So, remember in the last video nugget we talked about this idea of a belief model was essentially

a set of actual world states, but we don't really know, we're uncertain about what the

actual world state is, so we have to kind of deal with a set of possible world states.

And I would like to come back to this idea of sensorless planning, planning without observations,

and that we looked at with these and or searches. So I would like to see how this

meshes with this idea of belief state search. So, right, we are calling this conformant planning

or sensorless planning, and basically we're trying to find plans that work without any sensing.

So we have the sensorless vacuum cleaner world. In a sensorless world, we do not know our initial

state, so we have to actually deal with belief states. So we basically start searching in the

belief states, so we have no prior knowledge about what we are, and we may actually not see

what it, we don't have sensors, so we have no prior knowledge, so it might be one of the

eight possible states. And this plan here, right, going right actually eliminates all the left states

from this, we're still deterministic here, so we're left with the belief state 2, 4, 6, 8,

then suck actually removes the dirt deterministically from the right state that gives us

2 and 8, 4 and 8 as possible states, we go left, that gives us 3 and 7 as possible states,

and we suck, which removes the dirt in the other room, and that leaves us with one certain state,

and that just happens to be a goal state. So we've found a sensorless plan here, and we did it by

actually looking at the belief state. Okay, let's do the math. Say we have an agent problem, which

has a set of states, a set of actions, a tradition transition model, an initial state, and a goal

state, and so we just take this idea of a belief state and run with it. So remember the transition

model, that's what we're going to look at, is if we have an action and a state, that actually gives

us, we have a relation here, right, and an initial state. So what is the corresponding sensorless

belief state level problem? So we again have a five-tuple, and we take this five-tuple,

this problem, the actual problem, and lift it up. So the states are actually

subsets of states, right, we have belief spaces, and of course there are two to the, however many

states there are in the original problem, many of those. The initial state is just, since we are

sensorless, we have no information about this, is just the set of all states of the original problem.

The goal states are all the states in the belief space, which are subsets,

which are sets of states in the belief state. So we could just say the goal states are those

that are subsets of the goal. So the possible state must be, all the possible states in the belief

states must be goal states. So in this case, in this situation here, seven being one of many goal

states, we actually have a, that we actually find one of the singleton goal states is a nice side

effect of this example, but it's not actually needed. So the actions are just the actions,

and we now need a traditional transition model. And rather than having a transition model

that takes regular physical states into account, it actually takes sets of states into account.

So this is a little bit tricky what we want to do here. And so the problem is that not every action

is applicable to all the states, the physical states in the belief state. So we have a

transition model in the belief state. So we have to do something. And essentially we have two things

we can do. We can either take all the transitions here that we have, and remember those are sets.

We can do the union over those, and that's essentially if the actions are harmless,

all the possible things that can come out of an action and collect it up. If actions might do

harm, then we can be much more restrictive and kind of have a lower bound of what the effect

would be. This is kind of the safe bet. And you can see that there's in the prediction here,

we have in the belief space, we really have the, we may have some

slippage. We may have some incorrectness coming in. And of course, the bigger the incorrectness

is in here, the more pronounced we have. But in many cases, it doesn't really matter that

well, and we can have senseless plans that actually work in real world. For instance, here,

depends on the environment we're in. So essentially what we're doing is we're taking

physical problems and we're lifting them to the belief state level. Let's look at that in

terms of picture. You may remember that we have this state space for the physical world of the