EE290n Lecture 5 Notes

EE290N - Specification and Modeling of Reactive Real-Time Systems

Lecture 5 - September 10, 1996, Scribe: Dick Stevens.

Terms Defined :

two-sided discrete event system
one-sided discrete event system

These terms can be found in the glossary and in lecture 3, vugraph 5. The definition of one-sided discrete event system was modified in class to state that the bijection should be from a SUBSET of the natural numbers.

cut
Intuitively, a cut of a signal at a tag t is the subset of its events whose tags do not exceed t (i.e., are not later than t).
causality
Intuitively, a process is causal if for any input, the tags of the resulting output are no earlier than those of the input.
strict causality
Intuitively, a process is strictly causal if for any input, the tags of the resulting output are later than those of the input.
delta causality
Intuitively, a process is delta causal if there exists a uniform delay such that for any input, the tags of the resulting output are later than the tags of the input by at least the uniform delay.

For formal definitions of cut, causality, strict causality, and delta causality, see lecture 5, vugraphs 1, 2.

Where causality and strict causality can be defined for any discrete event system (where the tags in any signal are discrete), delta causality requires METRIC time for the tags. Cuts are used in the formal definitions of the three types of causality.

Much of the class was devoted to a lively discussion about these three types of causality and the differences between them. The lecturer remarked that the liveliness of the discussion served to illustrate how non-trivial the subject is.

Examples were given to illustrate how a system can be causal but not strictly causal and how a system can be strictly causal but not delta causal. The statements were made but not proved that delta causality implies strict causality and that strict causality implies causality.

Cantor metric
See lecture 5, vugraph 3.
Intuitively, the Cantor metric is a measure of how close two signals are in terms of "how long" they agree. The distance between two signals should decrease exponentially with the length of the maximum cuts of each that agree. During the class, some questions arose about this vugraph, leading to the conclusion that it must be revised. A sticking point arose about whether the definition should be based on the maximum cuts of agreement or the minimum cuts of disagreement. And how the case of full agreement (where the distance from a signal to itself is zero) should be handled.

The class was adjourned with the following promises:

A solution to the first homework assignment.
A better formal definition of the Cantor metric.

and proofs of the following:

The Cantor metric is an ultra-metric.
A delta causal process under the Cantor metric is a contraction mapping.
A delta causal process has a unique constructable fixed point that is its unique behavior.
A delta causal process is thus determinate.
Under the Cantor metric, if the space of signals is compact, then strict causality is sufficient for delta causality; hence the process has a unique behavior and is determinate.