Recently I came with the problem: define a metric for a space formed by nouns. Here is my formulation:
Let $W$ be the set of all words and let $S \subset W$ be the set of all nouns. Let $A$ be the set of sequences from $W$ such that, for each element $a \in A$, the range of $a$ contains at least one element from $S$.
Define the function $N: S \times S \rightarrow \mathbb{Z}$ given by $N(x, y)$ = the number of sequences on A that has both $x \in S$ and $y \in S$ in its range, where $\mathbb{Z}$ is the set of all integers. Define the map $dep: S \times S \rightarrow \mathbb{R}_+$ such that $dep(x,y) = N(x,x) - N(x,y)$, where $\mathbb{R}_+$ is the set of all non-negative real numbers.
Question: is $dep$ a metric? if not, is it a quasimetric?