Context: A.E. Prieditis 1993, where the length of shortest path from states was getting discussed and saw below formula for same.
I can make the min and sigma notation but the |,= and ^ are not making sense to me. I mean I know what typically they are used for like denoting OR and disjunction.
It is unnecessarily complicated notation.
The intention is simple & can be stated in simpler terms too.
Intention is that $k(s,t)$ is the minimum of the summation where each transition from state to state has some cost.
There are $n-1$ such transitions & $n-1$ associated costs.
We want the transitions to start with $s$ & terminate with $t$.
The unnecessarily completed part is after the $|$ , which just says that :
$s_1=s$ [ Starting State ] AND
$s_i \in S$ [ all states are in Universal Set $S$ ] AND
$s_n=t$ [ Terminal State ]
The middle condition is to cover all States.
Simpler to write it like these variations :
$\forall i : s_i \in S$
$s_i \in S , \forall i$
$s_i \in S$ for all $i$
$s_i \in S , 1 \le i \le n$
$s_i \in S , i \in [1,\cdots,n]$
When we try all ways to start with $s$ & terminate with $t$ , then we will have multiple Paths. We want to choose the minimum cost Path. Here $k(s,t)$ is the minimum cost of that Set of Paths.
The actual Path is not given by this formula.