On Transfinite Knuth-Bendix Orders

In this paper we discuss the recently introduced transfinite Knuth-Bendix orders. We prove that any such order with finite subterm coefficients and for a finite signature is equivalent to an order using ordinals below ω, that is, finite sequences of natural numbers of a fixed length. We show that this result does not hold when subterm coefficients are infinite. However, we prove that in this general case ordinals below ω ω suffice. We also prove that both upper bounds are tight. We briefly discuss the significance of our results for the implementation of firstorder theorem provers and describe relationships between the transfinite Knuth-Bendix orders and existing implementations of extensions of the Knuth-Bendix orders.