ABSTRACT
In this paper, a notation influenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation and we establish the desirable properties of our reduction modulo equivalence classes rather than single terms. Finally, we extend the cube consisting of eight type systems with class reduction and show that this extension satisfies all the desirable properties of type systems.
- H. P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, revised edition, 1984.Google Scholar
- H.P. Barendregt. λ-calculi with types. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume II, pages 118--310. Oxford University Press, 1992.Google Scholar
- R. Bloo, F. Kamareddine, and R. P. Nederpelt. The Barendregt Cube with Definitions and Generalised Reduction. Information and Computation, 126 (2):123--143, 1996. Google ScholarDigital Library
- P. de Groote. The conservation theorem revisited. In International Conference on Typed Lambda Calculi and Applications, LNCS 664. Springer-Verlag, 1993. Google ScholarDigital Library
- F. Kamareddine and R. Nederpelt. A useful λ-notation. Theoretical Computer Science, 155:85--109, 1996. Google ScholarDigital Library
- F. Kamareddine and R. P. Nederpelt. Refining reduction in the λ-calculus. Journal of Functional Programming, 5(4):637--651, 1995.Google ScholarCross Ref
- F. Kamareddine, A. Ríos, and J.B. Wells. Calculi of generalised βe-reduction and explicit substitution: Type free and simply typed versions. Journal of Functional and Logic Programming, 1998.Google Scholar
- M. Karr. Delayability in proofs of strong normalizability in the typed λ-calculus. In Mathematical Foundations of Computer Software, LNCS, 185. Springer-Verlag, 1985. Google ScholarDigital Library
- A.J. Kfoury, J. Tiuryn, and P. Urzyczyn. An analysis of ML typability. ACM, 41(2):368--398, 1994. Google ScholarDigital Library
- A.J. Kfoury and J.B. Wells. A direct algorithm for type inference in the rank-2 fragment of the second order λ-calculus. Proceedings of the 1994 ACM Conference on LISP and Functional Programming, 1994. Google ScholarDigital Library
- A.J. Kfoury and J.B. Wells. Addendum to new notions of reduction and non-semantic proofs of β-strong normalisation in typed λ-calculi. Technical report, Boston University, 1995. Google ScholarDigital Library
- A.J. Kfoury and J.B. Wells. New notions of reductions and non-semantic proofs of β-strong normalisation in typed λ-calculi. LICS, 1995. Google ScholarDigital Library
- Z. Khasidashvili. The longest perpetual reductions in orthogonal expression reduction systems. 3rdInternational Conference on Logical Foundations of Computer Science, Logic at St Petersburg, 813, 1994. Google ScholarDigital Library
- J. W. Klop. Combinatory Reduction Systems. Mathematical Center Tracts, 27, 1980. CWI.Google Scholar
- J.-J. Lévy. Optimal reductions. In J. Hindley and J. Seldin, editors, To H.B. Curry: Essays on combinatory logic, lambda-calculus and formalism, pages 159--191. Academic Press, 1980.Google Scholar
- R. P. Nederpelt, J. H. Geuvers, and R. C. de Vrijer. Selected papers on Automath. North-Holland, Amsterdam, 1994.Google Scholar
- L. Regnier. Lambda calcul et réseaux. PhD thesis, University Paris 7, 1992.Google Scholar
- L. Regnier. Une équivalence sur les lambda termes. Theoretical Computer Science, 126:281--292, 1994. Google ScholarDigital Library
- A. Sabry and M. Felleisen. Reasoning about programs in continuation-passing style. Proceedings of the 1992 ACM Conference on LISP and Functional Programming, pages 288--298, 1992. Google ScholarDigital Library
- M. H. Sørensen. Strong normalisation from weak normalisation in typed λ-calculi. Information and Computation, 133(1), 1997. Google ScholarDigital Library
- D. Vidal. Nouvelles notions de réduction en lambda calcul. PhD thesis, Université de Nancy 1, 1989.Google Scholar
- H. Xi. On weak and strong normalisations. Technical Report 96--187, Carnegie Mellon University, 1996.Google Scholar
Index Terms
- De Bruijn's syntax and reductional equivalence of λ-terms
Recommendations
Compositional Characterizations of lambda-Terms Using Intersection Types
MFCS '00: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer ScienceWe show how to characterize compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and ...
Is timed branching bisimilarity an equivalence indeed?
FORMATS'05: Proceedings of the Third international conference on Formal Modeling and Analysis of Timed SystemsWe show that timed branching bisimilarity as defined by van der Zwaag [14] and Baeten & Middelburg [2] is not an equivalence relation, in case of a dense time domain. We propose an adaptation based on van der Zwaag’s definition, and prove that the ...
Characterizing contextual equivalence in calculi with passivation
We study the problem of characterizing contextual equivalence in higher-order languages with passivation. To overcome the difficulties arising in the proof of congruence of candidate bisimilarities, we introduce a new form of labeled transition ...
Comments