EE290N - Specification and Modeling of Reactive Real-Time
Systems
Lecture 2 - August 29, 1996,
Scribe: Eugene Miloslavsky.
Embedded system - possible definitions:
- Uses fixed amount of resource.
- is a subsystem with a clearly defined job.
- is just a subsystem.
Various time models
- Continuous time.
- Discrete time.
- Multi-rate discrete time.
Totally ordered discrete events: event comes before, after of at the same time
as another event.
Partially ordered discrete events: not all ordering relations between events
are specified.
Not specifying all the ordering relations raises abstraction level and avoids
over-specification.
Events and Signals
The concepts of event, signal, functional signal, set of all signals and N-tuples
of signals were defined. Definitions
Examples of order
- Regular < on real numbers.
- Two cousins having a common grandfather is an example of partial order.
- 22/7 is a worse approximation to PI then 3.141592654. Order between
two real numbers in this scheme is the comparison of their distances to PI.
- Subset relation - is a partial order on sets.
Discrete time
What does it mean for a set of tags to constitute a discrete time?
I.e. what properties should a set of tags have to be called discrete time?
Intuitively, discrete set has to be countable and totally ordered. But is this
sufficient? - NO!
Note that total order and countability does not imply discreteness, since
rational numbers (with regular < order ) constitute a countable, totally ordered
set that is not discrete.
Definition:
A set A is countable if there exists a bijection
(a one-to-one and onto mapping) F: A->Z where Z is 0, +-1, +-2 ......
In fact, discreteness requires that the bijection in the definition of
countability preserve the order.
Processes, Behaviors, Connections
The definitions for this denotational view of systems are given in
The Tagged Signal Model - A Preliminary Version of a Denotational Framework for Comparing Models of Computation by Edward A. Lee and Alberto Sangiovanni-Vincentelli