Embedded Software for SoC - Grupo de Mecatrônica EESC/USP

268 Chapter 20
Let
be an acyclic path through PN nodes (places
and transitions). The direction of edges in the path L defines a sequence of
the transitions of L, in the order that appears in the path. We say that L is
fireable at a given marking if the sequence of transitions defined in this way
is fireable at the marking. We consider a sequence of markings of a path as
follows.
DEFINITION 2 (Marking sequence of a path). A marking sequence M(L) of
a path
is obtained by the consecutive application of
marking equations for firing transitions
starting from a marking
where
We call the intrinsic final marking of L.
DEFINITION 3 (Consumption index of a path). A consumption index of a
place p with respect to a path L is the maximum of over all that
belong to a marking sequence M(L), i.e.
The statement similar to Property 1 links consumption indexes of a path
with the path fireability.
PROPERTY 2. A path L is fireable from a marking M if and only if
The efficiency of the path-based compression strongly depends upon the
choice of the candidate paths. In general, a PN graph has too many paths to
be considered exhaustively. However the specification style might favour some
of the paths with respect to others. For example it is very common to encapsulate
the conditional constructs within a single block. The good specification
styles usually suggest entering such blocks through the head (modelled
in PN by a choice place) and exiting them through the tail (modelled in PN
by a merge place). These paths are clearly good candidates to try during
compression. Another potential advantage of paths from conditional constructs
is that several of them might start from the same place and end with the same
place. If they have the same consumption indexes and if their intrinsic final
markings are the same, then only one of the paths needs to be stored because
the conditions and final result of their firings are the same. Such alternative
paths in PN are called schedule-equivalent.
A simple example of a code with conditional is shown in Figure 20-3, where
the last parameter of each port access indicates the number of tokens to
read or write.
Paths and correspond to alternative
branches of computation and, as it is easy to see, are schedule-equivalent
because they both start from the same place end by the same place
and consume the same number of tokens from input port IN with the same