Abstract
A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and final coalgebras in a functor category and the Curry-Howard-correspondence. The definition of the system as a λ-calculus requires an appropriate definition of monotonicity to overcome subtle problems, most notably to ensure that the (co-) inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Thorsten Altenkirch. Representations of first order function types as terminal coalgebras. In Samson Abramsky, editor, Proceedings of TLCA 2001, volume 2044 of Lecture Notes in Computer Science, pages 8–21. Springer Verlag, 2001.
Thorsten Altenkirch and Bernhard Reus. Monadic presentations of lambda terms using generalized inductive types. In Jörg Flum and Mario Rodrýguez-Artalejo, editors, Computer Science Logic, 13th International Workshop, CSL’99, 8th Annual Conference of the EACSL, Madrid, Spain, September 20-25, 1999, Proceedings, volume 1683 of Lecture Notes in Computer Science, pages 453–468. Springer Verlag, 1999.
Richard S. Bird and Ross Paterson. De Bruijn notation as a nested datatype. Journal of Functional Programming, 9(1):77–91, 1999.
Richard Bird and Ross Paterson. Generalised folds for nested datatypes. Formal Aspects of Computing, 11(2):200–222, 1999.
Herman Geuvers. Inductive and coinductive types with iteration and recursion. In Bengt Nordström, Kent Pettersson, and Gordon Plotkin, editors, Proceedings of the 1992 Workshop on Types for Proofs and Programs, Bℴastad, Sweden, June 1992, pages 193–217, 1992. Only published electronically. Available at ftp://ftp.cs.chalmers.se/pub/cs-reports/baastad.92/proc.dvi.Z.
Neil Ghani. βη-equality for coproducts. In Mariangiola Dezani-Ciancaglini and Gordon Plotkin, editors, Proceedings of the Second International Conference on Typed Lambda Calculi and Applications (TLCA’ 95), Edinburgh, United Kingdom, April 1995, volume 902 of Lecture Notes in Computer Science, pages 171–185. Springer Verlag, 1995.
Jean-Yves Girard. Interprétation fonctionnelle et élimination des coupures dans l’arithmétique d’ordre supérieur. Thèse de Doctorat d’État, Universitè de Paris VII, 1972.
Brian Howard. Fixed Points and Extensionality in Typed Functional Programming Languages. PhD thesis, Stanford University, 1992.
Daniel Leivant. Contracting proofs to programs. In Piergiorgio Odifreddi, editor, Logic and Computer Science, volume 31 of APIC Studies in Data Processing, pages 279–327. Academic Press, 1990.
Ralph Matthes. Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. Doktorarbeit (PhD thesis), University of Munich, 1998.
Nax P. Mendler. Recursive types and type constraints in second-order lambda calculus. In Proceedings of the Second Annual IEEE Symposium on Logic in Computer Science, Ithaca, N.Y., pages 30–36. IEEE Computer Society Press, 1987.
Chris Okasaki. From fast exponentiation to square matrices: An adventure in types. In Proceedings of the fourth ACM SIGPLAN International Conference on Functional Programming (ICFP’ 99), Paris, France, September 27-29, 1999, volume 34 of SIGPLAN Notices, pages 28–35. ACM, 1999.
Christine Paulin-Mohring. Définitions Inductives en Théorie des Types d’Ordre Supérieur. Habilitation à diriger les recherches, ENS Lyon, 1996.
ZdzisSlaw SpSlawski and PaweSl Urzyczyn. Type Fixpoints: Iteration vs. Recursion. In Proceedings of the fourth ACM SIGPLAN International Conference on Functional Programming (ICFP’ 99), Paris, France, September 27-29, 1999, volume 34 of SIGPLAN Notices, pages 102–113. ACM, 1999.
Tarmo Uustalu and Varmo Vene. Least and greatest fixed points in intuitionistic natural deduction. Theoretical Computer Science. To appear.
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
Matthes, R. (2001). Monotone Inductive and Coinductive Constructors of Rank 2. 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_42
Download citation
DOI: https://doi.org/10.1007/3-540-44802-0_42
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