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Syntactic definitions of undefined: On defining the undefined

  • Zena Ariola
  • Richard Kennaway
  • Jan Willem Klop
  • Ronan Sleep
  • Fer-Jan de Vries
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 789)

Abstract

In the λ-calculus, there is a standard notion of what terms should be considered to be “undefined”: the unsolvable terms. There are various equivalent characterisations of this property of terms. We attempt to find a similar notion for orthogonal term rewrite systems. We find that in general the properties of terms analogous to the various characterisations of solvability differ.

We give two axioms that a notion of undefinedness should satisfy, and explore some of their consequences. The axioms lead to a concept analogous to the Böhm trees of the λ-calculus. Each term has a unique Böhm tree, and the set of such trees forms a domain which provides a denotational semantics for the rewrite system. We consider several particular notions of undefinedness satisfying the axioms, and compare them.

AMS subject Classification

68Q42 

CR subject Classification

F1.1 F4.1 F4.2 

Keywords & phrases

Orthogonal term rewriting systems Böhm Trees undefined terms lambda calculus solvability 

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References

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    J. R. Kennaway, J. W. Klop, M. R. Sleep, and F. J. de Vries. Transfinite reductions in orthogonal term rewriting systems. Technical Report SYS-C93-10, University of East Anglia, Norwich, U.K., 1993. Revised version of [?]. To appear in Information and Computation.Google Scholar
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    J. van Leeuwen, editor. Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics. North-Holland, Amsterdam, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Zena Ariola
    • 1
  • Richard Kennaway
    • 2
  • Jan Willem Klop
    • 3
  • Ronan Sleep
    • 2
  • Fer-Jan de Vries
    • 3
  1. 1.Computer and Information Science DepartmentUniversity of OregonEugene
  2. 2.School of Information SystemsUniversity of East AngliaNorwichUK
  3. 3.CWIGB Amsterdamthe Netherlands

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