Abstract
We give a fully constructive semantic proof of cut elimination for intuitionistic type theory with axioms. The problem here, as with the original Takeuti conjecture, is that the impredicativity of the formal system involved makes it impossible to define a semantics along conventional lines, in the absence, a priori, of cut, or to prove completeness by induction on subformula structure. In addition, unlike semantic proofs by Tait, Takahashi, and Andrews of variants of the Takeuti conjecture, our arguments are constructive.
Our techniques offer also an easier approach than Girard’s strong normalization techniques to the problem of extending the cut-elimination result in the presence of axioms. We need only to relativize the Heyting algebras involved in a straightforward way.
This is a preview of subscription content, log in via an institution to check access.
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
Andrews, P.: Resolution in type theory. Journal of Symbolic Logic 36(3) (1971)
Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)
de Swart, H.C.M̃.: Another intuitionistic completeness proof. Journal of Symbolic Logic 41, 644–662 (1976)
DeMarco, M., Lipton, J.: Completeness and cut elimination in the intuitionistic theory of types. Journal of Logic and Computation, 821–854 (November 2005)
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. Journal of Automated Reasoning 31, 33–72 (2003)
Dowek, G., Werner, B.: Proof normalization modulo. The Journal of Symbolic Logic 68(4), 1289–1316 (2003)
Friedman, H.: Equality between functionals. In: Parikh, R. (ed.) Logic Colloquium. Lecture Notes in Mathematics, vol. 453, pp. 22–37. Springer, Heidelberg (1975)
Girard, J.Y.: Une extension de l’interprétation de Gödel à l’analyse et son application à l’élimination de coupures dans l’analyse et la théorie des types. In: Fenstad, J.E. (ed.) Proceedings of the second Scandinavian proof theory symposium. North-Holland, Amsterdam (1971)
Girard, J.Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press, Cambridge (1998)
Kleene, S.C.: Permutability of inferences in Gentzen’s calculi LK and LJ. Memoirs of the American Mathematical Society 10, 1–26, 27–68 (1952)
Kreisel, G.: A remark on free choice sequences and the topological completeness proofs. Journal of Symbolic Logic 23, 369–388 (1958)
Kreisel, G.: On weak completeness of intuitionistic predicate logic. Journal of Symbolic Logic 27, 139–158 (1962)
Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51(1-2), 125–157 (1991)
Mitchell, J.: Foundations for Programming Languages. MIT Press, Cambridge (1996)
Okada, M.: Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic. Theoretical Computer Science 227, 333–396 (1999)
Okada, M.: A uniform semantic proof for cut-elimination and completeness of various first and higher order logics. Theoretical Computer Science 281, 471–498 (2002)
Plotkin, G.: Lambda definability in the full type hierarchy. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism. Academic Press, New York (1980)
Prawitz, D.: Hauptsatz for higher order logic. The Journal of Symbolic Logic 33(3), 452–457 (1968)
Schütte, K.: Syntactical and semantical properties of simple type theory. Journal of Symbolic Logic 25, 305–326 (1960)
Tait, W.: A non-constructive proof of Gentzen’s Hauptsatz for second-order predicate logic. Bulletin of the American Mathematical Society 72, 980–983 (1966)
Takahashi, M.-o.: A proof of cut-elimination in simple type theory. J. Math. Soc. Japan 19(4) (1967)
Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics: An Introduction, vol. 2. Elsevier Science Publishers, Amsterdam (1988)
van Dalen, D.: Lectures on Intuitionism. Lecture Notes in Mathematics, vol. 337, pp. 1–94. Springer, Heidelberg (1973)
Veldman, W.: An intuitionistic completeness theorem for intuitionistic predicate logic. Journal of Symbolic Logic 41, 159–166 (1976)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hermant, O., Lipton, J. (2008). A Constructive Semantic Approach to Cut Elimination in Type Theories with Axioms. In: Kaminski, M., Martini, S. (eds) Computer Science Logic. CSL 2008. Lecture Notes in Computer Science, vol 5213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87531-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-87531-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87530-7
Online ISBN: 978-3-540-87531-4
eBook Packages: Computer ScienceComputer Science (R0)