Abstract
We show the existence of an infinitary confluent and normalising extension of the finite extensional lambda calculus with beta and eta. Besides infinite beta reductions also infinite eta reductions are possible in this extension, and terms without head normal form can be reduced to bottom. As corollaries we obtain a simple, syntax based construction of an extensional Böhm model of the finite lambda calculus; and a simple, syntax based proof that two lambda terms have the same semantics in this model if and only if they have the same eta-Böhm tree if and only if they are observationally equivalent wrt to beta normal forms. The confluence proof reduces confluence of beta, bottom and eta via infinitary commutation and postponement arguments to confluence of beta and bottom and confluence of eta.
We give counterexamples against confluence of similar extensions based on the identification of the terms without weak head normal form and the terms without top normal form (rootactive terms) respectively.
Partially supported by IST-2001-322222 MIKADO; IST-2001-33477 DART.
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References
A. Arnold and M. Nivat. The metric space of infinite trees. algebraic and topolog-ical properties. Fundamenta Informaticae, 4:445–476, 1980.
S. van Bakel, F. Barbanera, M. Dezani-Ciancaglini, and F.J. de Vries. Intersection types for λ-trees. TCS, 272(1–2):3–40, 2002.
H. P. Barendregt. The Lambda Calculus Its Syntax and Semantics. North-Holland Publishing Co., Amsterdam, Revised edition, 1984.
H. P. Barendregt, J. Bergstra, J. W. Klop, and H. Volken. Degrees, reductions and representability in the lambda calculus. Technical report, Department of Mathematics, Utrecht University, 1976.
A. Berarducci. Infinite λ-calculus and non-sensible models. In Logic and algebra (Pontignano, 1994), pages 339–377. Dekker, New York, 1996.
M. Coppo, M. Dezani-Ciancaglini, and M. Zacchi. Type theories, normal forms, and D∞-lambda-models. Information and Computation, 72(2):85–116, 1987.
H.B. Curry and R. Feys. Combinatory Logic, volume I. North-Holland Publishing Co., Amsterdam, 1958.
N. Dershowitz, S. Kaplan, and D.A. Plaisted. Rewrite, rewrite, rewrite, rewrite, rewrite,.... Theoretical Computer Science, 83(1):71–96, 1991.
M. Dezani-Ciancaglini, P. Severi, and F.J. de Vries. Infinitary lambda calculus and discrimination of Berarducci trees. TCS, 200X. Draft available on http://www/mcs.le.ac.uk/~ferjan.
J. M. E. Hyland. A survey of some useful partial order relations on terms of the lambda calculus. In C Böhm, editor, Lambda Calculus and Computer Science Theory, volume 37 of LNCS, pages 83–93. Springer-Verlag, 1975.
J.R. Kennaway and F.J. de Vries. Infinitary rewriting. In TeReSe, editor, Term Rewriting Systems, Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 200X. Draft available on http://www/mcs.le.ac.uk/~ferjan.
J.R. Kennaway, J. W. Klop, R. Sleep, and F.J. de Vries. Infinitary lambda calculus. TCS, 175(1):93–125, 1997.
J. H. Morris Jr. Lambda calculus models of programming languages. PhD thesis, M.I.T., 1968.
R. Nakajima. Infinite normal forms for the λ-calculus. In C. Böhm, editor, Lambda calculus and Computer Science Theory, volume 37 of LNCS, pages 62–82. Springer-Verlag, 1975.
V. van Oostrom. Developing developments. TCS, 175(1):159–181, 1997.
D. Park. The Y-combinator in Scott’s lambda-calculus models (revised version). Technical report, Department of Computer Science, University of Warwick, 1976. Copy available at http://www.dcs.qmul.ac.uk/~ae/papers/others/ycslcm/.
G.D. Plotkin. Set-theoretical and other elementary models of the λ-calculus. TCS, 121:351–409, 1993.
C. P. Wadsworth. The relation between computational and denotational properties for Scott’s D∞-models of the lambda-calculus. SIAM J. Comput., 5(3):488–521, 1976.
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Severi, P., de Vries, FJ. (2002). An Extensional Böhm Model. In: Tison, S. (eds) Rewriting Techniques and Applications. RTA 2002. Lecture Notes in Computer Science, vol 2378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45610-4_12
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DOI: https://doi.org/10.1007/3-540-45610-4_12
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