Corrigendum to "A tutorial on EMPA: a theory of concurrent processes with nondeterminism, priorities, probabilities and time" - [TCS 202 (1998) 1-54]

In [4] we presented a tutorial on the stochastically timed process algebra EMPA together with the related theory: semantics, equivalence, and axiomatization. The purpose of this note is to remedy to two errors contained in that paper which are related to the strong extended Markovian bisimulation equivalence ∼EMB defined for EMPA. The first error is contained in the proof of the congruence property of ∼EMB w.r.t. recursive definitions but does not affect the validity of the result. The second error is contained in the proof of the congruence property of ∼EMB w.r.t. the parallel composition operator and affects the validity of the result, as it holds only for a class of terms allowing for a restricted form of nondeterminism. The first error, discovered by Mario Bravetti, is concerned with the proof of the congruence property w.r.t. recursive definitions of Thm. 5.19. The technique used in [4] consists of proving by induction that a certain relation over EMPA terms is a strong extended Markovian bisimulation up to ∼EMB. As recognized in [3], the proof of Thm. 5.19 contains two inaccuracies related to the employed notion of strong extended Markovian bisimulation up to ∼EMB and the structure of the inductive proof itself. The problem with the definition of strong extended Markovian bisimulation up to ∼EMB of Def. 5.12 is that, following the style of [7], it should be given w.r.t. the classes of an equivalence relation determined by ∼EMB and B, the relation over EMPA terms being defined. In Def. 5.12, instead, the classes of the relation ∼EMB B ∼EMB are considered which, in general, is not transitive even if B is supposed to be an equivalence relation. Similarly to [6], in [3] it is proposed to solve this problem by considering the classes of (B ∪ B−1∪ ∼EMB). The problem with the structure of the proof is the following. Given two terms E1 and E2, since the proof proceeds by induction on the maximum depth of the derivation of the transitions of E1 labeled with type a, having priority level l, and reaching an arbitrary equivalence class C w.r.t. B (the relation over EMPA terms that has to be proved to be a strong extended Markovian bisimulation up to ∼EMB), several cases arise depending on the outermost operator, as in the proof of the corresponding result in [8]. As far as static operators are concerned, the proof of Thm. 5.19 wrongly assumes that all the terms that, when applied the same static operator, belong to the same equivalence class C are equivalent, i.e. they form a single equivalence class. Instead, such terms form in general several equivalence classes, to each of which the induction hypothesis Rate(E1, a, l, C) = Rate(E2, a, l, C) should be applicable. Actually, this is possible only for those classes which are reachable from the subterm E′ 1 of E1 in the scope of the static operator, because for the other classes we would need a converse argument related to the transitions of the subterm E′ 2 of E2 in the scope of the static operator. Similarly to [5], in [3] it is proposed to solve the problem by