# 19.14 Continuous reachability in ordered data VAS

Data VAS is a finite set of finitely supported functions from ordered data set $$\mathbb{D}$$ to $$\mathbb{Z}^k$$, where k is a dimension. A configuration/marking is a finitely supported function from $$\mathbb{D}$$ to $$\mathbb{N}^k$$.

From a configuration $$m$$ there is a step to $$m'$$ if there is a vector $$x\in \text{VAS}$$ and $$\pi$$ an ordered preserving bijection (permutation) form $$\mathbb{D}$$ to $$\mathbb{D}$$ such that $$m+x\circ \pi=m'$$. The reachability relation is a transitive closure of the step relation. The reachability problem is undecidable, that is why we are looking for different relaxation of it.

Continuous reachability is a continuous version of the reachability. First, markings are finitely/supported functions from $$\mathbb{D}$$ to $$\mathbb{Q}_{\geq 0}^k$$. Further there is a continuous step from $$m$$ to $$m'$$ if there are: $$x\in \text{VAS}$$, an order preserving data permutation $$\pi$$, and a factor $$a\in \mathbb{Q}_{\geq 0}$$ such that $$m+a\cdot x\circ \pi=m'$$. A transitive closure of continuous step is a continuous reachability relation.