Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems 267
• Let α be a probabilistic trace in P. The probability of α starting in the state
s is inductively defined by
Pr ptrace
P (s,α)=
�
1 if α = ɛ,
�
s ′ ∈SP
Pr a P (s, s′ ,µ) · Pr ptrace
P (s ′ ,α ′ ) if α =(a,µ) α ′ .
• Let Pr ptrace
P (α) =Pr ptrace
P (sP,α)(wheresPis the initial state of P) denote
the probability of a probabilistic trace α in P.
Definition 9.26. [CSZ92] Let P, Q ∈ FPP.
P � ptrace Q iff for all probabilistic traces α : Pr ptrace
P
(α) � Pr ptrace
Q (α).
⊓⊔
Note that opposed to Cleaveland et al. we do not allow synchronization over τactions,
so � ptrace is slightly different from the corresponding relation in [CSZ92].
Theorem 9.27. [CSZ92] For all fully probabilistic processes P, Q:
P � ptrace QiffP⊑CL Q.
Proof sketch: The idea for proving ⊑CL ⊆� ptrace is to construct T (α) ∈T fp
τ
from a given probabilistic trace α such that the success probability in P�T (α)
(and Q�T (α), respectively) is equal to the probability of α in P (and Q, respectively).
The proof of � ptrace ⊆⊑CL relies heavily on the fact that only a subset of all
possible fully probabilistic test processes is necessary to decide whether P ⊑CL Q
or not. These test processes are called essential and an essential test process T
contains only paths that are successful in P�T (and Q�T , respectively) or ”stop”
after one step when reaching success is impossible. For details we refer to the
work of Cleaveland et al. [CSZ92]. Note that the set of essential test processes
of P is still infinite. Thus deciding P ⊑CL Q is not practical. ⊓⊔
Christoff gives a characterization by extended traces for his testing relations
[Chr90]. Bernardo and Cleaveland show that ⊑BC can also be characterized by
such traces [BC00]. This is not surprising since non-probabilistic test processes
(used by Christoff) and Markovian test processes used in [BC00] are very similar
and an extended trace ”simulates” in fact a non-probabilistic test process. Let
D ′ be the set of all weight functions σ over Actτ with σ(a) ∈{0, 1} for a �= τ
and σ(τ) =0.
Definition 9.28. An extended trace is a sequence
α =(a1,σ1) (a2,σ2) ...(an,σn), ai ∈ Act,σi ∈D ′ for 1 � i � n.
⊓⊔
Note that we restrict to τ-free extended traces here because for ⊑CH we apply
τ-free test processes and a characterization of ⊑BC including test processes that