  Burkart, Caucal, Steffen (1995) showed a procedure deciding bisimulation
equivalence of processes in Basic Process Algebra (BPA), i.e. of sequential
processes generated by context-free grammars. They improved the previous
decidability result of Christensen, H\"uttel, Stirling (1992), since their
procedure has obviously an elementary time complexity and the authors claim
that a close analysis would reveal a double exponential upper bound. Here a
self-contained direct proof of the membership in 2-ExpTime is given. This is
done via a Prover-Refuter game which shows that there is an alternating Turing
machine deciding the problem in exponential space. The proof uses similar
ingredients (size-measures, decompositions, bases) as the previous proofs, but
one new simplifying factor is an explicit addition of infinite regular strings
to the state space. An auxiliary claim also shows an explicit exponential upper
bound on the equivalence level of nonbisimilar normed BPA processes. The
importance of clarifying the 2-ExpTime upper bound for BPA bisimilarity has
recently increased due to the shift of the known lower bound from PSpace (Srba,
2002) to ExpTime (Kiefer, 2012).
