Abstract
The rewriting calculus, also called ρ-calculus, is a framework embedding λ-calculus and rewriting capabilities, by allowing abstraction not only on variables but also on patterns. The higher-order mechanisms of the λ-calculus and the pattern matching facilities of the rewriting are then both available at the same level. Many type systems for the λ-calculus can be generalized to the ρ-calculus: in this paper, we study extensively a first-order ρ-calculus à la Church, called \(\rho^{\rm stk}_\rightarrow\). The type system of \(\rho^{\rm stk}_\rightarrow\) allows one to type (object oriented flavored) fixpoints, leading to an expressive and safe calculus. In particular, using pattern matching, one can encode and typecheck term rewriting systems in a natural and automatic way. Therefore, we can see our framework as a starting point for the theoretical basis of a powerful typed rewriting-based language.
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
Abadi, M., Cardelli, L.: A Theory of Objects. Springer, Heidelberg (1996)
Barendregt, H.: Lambda Calculus: its Syntax and Semantics. North-Holland, Amsterdam (1984)
Barthe, G., Cirstea, H., Kirchner, C., Liquori, L.: Pure Patterns Type Systems. In: Proc. of POPL, pp. 250–261. ACM Press, New York (2003)
Bertolissi, C., Cirstea, H., Kirchner, C.: Translating Combinatory Reduction Systems into the Rewriting Calculus. In: Proc. of RULE. ENTCS (2003)
Borovansky, P., Kirchner, C., Kirchner, H., Moreau, P.-E.: ELAN from a rewriting logic point of view. Theoretical Computer Science 2(285), 155–185 (2002)
Byun, S., Kennaway, J., van Oostrom, V., de Vries, F.: Separability and Translatability of Sequential Term Rewrite Systems into the Lambda Calculus. Technical Report tr-2001-16, University of Leicester (2001)
Cirstea, H., Kirchner, C.: The Simply Typed Rewriting Calculus. In: Proc. of WRLA. ENTCS (2000)
Cirstea, H., Kirchner, C.: The rewriting calculus — Part I and II. Logic Journal of the Interest Group in Pure and Applied Logics 9(3), 427–498 (2001)
Cirstea, H., Kirchner, C., Liquori, L.: Matching Power. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 77–92. Springer, Heidelberg (2001)
Cirstea, H., Kirchner, C., Liquori, L.: The Rho Cube. In: Honsell, F., Miculan, M. (eds.) FOSSACS 2001. LNCS, vol. 2030, pp. 166–180. Springer, Heidelberg (2001)
Cirstea, H., Kirchner, C., Liquori, L.: Rewriting Calculus with(out) Types. In: Proc. of WRLA. ENTCS (2002)
Protheo, É.: The Elan Home Page (2003), http://elan.loria.fr
Kamin, S.N.: Inheritance in Smalltalk-80: A Denotational Definition. In: Proc. of POPL, pp. 80–87. The ACM press, New York (1988)
Kesner, D., Puel, L., Tannen, V.: A Typed Pattern Calculus. Information and Computation 124(1), 32–61 (1996)
Mendler, N.P.: Inductive Definition in Type Theory. PhD thesis, Cornell University, Ithaca, USA (1987)
The Maude Team. The Maude Home Page (2003), http://maude.cs.uiuc.edu/
van Oostrom, V.: Lambda Calculus with Patterns. Technical Report IR-228, Faculteit der Wiskunde en Informatica, Vrije Universiteit Amsterdam (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cirstea, H., Liquori, L., Wack, B. (2004). Rewriting Calculus with Fixpoints: Untyped and First-Order Systems. In: Berardi, S., Coppo, M., Damiani, F. (eds) Types for Proofs and Programs. TYPES 2003. Lecture Notes in Computer Science, vol 3085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24849-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-24849-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22164-7
Online ISBN: 978-3-540-24849-1
eBook Packages: Springer Book Archive