Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

22 CHAPTER 2. THE LANGUAGE CCS
a
p
b
p1
d
Figure 2.6: Labelled transition system with initial state p
p
p2
•
tea
coffee
•
•
1C=
2C=
collect
•
Remark 2.1 The definition of a labelled transition system permits situations like
that in Figure 2.6 (where p is the initial state). In that labelled transition system,
the state p2, where the action c can be performed in a loop, is irrelevant for the
behaviour of the process p since, as you can easily check, p2 can never be reached
from p. This motivates us to introduce the notion of reachable states. We say that
a state p ′ in the transition system representing a process p is reachable from p iff
there exists an directed path from p to p ′ . The set of all such states is called the set
of reachable states. In our example this set contains exactly two states, namely p
and p1.
Definition 2.1 [Labelled transition system] A labelled transition system (LTS) (at
times also called a transition graph) is a triple (Proc, Act, { α →| α ∈ Act}), where:
• Proc is a set of states (or processes);
• Act is a set of actions (or labels);
• α →⊆ Proc × Proc is a transition relation, for every α ∈ Act. As usual, we
shall use the more suggestive notation s α → s ′ in lieu of (s, s ′ ) ∈ α →, and
write s α (read ‘s refuses a’) iff s α → s ′ for no state s ′ .
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