EE290N - Specification and Modeling of Reactive Real-Time Systems

For a one sided discrete event system where without loss of generality define cantor metric

where is the smallest tag in such that , or if then .

An open neighbourhood in this metric space is the set of all the signals which have the same prefix. More formally, for , ,

where .

Problem: Let be the set of one sided discrete event signals and the cantor metric is defined. Show that is Huasdorff, i.e., given , show that there exist open sets and such that , and

We can also describe causality in the cantor metric space

• Causality
• Strict Causality
• Delta Causality such that ( is a contraction mapping)

We now investigate whether a causal system placed in feedback configuration deterministic. We will show that if the system is strictly causal then the feedback configuration is guaranteed to have at most one behaviour, i.e., it can have either a unique behaviour or no behaviour at all. We will also show that if the system is delta causal then the feedback configuration has a unique behaviour and we can systematically find it.

We will first show that the cantor metric is indeed a metric. We need to show that it satisfies the four conditions that any metric must satisfy. For ,

1. by definition
2. by definition
3. iff by definition
4. Triangle inequality

Proof Without loss of generality let . such that and . Since , . Hence such that . Hence .

Composing Functional Processes

• Parallel Composition The parallel composition of functional (causal, strictly causal or delta causal) processes is functional (causal, strictly causal or delta causal).
• Cascade Composition The cascade composition of functional (causal, strictly causal or delta causal) processes is functional (causal, strictly causal or delta causal).
• Source Composition The parallel composition of functional (causal, strictly causal or delta causal) processes and source processes is functional (causal, strictly causal or delta causal) if all the source processes are determinate (determinate, strictly causal, delta causal).
• Feedback Composition We will modify the general feedback configuration slightly. We replace the input with equivalent determinate source process. We also make all the output only signals as inputs. For , define the semantics to be a fixed point of , i.e., such that . If no or one signal satisfies this semantic, then the system is determinate otherwise it is indeterminate.
  • If is strictly causal, then it has at most one fixed point. Hence the feedback composition is determinate.
  • (Banach fixed point theorem) If the metric space is complete and is delta causal, then it has exactly one fixed point and that fixed point can be found by starting with any signal tuple and finding the limit of
  • If the metric space is compact (for instance if is finite and time is discrete), then only needs to be strictly causal to apply the Banach fixed point theorem.

Proof Assume then implies which is a contradiction. Hence .

Why do we have to restrict ourself to strict causality? Consider the following system with a delta causal process . Define an input output process pair such that output is the same as input. Now this composite process is no longer delta causal but it is causal nevertheless. Even though is delta causal but the overall system is causal and in the given feedback configuration, it is indeterminate.

Now consider the following example. The process has two inputs and and an output .
foreach , let
foreach , let
if collision, choose .
Now consider this process in a feedback configuration with the output fed back to and the input at . Clearly the system is a discrete event system and obeys strict causality but in the feedback configuration, the behaviour of the system is not discrete event. Hence strict causality need not preclude absence of behaviour.