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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?

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'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.

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  • $\begingroup$ If we are at $\{1, 3, 5, 7\}$ and through some seq of actions, we can get to a goal state of $\{1\}$ without going through $\{5, 7\}$ on the way. As for $\{5, 7\}$, we may find out, after a few expansions, that doesn't get us to goal state. What may be more concerning is that we could have generated $\{5, 7\}$ through some state that was not $\{1, 3, 5, 7\}$ in the first place. I'm concerned that if we prune away the search tree branch that is rooted at $\{1, 3, 5, 7\}$ because there is a state that is a subset of this, we are pruning away a path to what may be the only solution in this case. $\endgroup$ – namesake22 Jan 29 at 15:28

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