UNCERTAINTY INSENSING (AND ACTION)

AGENDA

Planning with belief states Nondeterministic sensing uncertainty Probabilistic sensing uncertainty

ACTION UNCERTAINTY

Each action representation is of the form:

Action:a(s) -> {s1,…,sr}

where each si, i = 1, ..., r describes one possible effect of the action in a state s

Right

AND/OR TREE

action nodes(world “decision” nodes)

state nodes(agent decision nodes)

Suck(R1)

loop

Right

AND/OR TREE

Suck(R1)

loopLeft Suck(R2)

goal goalloop

Suck(R2)

Right Suck(R1)

loop loop

OR SUB-TREE OR sub-tree : AND-OR tree :: path : classical search tree For each state node, only one

child is included For each action node, all

children are included It forms a part of a

potential solution ifnone of its nodes is closed

A solution is an OR sub-tree in which all leaves are goal states

BELIEF STATE

A belief state is the set of all states that an agent think are possible at any given time or at any stage of planning a course of actions, e.g.:

To plan a course of actions, the agent searches a space of belief states, instead of a space of states

SENSOR MODEL (DEFINITION #1) State space S The sensor model is a function

SENSE: S 2S

that maps each state s S to a belief state (the set of all states that the agent would think possible if it were actually observing state s)

Example: Assume our vacuum robot can perfectly sense the room it is in and if there is dust in it. But it can’t sense if there is dust in the other room

SENSE( ) =

SENSE( ) =

SENSOR MODEL (DEFINITION #2) State space S, percept space P The sensor model is a function

SENSE: S Pthat maps each state s S to a percept (the percept that the agent would obtain if actually observing state s)

We can then define the set of states consistent with the observation P

CONSISTENT(P) = { s if SENSE(s)=P }

SENSE( ) =

CONSISTENT( ) =

?

?

VACUUM ROBOT ACTION AND SENSOR MODEL

RightAppl if sIn(R1){s1 = s - In(R1) + In(R2), s2 = s}

[Right does either the right thing, or nothing]

LeftAppl if sIn(R2){s1 = s - In(R2) + In(R1),

s2 = s - In(R2) + In(R1) - Clean(R2)}[Left always move the robot to R1, but it may occasionally deposit dust in R2]

Suck(r)Appl sIn(r){s1 = s+Clean(r)}

[Suck always does the right thing]

• The robot perfectly senses the room it is in and whether there is dust in it

• But it can’t sense if there is dust in the other room

State s : any logical conjunction of In(R1), In(R2), Clean(R1), Clean (R2)(notation: + adds an attribute, - removes an attribute)

TRANSITION BETWEEN BELIEF STATES Suppose the robot is initially in state:

After sensing this state, its belief state is:

Just after executing Left, its belief state will be:

After sensing the new state, its belief state will be:

or if there is no dust in R1 if there is dust in R1

TRANSITION BETWEEN BELIEF STATES Suppose the robot is initially in state:

After sensing this state, its belief state is:

Just after executing Left, its belief state will be:

After sensing the new state, its belief state will be:

or if there is no dust in R1 if there is dust in R1

Left

Clean(R1) Clean(R1)

TRANSITION BETWEEN BELIEF STATES

How do you propagate the action/sensing operation to obtain the successors of a belief state?

Left

Clean(R1) Clean(R1)

COMPUTING THE TRANSITION BETWEEN BELIEF STATES

Given an action A, and a belief state S = {s1,…,sn}

Result of applying action, without sensing: Take the union of all SUCC(si,A) for i=1,…,n This gives us a pre-sensing belief state S’

Possible percepts resulting from sensing: {SENSE(si’) for si’ in S’} (using SENSE definition

#2) This gives us a percept set P

Possible states both in S’ AND consistent with each possible percept pj in P: Sj = {si | SENSE(si’)=pj for si’ in S’}

i.e.,Sj = CONSISTENT(pj) ∩ S’

AND/OR TREE OF BELIEF STATES

Left

Suck

Suck

goal

A goal belief state is one in which all states are goal states

An action is applicable to a belief state B if its precondition is achieved in all states in B

Right

loop goal

AND/OR TREE OF BELIEF STATES

Left

Suck

Right

loopSuck

goal

Right

loop goal

AND/OR TREE OF BELIEF STATES

Left

Suck

Right

Suck

goal

Right

goal

BELIEF STATE REPRESENTATION

Solution #1: Represent the set of states explicitly

Under the closed world assumption, if states are described with n propositions, there are O(2n) states

The number of belief states is

A belief state may contain O(2n) states

This can be hugely expensive

n2O(2 )

BELIEF STATE REPRESENTATION

Solution #2: Represent only what is known For example, if the vacuum robot knows that

it is in R1 (so, not in R2) and R2 is clean, then the representation is K(In(R1)) K(In(R2)) K(Clean(R2))where K stands for “Knows that ...”

