Embedded Software for SoC - Grupo de Mecatrônica EESC/USP

State Space Compression in History 267
Two questions need to be answered in applying the above compression
technique.
1.
2.
How to choose a sequence of transitions which should be fired at once
How to determine upfront that this sequence of transitions is fireable
Let us first consider the second question, while the first question will be
addressed in the Section 4.2. We use marking equations to check if a sequence
of transitions is fireable. Since self-loops introduce inaccuracy in calculating
the fireable transitions in the marking equations, let us assume in the rest of
this section that a specification is provided as a PN without self-loops. 1 In
the sequel, we may use a term trace to refer to a sequence of transitions.
DEFINITION 1 (Consumption index of a trace). Let a trace S, firing from a
marking produce a sequence of markings A
consumption index of a place p with respect to is the maximum
across all
Informally,
shows how much the original token count for
a place p deviates while executing a trace S from a marking Applying
the marking equations, the following is immediately implied:
PROPERTY 1. A trace S, fireable from a marking
marking if and only if
is fireable from a
Let us illustrate the application of this trace-based compression with the
producer-consumer example in Figure 20-2. When a scheduler has executed
a trace t3, t4 from a marking
it could compute the consumption
indexes for that trace as:
From this a scheduler can conclude that t3, t4 is fireable from
because according to Property 1 this marking has enough token resources for
executing the considered trace. Note that once a trace is known to be fireable
at a given marking, the marking obtained after firing the trace can be
analytically obtained by the marking equation. The scheduler can therefore
fire the trace at once and compute the resulting marking, as shown in Figure
20-2(c).
4.2. Path-based compression
The technique shown in the previous section relies on a scheduler to find
candidate traces for compressing a schedule. This leaves the first question
unanswered: how to find such candidate traces in the first place. This could
be addressed through the analysis of the structure of the specification PN.