Undecidability of BPP Equivalences Revisited

Hüttel [3] gave a proof of the undecidability of BPP equivalences. In this note, we point out some of the flaws in his proof, and provide a new, actually simpler proof. 1 Flaws in Hüttel’s Proof Hüttel [3] proves that all BPP equivalences between ready simulation equivalence and trace equivalence are undecidable by encoding a Minsky machine M into two BPP processes L and L, so that (i) If M halts, then the trace set of L is not a subset of the trace set of L, and (ii) If M does not halt, then L ready-simulates L. The proof contains the following major flaws. 1. The proof of Theorem 6 (the main theorem) says: “If M does not halt, Lk = Lstop is unreachable, and R = · · · is seen to be a readysimulation.” This is wrong, because L may reach Lstop by invalid transitions. Therefore, the definition of the relation R must be modified, at least to include a pair having Lstop on the left-hand side. Moreover, the first pair: (Lj |Cm i |Cn i+1 |Lc0 |Lc1, Lj |Cm i |Cn i+1 |Lc0 |Lc1) does not satisfy the conditions required for R to be a ready-simulation. For example, if j is a type 1 instruction, the lefthand side can make the following transition: Lj |Cm i |Cn i+1 |Lc0 |Lc1 l′′ 2i −→ l′′ 2iG |Cm i |Cn i+1 |Lc0 |Lc1. It should be matched by: Lj |Cm i |Cn i+1 |Lc0 |Lc1 l′′ 2i −→ HS |Cm i |Cn i+1 |Lc0 |Lc1 for some S ⊆ Act2. However, l′′′ 2i is in the ready set of HS |Cm i |Cn i+1 |Lc0 |Lc1 but not in that of l′′ 2iG |Cm i |Cn i+1 |Lc0 |Lc1.