Some Notions of Equivalence

6.3 Some Notions of Equivalence

In this section, we introduce three notions of equivalence among CCC processes. A notion of equivalence defines the circumstances in which we can deem two CCC pro- cesses to be equivalent in their behaviors. The first two, strong and weak-bisimulation equivalences, are closely related to Markovian strong bisimulation [Buc94, Hil96] and Markovian weak bisimulation [Bra02], respectively. Both of them operate on the level of the transition relations of the processes in question. The third notion operates on the level of the completion times of the processes.

6.3.1 Bisimulations

For the purpose of defining strong and weak-bisimulation relations, we first define

a generalization of the function γ defined in Equation (6.2). The new function γ c accumulates the rates of transitions from a process P ∈ L to all processes in C ⊆ L.

Definition 6.13. Let the function γ c :(

L×2 L ) →R

be defined as follows

(λ,w)∈{(λ,w)|P λ,w −−−→ Q }

where C ⊆ L. If R is an equivalence relation—i.e., a relation that is reflexive, symmetric, and

transitive—on L, then let L/R be the partitioning of L induced by R, and for P ∈ L, let [P ] R be the partition (class) that contains P . We write P RQ when (P, Q) ∈ R. The two notations are interchangeable.

Strong Bisimulation We define the strong-bisimulation relation in a same style as that of [BKHW05, Her02], but for the absence of labelling on processes. We explicitly restrict process stop to be the solitary member of its class to simplify subsequent proofs.

Definition 6.14. An equivalence relation S ⊆ L × L is a strong bisimulation on L if and only if (1) [stop] S = {stop} and (2) (P, Q) ∈ S implies that

∀C ∈ L/S : γ c (P, C) = γ c (Q, C).

6.3. Some Notions of Equivalence 113

Two processes P and Q are strongly bisimilar (denoted by P ∼ Q) if there exists a strong bisimulation S such that (P, Q) ∈S.

Example 6.15. Let process P ∈ L be defined by P = ((λ 1 ) k(λ 1 )).(λ 2 ). Figure 6.5 shows the semantics of the process in M LTS .

A relation S that identifies processes P 1 := ((λ 1 ) kstop).(λ 2 ) and P 2 := (stop k(λ 1 )).(λ 2 ), processes P , (λ 2 ), and stop with themselves, respectively, is a strong bisimulation on L. This can be verified as γ c (P 1 , [(λ 2 )] S )=γ c (P 2 , [(λ 2 )] S )=λ 1 , and for all other C ∈ L/S it holds that γ c (P 1 , C) = γ c (P 2 , C) = 0. Therefore we conclude that P 1 ∼P 2 .

((λ 1 ) kstop).(λ 2 )

stop P 2

(stop k(λ 1 )).(λ 2 )

Figure 6.5: The M LTS of P = ((λ 1 ) k(λ 1 )).(λ 2 )

Weak Bisimulation The weak-bisimulation relation is also defined in a similar style as that of [BKHW05, Her02]. However, the absence of the process labelling enforces us

to put process stop in its own partition to avoid identifying the whole set of processes in a single equivalence class.

Definition 6.16. An equivalence relation W ⊆ L × L is a weak bisimulation on L if and only if (1) [stop] W = {stop} and (2) (P, Q) ∈ W implies that

(6.6) Two processes P and Q are weakly bisimilar (denoted by P ≈ Q) if there exists a weak

∀C ∈ L/W , C 6= [P ] W :γ c (P, C) = γ c (Q, C).

bisimulation W such that (P, Q) ∈W. Example 6.17. Let process Q ∈ L be defined by Q = (2λ 1 ).((λ 2 ) ⊳ (λ 1 )(λ 2 )).(λ 1 ). The

M LTS semantics of the process Q is depicted in Figure 6.6.

A relation W that identifies process Q 1 := ((λ 2 ) ⊳ (λ 1 )(λ 2 )).(λ 1 ) with process Q 2 := (λ 1 ).(λ 2 ), processes Q, (λ 1 ), and stop with themselves, respectively, is a weak bisimulation on L. This can be verified as γ c (Q 1 , [(λ 1 )] W )=γ c (Q 2 , [(λ 1 )] W )=λ 2 , and for all other

C ∈ L/W , such that C 6= [Q 1 ] W , it holds that γ c (Q 1 , C) = γ c (Q 2 , C) = 0. Therefore we

conclude that Q 1 ≈Q 2 . The two notions of equivalence defined above provide a compositional notion of

semantics for CCC that is consistent with the structural operational semantics defined in Table 6.1. In short, both notions of equivalence are congruences, as shown in the following lemma.

114 Chapter 6. A Simple Stochastic Calculus

Figure 6.6: The M LTS of Q = (2λ 1 ).((λ 2 ) ⊳ (λ 1 )(λ 2 )).(λ 1 )

Lemma 6.18. Each of strong bisimilarity ∼ and weak bisimilarity ≈ is a congruence with respect to all operators of language L.

A proof sketch of this lemma is available in Appendix B.4. The congruent nature of both weak and strong bisimilarities is important, for it enables us to substitute a process with an equivalent one during compositions of pro- cesses. The substitution is useful if, for instance, the replacing process possesses some

“better” properties—say, having simpler structure or having smaller set of reachable processes—than the replaced one.

We have mentioned that strong and weak-bisimulation relations are closely related to Markovian strong and weak-bisimulation relations as defined in [Buc94, Hil96, Bra02]. This is straightforward from the fact that the semantics of CCC processes is absorbing Markov chains. Our definitions extend the traditional ones by explicitly providing a different handling of the absorbing state.

6.3.2 PH-Equivalence

In this section, we define a new notion of equivalence among CCC processes based on the PH distributions they represent. We will also clarify the relationship between this notion of equivalence and the previously defined weak and strong bisimilarities.

Definition 6.19. Two processes P, Q ∈ L are PH -equivalent (denoted by P ≈ PH Q) if and only if

(6.7) Example 6.20. Consider process P in Example 6.15 and process Q in Example 6.17.

PH(P ) = PH(Q).

Once we obtain the absorbing CTMC semantics of both processes, we can use Algorithm 3.13 to obtain the canonical representations of PH(P ) and PH(Q), and, at the same time, to reduce the size of the representations. The algorithm produces the same Cox representa-

tion for both, hence PH(P ) = PH(Q), and therefore P ≈ PH Q. Assuming that λ 2 <λ 1 , Figure 6.7 depicts the Cox representation we obtain.

PH -equivalence is also a congruence, as shown in the following lemma. Thus we are allowed to interchange processes that are equivalent during compositions of processes. This is especially important, because, then, we can use the reduction algorithm of