Abstract
After an introduction to the syntax of Gödel systemT, we present its naive denotational semantics in the domain of lazy natural numbers and show an adequacy property relating syntax and semantics. We recall the natural restrictions of systemT delineating primitive recursion as a subsystem. We discuss the weakness of primitive recursion by exhibiting a simple unary algorithm whose denotation is not the semantics of a primitive recursive algorithm. This algorithm can nevertheless be programmed in systemT by using the power of higher-order (functional) definitions. Generalizing this example, we obtain a representation theorem, asserting that every “reasonable” algorithm of typeN →N can be programmed in systemT. We conclude by discussing what is known in the case of higher arities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99(1928)1–36, English translation in: J. van Heijenoort,From Frege to Gödel (Harvard University Press, 1981) pp. 493–507.
A. Church,The Calculi of Lambda-Conversion (Princeton University Press, Princeton, NJ, 1941).
L. Colson, About primitive recursive algorithms,Proc. 16th Int. Colloq. on Automata, Languages and Programming, eds. G. Ausiello, M. Dezani-Ciancaglini and S. Ronchi Della Rocca, LNCS 372 (Springer, 1989) 194–206. Journal version in TCS 83(1991)57–69.
L. Colson, On list prinitive recursion and the complexity of computinginf, BIT 32(1992)5–9.
L. Colson, Représentation intentionnelle d'algorithmes dans les systèmes fonctionnels: une étude de cas, Ph.D. Thesis, Université Paris 7 (1991).
T. Coquand, Une preuve directe du théorème d'ultime obstination, Compt. Rend. de l'Acad. des Sci. Paris I 314(1992)489–492.
S. Fortune, D. Leivant and M. O'Donnell, The expressiveness of simple and second-order type structures, J. ACM 30(1)(1983).
J.Y. Girard,Proof Theory and Logical Complexity, Vol. 1. Studies in Proof Theory (Bibliopolis, Napoli, 1988).
J.Y. Girard, Y. Lafont and P. Taylor, Proofs and types,Cambridge Tracts in Theor. Comp. Sci. 7 (Cambridge University Press, 1989).
K. Gödel, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica 12 (1958)280–287, English translation: On a hitherto unexploited extension of the finitary standpoint, J. Philos. Logic 9 (1980).
S.C. Kleene,Introduction to Metamathematics (North-Holland, 1952).
J.L. Krivine, Un algorithme non typable dans le systèmeF, Compt. Rend. de l'Acad. des Sci. Paris I 304(5)(1987).
P. Martin-Löf, Constructive mathematics and computer programming, in:Logic, Methodology and Philosophy of Science 6 (North-Holland, 1980) pp. 153–175.
P. Martin-Löf, Intuitionistic type theory,Studies in Proof Theory (Bibliopolis, Napoli, 1984).
G. Plotkin, LCF considered as a programming language, Theor. Comp. Sci. 5(1977)223–255.
W.W. Tait, Intensional interpretation of functionals of finite type I, JSL 32(1967)198–212.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Colson, L. A unary representation result for systemT . Ann Math Artif Intell 16, 385–403 (1996). https://doi.org/10.1007/BF02127804
Issue Date:
DOI: https://doi.org/10.1007/BF02127804