Transfinite reductions in orthogonal term rewriting systems

  • J. R. Kennaway
  • J. W. Klop
  • M. R. Sleep
  • F. J. de Vries
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 488)


Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, which we allow to be infinite, are unique, in contrast to ω-normal forms. Strongly converging fair reductions result in normal forms.

In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for Böhm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by ⊥) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows. The top-terminating OTRSs of Dershowitz c.s. are examples of non-unifiable OTRSs.

1985 Mathematics Subject Classification


1987 CR Categories

F4.1 F4.2 

Keywords and Phrases

orthogonal term rewriting systems infinitary rewriting strong converging reductions infinite Church-Rosser Properties normal forms Böhm Trees head normal forms non-unifiable term rewriting systems 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • J. R. Kennaway
    • 2
  • J. W. Klop
    • 1
  • M. R. Sleep
    • 2
  • F. J. de Vries
    • 1
  1. 1.CWI, Centre for Mathematics and Computer ScienceAmsterdam
  2. 2.School of Information SystemsUniversity of East AngliaNorwich

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