The decidability of the generalized confluence problem for context-free languages

A language $$L \subseteq \sum {^* }$$is confluent with respect to a given quasi order ≤ on σ* if, for any x,y ε L, there is z ε L such that x ≤ z and y ≤ z. L is confluent with respect to ≤ in generalized sense if it is a finite union of languages confluent with respect to ≤. We investigate in this paper the decidability of the generalized confluence problem with respect to the prefix partial order and the factor partial order for context-free languages, thus generalizing previous results concerning the decidability of the ordinary confluence problem.