Bisimilarity on Basic Process Algebra is in 2-ExpTime (an explicit proof)

Abstract

Abstract. 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, Hüttel, 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).

Equivalences of Pushdown Systems Are Hard

“…An interesting subclass is BPA (Basic Process Algebra), corresponding to real-time PDA with a single control state. In the so called normed case bisimilarity is polynomial (see the above mentioned [13], [8]), but in general bisimilarity is in 2-ExpTime (claimed in [6] and explicitly proven in [16]) and ExpTime-hard [17].…”

Bisimilarity of Pushdown Automata is Nonelementary

“…On the other hand, the best known lower bound for this problem is EXPTIME shown by Kučera and Mayr [11]. In [9] EXPTIME-hardness has been established even for the subclass of basic process algebras, for which a 2EXPTIME upper bound is known [3] (in [4] a simpler proof has recently been announced). Such complexity gaps are typical in the context of infinite-state systems.…”

Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

“…7 and eq. (27) are in terms of |G| (recall that the grammatical constant g is exponential and n, s, and c are doubly exponential in terms of |G|), there exists a constant d independent from G such that |G| ≤ N 0 ≤ H ω 2 ·d (|G|) and G G (x) ≤ H ω 2 ·d (max{x, |G|}) for all G and x, where according to (16) H ω 2 ·d is the dth iterate of H ω 2 (x) = 2 x+1 (x+1)−1. Then by (17), h(x) def = H ω 2 ·d (x) is a suitable control function that satisfies (20) and therefore (22).…”