Dividing Line between Decidable PDA's and Undecidable Ones

Sénizergues has proved that language equivalence is decidable for disjoint ǫ-deterministic PDA. Stirling has showed that strong bisimilarity is decidable for PDA. On the negative side Srba demonstrated that the weak bisimilarity is undecidable for normed PDA. Later Jančar and Srba established the undecidability of the weak bisimilarity for disjoint ǫ-pushing PDA and disjoint ǫ-popping PDA. These decidability and undecidability results are extended in the present paper. The extension is accomplished by looking at the equivalence checking issue for the branching bisimilarity of several variants of PDA.