Abstract
We will construct from every partial combinatory algebra (pca, for short) A a pca a-lim(A) s.t. (1) every representable numeric function ϕ(n) of a-lim(A) is exactly of the form limt ξ(t, n) with ξ(t, n) being a representable numeric function of A, and (2) A can be embedded into a-lim(A) which has a synchronous application operator. Here, a-lim(A) is A equipped with a limit structure in the sense that each element of a-lim(A) is the limit of a countable sequence of A-elements. We will discuss limit structures for A in terms of Barendregt’s range property. Moreover, we will repeat the construction lim(−) transfinite times to interpret infinitary λ-calculi. Finally, we will interpret affine type-free λµ-calculus by introducing another partial applicative structure which has an asynchronous application operator and allows a parallel limit operation. keywords: partial combinatory algebra, limiting recursive functions, realizability interpretation, discontinuity, infinitary lambda-calculi, λµ-calculus. In the interpretation, µ-variables(=continuations) are interpreted as streams of λ-terms.
The author acknowledges Susumu Hayashi, Mariko Yasugi, Stefano Berardi, and Ken-etsu Fujita. The comment by anonymous referees was useful to partly improve the presentation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H.P. Barendregt. The Lambda Calculus. 1984.
S. Berardi. Classical logic as limit completion I&II, submitted, 2001.
A. Berarducci. Infinite λ-calculus and non-sensible models. In Logic and algebra (Pontignano, 1994), pp. 339–377. Dekker, New York, 1996.
K. Fujita. On proof terms and embeddings of classical substructural logics. Studia Logica, 61(2):199–221, 1998.
E. Mark Gold. Limiting recursion. JSL, 30:28–48, 1965.
S. Hayashi and M. Nakata. Towards limit computable mathematics. In TYPES2000, LNCS. Springer, 2001.
S. Hayashi, R. Sumitomo, and K. Shii. Towards animation of proofs-testing proofs by examples-. TCS, 2001. to appear.
J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.J. de Vries. Infinitary lambda calculus. TCS, 175(1):93–125, 1997.
M. Nakata and S. Hayashi. Limiting first order realizability interpretation. Sci. Math. Japonicae, 2001. to appear.
M. Parigot. λµ-calculus: an algorithmic interpretation of classical natural deduction. In Logic programming and automated reasoning. Springer, 1992.
H. Schwichtenberg and S. S. Wainer. Infinite terms and recursion in higher types. In J. Diller and G. H. Müller, eds., =ISILC Proof Theory Symposion, pp. 341–364. LNM., Vol. 500, 1975. Springer.
R. I. Soare. Recursively Enumerable Sets and Degrees Springer, 1987.
Th. Streicher and B. Reus. Classical logic, continuation semantics and abstract machines. J. Funct. Programming, 8(6):543–572, 1998.
H.R. Strong. Algebraically generalized recursive function theory. IBM Journal of Research and Development, 12:465–475, 1968.
W.W. Tait. Infinitely long terms of transfinite type. In Crossley and Dummet, eds., Formal Systems and Recursive Functions, pp. 176–185. North-Holland, 1965.
A. S. Troelstra and D. van Dalen. Constructivism in Mathematics, 1988.
E. G. Wagner. Uniformly reflexive structures: On the nature of Gödelizations and relative computability. Trans. of the Amer. Math. Soc., 144:1–41, 1969.
M. Yasugi, V. Brattka, and M. Washihara. Computability aspects of some discontinuous functions. http://www.kyoto-su.ac.jp/~yasugi/Recent/gaussnew.ps, 2001.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Akama, Y. (2001). Limiting Partial Combinatory Algebras towards Infinitary Lambda-Calculi and Classical Logic. In: Fribourg, L. (eds) Computer Science Logic. CSL 2001. Lecture Notes in Computer Science, vol 2142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44802-0_28
Download citation
DOI: https://doi.org/10.1007/3-540-44802-0_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42554-0
Online ISBN: 978-3-540-44802-0
eBook Packages: Springer Book Archive