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

2.2. CCS, FORMALLY 23
A labelled transition system is finite if its sets of states and actions are both finite.
For example, the LTS for the process SmUni defined by equation 2.4 on page 13
(see page 19) is formally specified thus:
Proc = {SmUni, (CM | CS1) \ coin \ coffee, (CM1 | CS2) \ coin \ coffee,
Act = {pub, τ}
(CM | CS) \ coin \ coffee}
pub
→ = { SmUni, (CM | CS1) \ coin \ coffee ,
(CM | CS) \ coin \ coffee, (CM | CS1) \ coin \ coffee } , and
τ
→ = { (CM | CS1) \ coin \ coffee, (CM1 | CS2) \ coin \ coffee ,
(CM1 | CS2) \ coin \ coffee, (CM | CS) \ coin \ coffee } .
As mentioned above, we shall often distinguish a so called start state (or initial
state), which is one selected state in which the system initially starts. For example,
the start state for the process SmUni presented above is, not surprisingly, the
process SmUni itself.
Remark 2.2 Sometimes the transition relations α → are presented as a ternary relation
→⊆ Proc × Act × Proc and we write s α → s ′ whenever (s, α, s ′ ) ∈→. This is
an alternative way to describe a labelled transition system and it defines the same
notion as Definition 2.1.
Notation 2.1 Let us now recall a few useful notations that will be used in connection
with labelled transitions systems.
• We can extend the transition relation to the elements of Act ∗ (the set of all
finite strings over Act including the empty string ε). The definition is as
follows:
– s ε → s for every s ∈ Proc, and
– s αw → s ′ iff there is a state t ∈ Proc such that s α → t and t w → s ′ , for
every s, s ′ ∈ Proc, α ∈ Act and w ∈ Act ∗ .
In other words, if w = α1α2 · · · αn for α1, α2 . . . , αn ∈ Act then we write
s w → s ′ whenever there exist states s0, s1, . . . , sn−1, sn ∈ Proc such that
s = s0
α1 α2 α3 α4
→ s1 → s2 → s3 → · · · αn−1
→ sn−1 αn
→ sn = s ′ .
For the transition system in Figure 2.6 we have, for example, that p ε → p,
p ab
→ p and p1 bab
→ p.