How many belief states can be represented? Only 3n, instead of

n2O(2 )

SUCCESSOR OF A BELIEF STATE THROUGH AN ACTION

LeftAppl if s In(R2){s1 = s - In(R2) + In(R1),

s2 = s - In(R2) + In(R1) - Clean(R2)}

K(In(R2))K(In(R1)) K(Clean(R2))

s1 K(In(R2))K(In(R1))K(Clean(R2))

s2 K(In(R2))K(In(R1))K(Clean(R2))

K(In(R2))K(In(R1))

An action does not dependon the agent’s belief state K does not appear inthe action description(different from R&N, p. 440)

SENSORY ACTIONS So far, we have assumed a unique sensory operation

automatically performed after executing of each action of a plan

But an agent may have several sensors, each having some cost (e.g., time) to use

In certain situations, the agent may like better to avoid the cost of using a sensor, even if using the sensor could reduce uncertainty

This leads to introducing specific sensory actions, each with its own representation active sensing

Like with other actions, the agent chooses which sensory actions it want to execute and when

EXAMPLE

Check-Dust(r):Appl if s In(r){when Clean(r)

b’ = b - K(Clean(r))

+ K(Clean(r)) } {when Clean(r) b’ = b - K(Clean(r))

+ K(Clean(r))}

K(In(R1))K(In(R2)) K(Clean(R2))

K(In(R1))K(In(R2)) K(Clean(R2))K(Clean(R1))

K(In(R1))K(In(R2)) K(Clean(R2))K(Clean(R1))

Check-Dust(R1):

A sensory action maps a stateinto a belief stateIts precondition is about the stateIts effects are on the belief state

INTRUDER FINDING PROBLEM

A moving intruder is hiding in a 2-D workspace The robot must “sweep” the workspace to find the

intruder Both the robot and the intruder are points

robot’s visibilityregion

hidingregion 1

cleared region

2 3

4 5 6

robot

DOES A SOLUTION ALWAYS EXIST?

Easy to test: “Hole” in the workspace

Hard to test:No “hole” in the workspace

No !

INFORMATION STATE

Example of an information state = (x,y,a=1,b=1,c=0) An initial state is of the form (x,y,1, 1, ..., 1) A goal state is any state of the form (x,y,0,0, ..., 0)

(x,y)

a = 0 or 1

c = 0 or 1

b = 0 or 1

0 cleared region1 hidding region

CRITICAL LINE

a=0b=1

a=0b=1

Information state is unchanged

a=0b=0

Critical line

A

B C D

E

CRITICALITY-BASED DISCRETIZATION

Each of the regions A, B, C, D, and E consists of “equivalent” positions of the robot,so it’s sufficient to consider a single positionper region

CRITICALITY-BASED DISCRETIZATION

A

B C D

E

(C, 1, 1)

(D, 1)(B, 1)

CRITICALITY-BASED DISCRETIZATION

A

B C D

E

(C, 1, 1)

(D, 1)(B, 1)

(E, 1)(C, 1, 0)

CRITICALITY-BASED DISCRETIZATION

A

B C D

E

(C, 1, 1)

(D, 1)(B, 1)

(E, 1)(C, 1, 0)

(B, 0) (D, 1)

CRITICALITY-BASED DISCRETIZATION

A

C D

E

(C, 1, 1)

(D, 1)(B, 1)

(E, 1)(C, 1, 0)

(B, 0) (D, 1)Much smaller search tree than with grid-based discretization !

B

SENSORLESS PLANNING

PLANNING WITH PROBABILISTIC UNCERTAINTY IN SENSING

No motion

Perpendicular motion

PARTIALLY OBSERVABLE MDPS

Consider the MDP model with states sS, actions aA Reward R(s) Transition model P(s’|s,a) Discount factor g

With sensing uncertainty, initial belief state is a probability distributions over state: b(s)b(si) 0 for all siS, i b(si) = 1

Observations are generated according to a sensor model Observation space oO Sensor model P(o|s)

Resulting problem is a Partially Observable Markov Decision Process (POMDP)

POMDP UTILITY FUNCTION

A policy p(b) is defined as a map from belief states to actions

Expected discounted reward with policy p:

Up(b) = E[t gt R(St)]

where St is the random variable indicating the state at time t

P(S0=s) = b0(s)

P(S1=s) = ?

