In an unpublished paper I have read, the concept of a signal trajectory is used.
A signal trajectory of a Boolean circuit is defined to be a non-zeno and right-continuous function $h : \mathbb{R^+} \to \mathbb{B}^{Var}$ which assigns a boolean value $h(t)(p) \in \mathbb{B}^{Var}$ to each (signal) variable $x \in Var$ at each point in time $t \in \mathbb{R^+}$.
It is non-zeno in the following sense:
- Non-zeno: In each bounded interval of $\mathbb{R}^+$ the function $h$ only has a finite number of value changes. In other words, it cannot switch the valuation of a signal variable with unbounded frequency.
From wikipedia, I see that a function $f$ is right-continuous at a point $c$ if and only if for any number $\epsilon > 0$ there exists some number $\delta > 0$ such that for all $x$ in the domain with $c<x<c+\delta$, the value of $f(x)$ will satisfy $|f(x) - f(c)| < \epsilon$.
My questions are:
- How does the definition of right-continuity apply to a signal trajectory, since when we apply $f: \mathbb{R^+} \to \mathbb{B}^{Var}$ to a real number, this returns a set of variables, whence how are we to understand $|f(x) - f(c)|$ (i.e, subtraction on sets of variables)? Is it just set complementation? Can anyone give an example of how the right-continuity condition as regards signal functions?
- What is meant by the non-zeno condition that ``In each bounded interval of $\mathbb{R}^+$ the function $h$ only has a finite number of value changes.'' How would this be expressed mathematically?