Trans nite Reductions in Orthogonal Term Rewriting

We deene the notion of transsnite term rewriting: rewriting in which terms may be innnitely large and rewrite sequences may be of any ordinal length. For orthogonal rewrite systems, some fundamental properties known in the nite case are extended to the transsnite case. Among these are the Parallel Moves lemma and the Unique Normal Form property. The transsnite Church-Rosser property (CR 1) fails in general, even for orthogonal systems, including such well-known systems as Combinatory Logic. Syntactic characterisations are given of some classes of orthogonal TRSs which do satisfy CR 1. We also prove a weakening of CR 1 for all orthogonal systems, in which the property is only required to hold up to a certain equivalence relation on terms. Finally, we extend the theory of needed reduction from the nite to the transsnite case. The reduction strategy of needed reduction is normalising in the nite case, but not in the transsnite case. To obtain a normalising strategy, it is necessary and suucient to add a requirement of fairness. Parallel outermost reduction is such a strategy.