Summary
The paper shows that in spaces where elements are arranged in a clear order and spaced apart, any segment you pick out is actually limited in size. This helps clarify when a system's states are both limited and contained within bounds. The finding also holds true when combining several such ordered spaces into one. The authors focus especially on how this applies to models called transition systems used to describe state changes.
What this means in practice
- •For software engineers: Determine whether state spaces in system models are finite by checking boundedness in ordered discrete domains.
- •For control system designers: Use properties of well-ordered discrete state spaces to guarantee finiteness of reachable states in discrete control transitions.
A theory result. No direct application yet.
Abstract
In this short note, we prove that any closed interval in a well-ordered and discrete space is finite. The coincidence between finiteness and boundedness can then be established for transition systems where the state space is well-ordered and discrete. Finally, this result is also proven when the state space is a Cartesian product of $d$ discrete and well-ordered sets.