POMDP UTILITY FUNCTION

A policy p(b) is defined as a map from belief states to actions

Expected discounted reward with policy p:

Up(b) = E[t gt R(St)]

where St is the random variable indicating the state at time t

P(S0=s) = b0(s)

P(S1=s) = P(s|p0(b),b0) = s’ P(s|s’,p(b0)) P(S0=s’) = s’ P(s|s’,p(b0)) b0(s’)

POMDP UTILITY FUNCTION

A policy p(b) is defined as a map from belief states to actions

Expected discounted reward with policy p:

Up(b) = E[t gt R(St)]

where St is the random variable indicating the state at time t

P(S0=s) = b0(s)

P(S1=s) = s’ P(s|s’,p(b)) b0(s’)

P(S2=s) = ?

POMDP UTILITY FUNCTION

A policy p(b) is defined as a map from belief states to actions

Expected discounted reward with policy p:

Up(b) = E[t gt R(St)]

where St is the random variable indicating the state at time t

P(S0=s) = b0(s)

P(S1=s) = s’ P(s|s’,p(b)) b0(s’) What belief states could the robot be after 1

step?

b0

Predictb1(s)=s’ P(s|s’,(b0)) b0(s’)

Choose action p(b0)

b1

b0

oA oB oC oD

Predictb1(s)=s’ P(s|s’,(b0)) b0(s’)

Choose action p(b0)

b1

Receiveobservation

b0

P(oA|b1)

Predictb1(s)=s’ P(s|s’,(b0)) b0(s’)

Choose action p(b0)

b1

Receiveobservation

b2

P(oB|b1) P(oC|b1) P(oD|b1)

b3 b4 b5

b0

Predictb1(s)=s’ P(s|s’,(b0)) b0(s’)

Choose action p(b0)

b1

Update belief b2 b3 b4 b5

b2(s) = P(s|b1,oA)b3(s) = P(s|b1,oB)

b4(s) = P(s|b1,oC)

P(oA|b1) P(oB|b1) P(oC|b1) P(oD|b1)

b5(s) = P(s|b1,oD)

Receiveobservation

b0

Predictb1(s)=s’ P(s|s’,(b0)) b0(s’)

Choose action p(b0)

b1

Update belief b2 b3 b4 b5

b2(s) = P(s|b1,oA)b3(s) = P(s|b1,oB)

b4(s) = P(s|b1,oC)

P(oA|b1) P(oB|b1) P(oC|b1) P(oD|b1)

b5(s) = P(s|b1,oD)

Receiveobservation

P(o|b) = sP(o|s)b(s)

P(s|b,o) = P(o|s)P(s|b)/P(o|b)

= 1/Z P(o|s) b(s)

BELIEF-SPACE SEARCH TREE Each belief node has |A| action node successors Each action node has |O| belief successors Each (action,observation) pair (a,o) requires

predict/update step similar to HMMs

Matrix/vector formulation: b(s): a vector b of length |S| P(s’|s,a): a set of |S|x|S| matrices Ta

P(ok|s): a vector ok of length |S|

b’ = Tab (predict)

P(ok|b’) = okT b’ (probability of observation)

bk = diag(ok) b’ / (okT b’) (update)

Denote this operation as ba,o

RECEDING HORIZON SEARCH

Expand belief-space search tree to some depth h

Use an evaluation function on leaf beliefs to estimate utilities

For internal nodes, back up estimated utilities:U(b) = E[R(s)|b] + g maxaA oO P(o|ba)U(ba,o)

QMDP EVALUATION FUNCTION

One possible evaluation function is to compute the expectation of the underlying MDP value function over the leaf belief states f(b) = s UMDP(s) b(s)

“Averaging over clairvoyance” Assumes the problem becomes instantly fully

observable Is optimistic: U(b) f(b) Approaches POMDP value function as state and

sensing uncertainty decreases In extreme h=1 case, this is called the QMDP

policy

QMDP POLICY (LITTMAN, CASSANDRA, KAELBLING 1995)

WORST-CASE COMPLEXITY

Infinite-horizon undiscounted POMDPs are undecideable (reduction to halting problem)

Exact solution to infinite-horizon discounted POMDPs are intractable even for low |S|

Finite horizon: O(|S|2 |A|h |O|h) Receding horizon approximation: one-step

regret is O(gh) Approximate solution: becoming tractable for

|S| in millions a-vector point-based techniques Monte Carlo tree search …Beyond scope of course…

NEXT TIME

Is it possible to learn how to make good decisions just by interacting with the environment?

Reinforcement learning R&N 21.1-2

DUE TODAY

HW6 Midterm project report

HW7 available