Lecture Notes in Computer Science 3472

122 Stefan D. Bruda
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Fig. 5.1. Representation of infinite process trees: an infinite tree (a), and its graph
representation (b).
from Figure 5.6 on page 133 (again, states are depicted by nodes, and the relation
−→ is represented by arrows between nodes, labeled with actions). It may
be the case that y is a nice, convergent state in the respective labeled transition
system (i.e., y ↓B). Still, it is obviously the case that y ↑ in the derived transition
system (we refer to this as “y may diverge” instead of “y diverges,” given that y
may decide at some time to perform action b and get out of the loop of internal
actions).
Again, we shall use in what follows natural extensions of the relation ⇒
such as p a ⇒ and �⇒ . We also use by abuse of notation the same operator for
the reflexive and transitive closure of ⇒ (in the same way as we did for −→ ).
A transition system gives a description of the actions that can be performed
by a process depending on the state that process is in. A process does in addition
start from an initial state. In other words, a process is fully described by a
transition system and an initial state. In most cases we find it convenient to
fix a global transition system for all the processes under consideration. In this
setting, a process is then uniquely defined by its initial state. We shall then blur
the distinction between a process and a state, often referring to “the process
p ∈ Q.”
Finally, a process can be represented as a tree in a natural way: Tree nodes
represent states. The root node is the initial state. The edges of the tree will
be labeled by actions, and there exists an edge between nodes p and q labeled
with a iff it holds that p a
−→ q in the given transition system (or that p a ⇒ q
if we talk about a derived transition system). We shall not make use of this
representation except when we want to represent a process (or part thereof)
graphically for illustration purposes. Sometimes we find convenient to “abbreviate”
tree representation by drawing a graph rather than a tree when we want
to represent infinite trees with states whose behavior repeats over and over (in
which case we join those states in a loop). The reader should keep in mind that
this is just a convenient representation, and that in fact she is in front of a finite
representation of an infinite tree. As an example, Figure 5.1 shows such a graph
together with a portion of the unfolded tree represented by the graph.
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