Abstract
There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the data have an associated indices which can be used to store important computational information. The crucial and salient feature of dependent polynomials and indexed containers is that the index types are defined in advance of the data. At the most sophisticated level, induction-recursion allows us to define data and indices simultaneously.
This work investigates the relationship between the theory of small inductive recursive definitions and the theory of dependent polynomials and indexed containers. Our central result is that the expressiveness of small inductive recursive definitions is exactly the same as that of dependent polynomials and indexed containers. A second contribution of this paper is the definition of morphisms of small inductive recursive definitions. This allows us to extend our main result to an equivalence between the category of small inductive recursive definitions and the category of dependent polynomials/indexed containers. We comment on both the theoretical and practical ramifications of this result.
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.
Similar content being viewed by others
References
Abbott, M., Altenkirch, T., Ghani, N.: Containers. Constructing Strictly Positive Types. TCS 342, 3–27 (2005)
Altenkirch, T., Morris, P.: Indexed containers. In: Procs. of the 24th Annual IEEE Symposium on Logic in Computer Science (LICS 2009). IEEE Computer Society (2009)
Aczel, P.: An introduction to inductive definition. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 739–782. North-Holland, Amsterdam (1977)
Bove, A., Capretta, V.: Nested General Recursion and Partiality in Type Theory. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 121–135. Springer, Heidelberg (2001)
Clairambault, P., Dybjer, P.: The Biequivalence of Locally Cartesian Closed Category and Martin Löf Type Theories. Arxiv:1112.3456v1 [cs.LO] (December 15, 2011)
Coquand, T., Dybjer, P.: Inductive Definitions and Type Theory an Introduction. In: Thiagarajan, P.S. (ed.) FSTTCS 1994. LNCS, vol. 880, pp. 60–76. Springer, Heidelberg (1994)
Curien, P.-L.: Substitution up to isomorphism. Fundamenta Informaticae 19(1-2), 51–86 (1993)
Dybjer, P.: A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic 65(2), 525–549 (2000)
Dybjer, P., Setzer, A.: A Finite Axiomatization of Inductive-Recursive Definitions. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999)
Dybjer, P., Setzer, A.: Induction-recursion and initial algebras. Annales of Pure and Applied Logic 124, 1–47 (2003)
Dybjer, P., Setzer, A.: Indexed Induction-Recursion. Journal of Logic and Algebraic Programming 66(1), 1–49 (2006)
Gambino, N., Hyland, M.: Wellfounded trees and dependent polynomial functors. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 210–225. Springer, Heidelberg (2004)
Gambino, N., Kock, J.: Polynomial functors and polynomial monads. Arxiv:0906.4931v2 [math.CT] (March 6, 2010)
Hofmann, M.: On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 427–441. Springer, Heidelberg (1995)
Kock, J.: Notes on Polynomial functors, http://www.mat.uab.es/~kock/cat/polynomial.html
Mac Lane, S.: Categories for the working mathematician, 2nd edn. Springer, New York (1998)
Martin-Löf, P.: An intuitionistic theory of types: Predicative part. In: Logic Colloquium 1973, pp. 73–118. North-Holland, Amsterdam (1973)
Moerdijk, I., Palmgren, E.: Wellfounded trees in categories. Annals of Pure and Applied Logic 104, 189–218 (2000)
Seely, R.A.G.: Locally cartesian closed categories and type theory. Math. Proc. Cambridge Philos. Soc. (95), 33–48 (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hancock, P., McBride, C., Ghani, N., Malatesta, L., Altenkirch, T. (2013). Small Induction Recursion. In: Hasegawa, M. (eds) Typed Lambda Calculi and Applications. TLCA 2013. Lecture Notes in Computer Science, vol 7941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38946-7_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-38946-7_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38945-0
Online ISBN: 978-3-642-38946-7
eBook Packages: Computer ScienceComputer Science (R0)