  A decidability proof for bisimulation equivalence of first-order grammars
(finite sets of labelled rules for rewriting roots of first-order terms) is
presented. The equivalence generalizes the DPDA (deterministic pushdown
automata) equivalence, and the result corresponds to the result achieved by
Senizergues (1998, 2005) in the framework of equational graphs, or of PDA with
restricted epsilon-steps. The framework of classical first-order terms seems
particularly useful for providing a proof that should be understandable for a
wider audience. We also discuss an extension to branching bisimilarity,
announced by Fu and Yin (2014).
