Abstract
This paper introduces a notion of partial type assignment on applicative term rewriting systems that is based on a combination of an essential intersection type assignment system, and the type assignment system as defined for ML [16], both extensions of Curry's type assignment system [11]. Terms and rewrite rules will be written as trees, and type assignment will consists of assigning intersection types function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes. The only constraints on this system are local: they are imposed by the relation between the type assigned to a node and those assigned to its incoming and out-going edges. In general, given an arbitrary typeablc applicative term rewriting system, the subject reduction property does not hold. We will formulate a sufficient but undecidable condition typeable rewrite rules should satisfy in order to obtain this property.
Partially supported by the Esprit Basic Research Action 3074 “Semagraph” and the Netherlands Organisation for the advancement of pure research (N.W.O.).
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References
S. van Bakel. Principal type schemes for the Strict Type Assignment System. Technical Report 91-6, Department of Computer Science, University of Nijmegen, 1991.
S. van Bakel. Complete restrictions of the Intersection Type Discipline. Theoretical Computer Science, 102:135–163, 1992.
S. van Bakel. Partial Intersection Type Assignment of Rank 2 in Applicative Term Rewriting Systems. Technical Report 92-03, Department of Computer Science, University of Nijmegen, 1992.
S. van Bakel, S. Smetsers, and S. Brock. Partial Type Assignment in Left Linear Applicative Term Rewriting Systems. In J.-C. Raoult, editor, Proceedings of CAAP '92. 17th Colloquim on Trees in Algebra and Programming, Rennes, France, volume 581 of Lecture Notes in Computer Science, pages 300–321. Springer-Verlag, 1992.
H. Barendregt. The lambda calculus: its syntax and semantics. North-Holland, Amsterdam, revised edition, 1984.
H. Barendregt, M. Coppo, and M. Dezani-Ciancaglini. A filter lambda model and the completeness of type assignment. The Journal of Symbolic Logic, 48(4):931–940,1983.
H.P. Barendregt, M.C.J.D. van Eekelen, J.R.W. Glauert, J.R. Kennaway, M.J. Plasmeijer, and M.R. Sleep. Term graph rewriting. In Proceedings of PARLE, Parallel Architectures and Languages Europe, Eindhoven, The Netherlands, volume 259-II of Lecture Notes in Computer Science, pages 141–158. Springer-Verlag, 1987.
T. Brus, M.C.J.D. van Eekelen, M.O. van Leer, and M.J. Plasmeijer. Clean — A Language for Functional Graph Rewriting. In Proceedings of the Third International Conference on Functional Programming Languages and Computer Architecture, Portland, Oregon, USA, volume 274 of Lecture Notes in Computer Science, pages 364–368. Springer-Verlag, 1987.
M. Coppo and M. Dezani-Ciancaglini. An Extension of the Basic Functionality Theory for the λ-Calculus. Notre Dame, Journal of Formal Logic, 21(4): 685–693,1980.
M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Principal type schemes and λ-calculus semantics. In J. R. Hindley and J. P. Seldin, editors, To H. B. Curry, Essays in combinatory logic, lambda-calculus and formalism, pages 535–560. Academic press, New York, 1980.
H.B. Curry and R. Feys. Combinatory Logic, volume 1. North-Holland, Amsterdam, 1958.
N. Dershowitz and J.P. Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 6, pages 245–320. North-Holland, 1990.
M. Dezani-Ciancaglini and J.R. Hindley. Intersection types for combinatory logic. Theoretical Computer Science, 100:303–324, 1992.
K. Futatsugi, J. Goguen, J.P. Jouannaud, and J. Meseguer. Principles of OBJ2. In Proceedings 12th ACM Symposium on Principles of Programming Languages, pages 52–66, 1985.
J.W. Klop. Term Rewriting Systems. Report CS-R9073, Centre for Mathematics and Computer Science, Amsterdam, 1990.
R. Milner. A theory of type polymorphism in programming. Journal of Computer and System Sciences, 17:348–375, 1978.
E.G.J.M.H. Nöcker, J.E.W. Smetsers, M.C.J.D. van Eekelen, and M.J. Plasmeijer. Concurrent Clean. In Proceedings of PARLE '91, Parallel Architectures and Languages Europe, Eindhoven, The Netherlands, volume 506-II of Lecture Notes in Computer Science, pages 202–219. Springer-Verlag, 1991.
S. Peyton Jones and J. Salkild. The spineless tagless G-machine. In Functional Programming Languages and Computer Architecture, pages 184–201. ACM press, 1989.
F. Pfenning. Partial Polymorphic Type Inference and Higher-Order Unification. In Proceedings of the 1988 conference on LISP and Functional Programming Languages, volume 201 of Lecture Notes in Computer Science, pages 153–163. Springer-Verlag, 1988.
B.C. Pierce. Programming with Intersection Types and Bounded Polymorphism. PhD thesis, Carnegie Mellon University, School of Computer Science, Pitssburgh, 1991. CMU-CS-91-205.
S. Ronchi della Rocca and B. Venneri. Principal type schemes for an extended type theory. Theoretical Computer Science, 28:151–169, 1984.
D.A.Turner. Miranda: A non-strict functional language with polymorphic types. In Proceedings of the conference on Functional Programming Languages and Computer Architecture, volume 201 of Lecture Notes in Computer Science, pages 1–16. Springer-Verlag, 1985.
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van Bakel, S. (1993). Partial intersection type assignment in applicative term rewriting systems. In: Bezem, M., Groote, J.F. (eds) Typed Lambda Calculi and Applications. TLCA 1993. Lecture Notes in Computer Science, vol 664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0037096
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DOI: https://doi.org/10.1007/BFb0037096
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