Abstract
The intersection type assignment system IT uses the formulas of the negative fragment of the predicate calculus (LJ) as types for the λ-terms. However, the deductions of IT only correspond to the proper sub-set of the derivations of LJ, obtained by imposing a meta-theoretic condition about the use of the conjunction of LJ. This paper proposes a logical foundation for IT. This is done by introducing a logic IL. Intuitively, a derivation of IL is a set of derivations in LJ such that the derivations in the set can be thought of as writable in parallel. This way of looking at LJ, by means of IL, allows to transform the meta-theoretic condition, mentioned above, into a purely structural property of IL. The relation between IL and LJ surely has a first main benefit: the strong normalization of LJ directly implies the same property on IL, which translates in a very simple proof of the strong normalizability of the λ-terms typable with IT.
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
Samson Abramsky. Domain theory in logical form. Ann. Pure Appl. Logic, 51(1–2):1–77, 1991.
F. Alessi and F. Barbanera. Strong conjunction and intersection types. In 16h International Symposium on Mathematical Foundation of Computer Science (MFCS91), volume Lecture Notes in Computer Science 520. Springer-Verlag, 1991.
F. Barbanera and S. Martini. Proof-functional connectives and realizability. Archive for Mathematical Logic, 33:189–211, 1994.
Henk Barendregt, Mario Coppo, and Mariangiola Dezani-Ciancaglini. A filter lambda model and the completeness of type assignment. J. Symbolic Logic, 48(4):931–940 (1984), 1983.
B. Capitani, M. Loreti, and Venneri B. Hyperformulae, parallel deductions and intersection types. To appear in “Workshop on Bohm Theorem”, IC ALP 2001, Creta (Greece), 2001.
Mario Coppo and Mariangiola Dezani-Ciancaglini. An extension of the basic functionality theory for the λ-calculus. Notre Dame J. Formal Logic, 21(4):685–693, 1980.
Mario Coppo, Mariangiola Dezani-Ciancaglini, Furio Honsell, and Giuseppe Longo. Extended type structures and filter lambda models. In Logic colloquium’ 82 (Florence, 1982), pages 241–262. North-Holland, Amsterdam, 1984.
H. Curry, R. Feys, and W. Craig. Combinatory Logic, volume 1. North Holland, 1958.
M. Dezani-Ciancaglini, S. Ghilezan, and B. Venneri. The “relevance” of intersection and union types. Notre Dame J. Formal Logic, 38(2):246–269, 1997.
Lavinia Egidi, Furio Honsell, and Simona Ronchi Della Rocca. Operational, denotational and logical descriptions: a case study. Fund. Inform., 16(2):149–169, 1992. Mathematical foundations of computer science’ 91 (Kazimierz Dolny, 1991).
Jean-Yves Girard. Locus solum: From the rules of logic to the logic of rules. Internal Report, IML, Marseille, 2001.
J.Y. Girard. Interpretation Fonctionelle et Elimination des Coupures de l’Arithmetique d’Ordre Superieur. PhD thesis, Université Paris VII, 1972.
J. Roger Hindley. Coppo Dezani types do not correspond to propositional logic. Theoret. Comput. Sci., 28(1–2):235–236, 1984.
Furio Honsell and Simona Ronchi Della Rocca. Reasoning about interpretations in qualitative lambda-models. In Programming Concepts and Methods, pages 505–522. North Holland, 1990.
Furio Honsell and Simona Ronchi Della Rocca. An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus. J. Comput. System Sci., 45(1):49–75, 1992.
Assaf Kfoury. Beta-reduction as unification. In Logic Algebra and Computer Science, pages 241–262. Polish Academy of Science, Warsaw, 1999.
Daniel Leivant. Polymorphic type inference. Symposium on Principles of Programming Languages, 1983.
G. E. Mints. The completeness of provable realizability. Notre Dame J. Formal Logic, 30(3):420–441, 1989.
Garrel Pottinger. A type assignment for the strongly normalizable λ-terms. In To H. B. Curry: essays on combinatory logic, lambda calculus and formalism, pages 561–577. Academic Press, London, 1980.
Dag Prawitz. Natural Deduction. Almquist & Wiksell.
J. C. Reynolds. Design of the programming language Forsythe. In P. O’Hearn and R.D. Tennent, editors, Algol-like Languages. Birkhauser, 1996.
L. Roversi. a Type-Free Resource-Aware λ-Calculus. In Fifth Annual Conference of the EACSL (CSL’ 96), volume 1258 of Lecture Notes in Computer Science, pages 399–413, Utrecht (The Nederland), September 1996. Springer-Verlag.
Betti Venneri. Intersection types as logical formulae. J. Logic Comput., 4(2):109–124, 1994.
J. Wells, Allyn Dimock, Robert Muller, and Franklyn Turbak. A typed intermediate language for flow-directed compilation. In 7th International Joint Conference on Theory and Practice of Software Development (TAPSOFT97), pages 757–771.
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
Della Rocca, S.R., Roversi, L. (2001). Intersection 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_29
Download citation
DOI: https://doi.org/10.1007/3-540-44802-0_29
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