foundations of functional programming, Ph.D. thesis, Imperial College of Science and Technology, University of London, 1988.
A.M. Pitts, Relational Properties of Recursively Defined Domains, 8th LICS Conference Proceedings, IEEE Computer Society Press:86–97, 1993.
G.D. Plotkin, Call-by-name, Call-by-value and the λ-calculus, Theoretical Computer Science (1):125–159, 1975.
S. Ronchi Della Rocca, International Summer School in Logic for Computer Science, Chambery 28/6-9/7 1993, lecture notes.
J.J.M.M. Rutten, Processes as terms: non-wellfounded models for bisimulation, Math.Struct.Comp.Sci., 2(3):257–275, 1992.
J.J.M.M. Rutten, D. Turi, On the Foundations of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders, REX Conference Proceedings, J. de Bakker et al. eds., Springer LNCS n.666:477–530, 1993.
D. Turi, B. Jacobs, On final Semantics for applicative and non-deterministic languages, Fifth Biennial Meeting on Category Theory and Computer Science, Amsterdam, 1993.
C.P. Wadsworth, The relation between computational and denotational properties for Scott's D ∞-models of the λ-calculus, SIAM J. of Computing, 5(3):488–521, 1976.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Honsell, F., Lenisa, M. (1995). Final semantics for untyped λ-calculus. In: Dezani-Ciancaglini, M., Plotkin, G. (eds) Typed Lambda Calculi and Applications. TLCA 1995. Lecture Notes in Computer Science, vol 902. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014057
Download citation
DOI: https://doi.org/10.1007/BFb0014057
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59048-4
Online ISBN: 978-3-540-49178-1
eBook Packages: Springer Book Archive