Abstract

We present abstract hyper-normalisation results for strategies. These
results are then applied to term rewriting systems, both first and
higher-order. For example, we show hyper-normalisation of the
left—outer strategy for, what we call, left—outer pattern rewrite
systems, a class comprising both Combinatory Logic and the
lambda-calculus but also systems with critical pairs. Our results apply
to strategies that need not be deterministic but do have Newman’s random
descent property: all reductions to normal form have the same length,
with Huet and Lévy’s external strategy being an example. Technically, we
base our development on supplementing the usual notion of commutation
diagram with a notion of order, expressing that the measure of its right
leg does not exceed that of its left leg, where measure is an
abstraction of the usual notion of length. We give an exact
characterisation of such global commutation diagrams, for pairs of
reductions, by means of local ones, for pairs of steps, we dub Dyck
diagrams.

BibTeX

@inproceedings{VvOYT-FSCD16,
  author    = "Vincent van Oostrom and Yoshihito Toyama",
  title     = "Normalisation by Random Descent",
  booktitle = "Proceedings of the 1st International Conference on Formal
               Structures for Computation and Deduction (FSCD 2016)",
  pages     = "32:1-32:18",
  series    = "Leibniz International Proceedings in Informatics (LIPIcs)",
  volume    = 52,
  editor    = "Delia Kesner and Brigitte Pientka",
  publisher = "Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik",
  year      = 2016
}