Search on Conformant Problem: solution for subset of a belief state I am having trouble understanding the following statement.

I have understood why in a sensorless/conformant problem, if there exists a solution (a sequence of actions) for a belief state $b$, then it is also a solution for any $b'$ that is a subset $b$.

Next claim is what is confusing me.

Then in a standard graph search, we can save time by pruning some branches from the search tree.

Example from the Sensorless Vacuum World problem in Russel Norvig.

When we have a successor $\{1,3,5,7\}$ generated during graph search, we do not add it to the frontier if we have already expanded the belief state $\{5, 7\}$ which is a subset.

I thought those cases are totally unrelated since each belief state may encode a different path from the start state. If $\{5, 7\}$ does give us a solution, we have no guarantee it will work for $\{1, 3, 5, 7\}$. It's also possible $\{1, 3, 5, 7\}$ can lead to a solution through a different path.

I must be missing something.. can someone please explain?
'different path from the start state': The sensorless vacuum world is by definition a 'single-state problem', that is, all variables in the world are incorporated into a single state variable. In that case, history doesn't matter. If we are in state 5 by one path, the world is exactly as it would be if we had got there by another path.

For an example of something else represented as a single-state problem, consider chess: you might say history matters because whether you can castle, depends on whether the king has moved, not just the current board position. Okay so if we represent chess as a single-state problem, then the state would encode not only the current board position but also other relevant variables like whether the king has moved.

Subsets: if we know how to get to {5,7}, we don't need to be interested in {1,3,5,7}. Getting to {5,7} gives us all the same information plus more besides.