Bisimulation of Context-free Grammars and of Pushdown Automata

We consider the bisimulation on the transition graphs of pushdown automata. First we give a characterization of bisimulation using the unfolded trees. We recall that the bisimulation is decidable for the subclass of preex transition graphs of context-free grammars. Furthermore any pushdown transition graph is a regular graph: it can be generated by iterated parallel rewritings of a deterministic graph grammar. Then we show that the rational restrictions of pushdown transition graphs are exactly and eeectively the regular graphs of nite degree. We restrict to the class of graphs which are regular by increasing valuation : the length to a terminal coroot. This subclass strictly contains the preex transition graphs of reduced context-free grammars. And we show that the bisimulation is decidable on any graph regular by increasing valuation. Finally we show that the bisimulation on the regular graphs of nite out-degree is equivalent to the bisimulation on the preex transition graphs of context-free grammars on terms. For the deterministic case, these problems are inter-reducible to the well-known equivalence problem of dpda.