Hostname: page-component-857557d7f7-nfgnx Total loading time: 0.001 Render date: 2025-12-10T11:42:19.304Z Has data issue: false hasContentIssue false
  • English
  • Français

A typed, algebraic, computational lambda-calculus†

Published online by Cambridge University Press:  28 February 2013

BENOÎT VALIRON
BENOÎT VALIRON*
Affiliation:
LIPN – UMR 7030 CNRS – Université Paris 13, 99 avenue J-B Clément, F-93430 Villetaneuse, France Email: benoit.valiron@monoidal.net
Get access Rights & Permissions [Opens in a new window]

Abstract

Lambda-calculi with vectorial structures have been studied in various ways, but their semantics remain largely uninvestigated. The main contribution of this paper is to provide a categorical framework for the semantics of such algebraic lambda-calculi. We first develop a categorical analysis of a general simply typed lambda-calculus endowed with the structure of a module. We study the problems arising from the addition of a fixed-point combinator and show how to modify the equational theory to solve them. The categorical analysis carries nicely over to the modified language. We provide various concrete models for both the case without fixpoints and for the case with them.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Special Issue 2: Developments In Computational Models 2010 , April 2013 , pp. 504 - 554
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

A preliminary version of this work appeared as Valiron (2010).

References

Arrighi, P. and Díaz-Caro, A. (2009) Scalar system F for linear-algebraic λ-calculus: Towards a quantum physical logic. In: Coecke, B., Panangaden, P. and Selinger, P. (eds.) Proceedings of QPL'09. Electronic Notes in Theoretical Computer Science 270 (2)219–229.CrossRefGoogle Scholar
Arrighi, P. and Dowek, G. (2008) Linear-algebraic lambda-calculus: higher-order, encodings, and confluence. In: Voronkov, A. (ed.) Proceedings of the 19th International Conference on Rewriting Techniques and Applications. Springer-Verlag Lecture Notes in Computer Science 5117 17–31.CrossRefGoogle Scholar
Barbanera, F. and Fernández, M. (1993) Combining first and higher-order rewrite systems with type assignment systems. In: Bezem, M. and Groote, J. E. (eds.) Proceedings of the International Conference on Typed Lambda Calculi and Applications, TLCA'93. Springer-Verlag Lecture Notes in Computer Science 664 60–74.CrossRefGoogle Scholar
Blanqui, F., Jouannaud, J.-P. and Okada, M. (1999) The calculus of algebraic constructions. In: Narendran, P. and Rusinowitch, M. (eds.) Rewriting Techniques and Applications: 10th International Conference, RTA-99. Springer-Verlag Lecture Notes in Computer Science 1631 301–316.CrossRefGoogle Scholar
Breazu-Tannen, V. and Gallier, J. (1991) Polymorphic rewriting conserves algebraic strong normalization. Theoretical Computer Science 83 (1)3–28.CrossRefGoogle Scholar
Bucciarelli, A., Ehrhard, T. and Manzonetto, G. (2009) A relational model of a parallel and non-deterministic lambda-calculus. In: Artemov, S. and Nerode, A. (eds.) Logical Foundations of Computer Science. Springer-Verlag Lecture Notes in Computer Science 5407 107–121.CrossRefGoogle Scholar
Ehrhard, T. (2005) Finiteness spaces. Mathematical Structures in Computer Science 15 615–646.CrossRefGoogle Scholar
Ehrhard, T. and Regnier, L. (2003) The differential lambda-calculus. Theoretical Computer Science 309 (1–2)1–41.CrossRefGoogle Scholar
Filinski, A. (1996) Representing monads. In: Proceedings of the 21st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages 446–457.Google Scholar
Heckmann, R. (1990) Power Domain Constructions, Ph.D. thesis, Universität der Saarlandes, Saabrücken, Germany.Google Scholar
Jones, C. and Plotkin, G. D. (1989) A probabilistic powerdomain of evaluations. In: Proceedings of the 4th Symposium on Logic in Computer Science 186–195.Google Scholar
Kelly, G. M. (1982) Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series 64, Cambridge University Press. (Available in Reprints in Theory and Application of Categories 10.)Google Scholar
Lambek, J. and Scott, P. (1989) Introduction to Higher Order Categorical Logic, Cambridge University Press.Google Scholar
MacLane, S. Lane, S. (1998) Categories for the Working Mathematician, Springer-Verlag.Google Scholar
Moggi, E. (1991) Notions of computation and monads. Information and Computation 93 55–92.CrossRefGoogle Scholar
Pierce, B. C. (2002) Types and Programming Languages, MIT Press.Google Scholar
Plotkin, G. D. (1977) LCF considered as a programming language. Theoretical Computer Science 5 223–255.CrossRefGoogle Scholar
Plotkin, G. D. (1983) Pisa notes (on domain theory). Unpublished set of notes.Google Scholar
Selinger, P. (2003) Order-incompleteness and finite lambda-reduction models. Theoretical Computer Science 309 43–63.CrossRefGoogle Scholar
Valiron, B. (2010) Semantics of a typed algebraic lambda-calculus. In: Cooper, S. B., Panangaden, P. and Kashefi, E. (eds.) Proceedings of the 6th Workshop on Developments in Computational Models. Electronic Proceedings in Theoretical Computer Science 26 147–158.CrossRefGoogle Scholar
Tonder, A. v. (2004) A lambda calculus for quantum computation. SIAM Journal of Computing 33 1109–1135.CrossRefGoogle Scholar
Vaux, L. (2009) The algebraic lambda-calculus. Mathematical Structures in Computer Science 19 1029–1059.CrossRefGoogle Scholar
Winskell, G. (1983) A note on powerdomains and modality. In: Karpinski, M. (ed.) Foundations of Computation Theory. Springer-Verlag Lecture Notes in Computer Science 158 505–514.CrossRefGoogle Scholar