Knowledge Repn. & Reasoning Lec #26: Filtering with Logic

Knowledge Repn. & ReasoningLec #26: Filtering with Logic

UIUC CS 498: Section EA

Professor: Eyal AmirFall Semester 2004

Last Time

• Dynamic Bayes Nets– Forward-backward algorithm– Filtering

• Approximate inference via factoring and sampling

Filtering Stochastic Processes

• Dynamic Bayes Nets (DBNs): factored representation

s1 s4s3s2 s5

s1 s4s3s2 s5

s1 s4s3s2 s5

s1 s4s3s2 s5

Filtering Stochastic Processes

• Dynamic Bayes Nets (DBNs): factored representation

s4s3s2 s5

s4s3s2 s5

s4s3s2 s5

s4s3s2 s5

Filtering Stochastic Processes

• Dynamic Bayes Nets (DBNs): factored representation

s4s3 s5

s4s3 s5

s4s3 s5

s4s3 s5

Filtering Stochastic Processes

• Dynamic Bayes Nets (DBNs): factored representation

s4 s5

s4 s5

s4 s5

s4 s5O(2O(2nn) space) spaceO(2O(22n2n) time) time

Filtering Stochastic Processes

• Dynamic Bayes Nets (DBNs): factored representation: O(2n) space, O(22n) time

• Kalman Filter: Gaussian belief state and linear transition model

s1 s4s3s2 s5

s1 s4s3s2 s5

s1 s4s3s2 s5

Filtering Stochastic Processes

• Dynamic Bayes Nets (DBNs): factored representation: O(2n) space, O(22n) time

• Kalman Filter: Gaussian belief state and linear transition model

s4 s5

s4 s5

s4 s5

O(nO(n22) space) spaceO(nO(n33) time) time

Complexity Results

• Filtering for deterministic systems is NP-hard when the initial state is not fully known [Liberatore ’97]

• [Amir&Russell’03][Amir&Russell’03]: Every representation: Every representation of of

belief states belief states grows exponentiallygrows exponentially for for

some some deterministicdeterministic systems systems

Today

• Tracking and filtering logical knowledge

• Foundations for efficient filtering

• Compact representation indefinitely

• Possible projects

Logical Filtering• Belief state = logical formula

Logical Filtering• Belief state = logical formula

• Observations = logical formulae

Logical Filtering• Belief state = logical formula

• Observations = logical formulae

• Actions = effect rules– e.g., “fetch(X,Y) causes has(X) if in(X,Y)”

Logical Filtering• Belief state = logical formula

• Observations = logical formulae

• Actions = effect rules– e.g., “fetch(X,Y) causes has(X) if in(X,Y)”

• Actions may be nondeterministic

• Partial observations

Example: A Cleaning Robot• Initial Knowledge:

?• Apply action

fetch(broom,closet)• Resulting knowledge

in(broom,closet)• Reason:

– If initially in(broom,closet), then still in(broom,closet)– If initially in(broom,closet), then now in(broom,closet)

Filtering with Possible Worlds

Problem: n world features Problem: n world features 2 2nn states states

Filtering Possible Worlds

• Initially we are in {s1,…,sk}

• Action a

• Filter[a]({s1,…,sk})=

{s’ | R(s1,a,s’) or … or R(sk,a,s’)}

• observing o

• Filter[o]({s1’,…,su’})=

{s1’,…,su’} {s | o holds in s}

Filtering with Logical Formulae

• Action-Definition(a)t,t+1

(Precondi(a)t Effecti(a)t+1)

i Frame-Axioms(a)

Filtering with Logical Formulae

• Belief state S represented by • Actions: Filter[a](t) logical resultst+1 of

t Action-Definition(a)t,t+1

AAtt v B v Btt BBttCCt+1t+1 AAt+1t+1 v (B v (Bt+1 t+1 CCt+1t+1))

Filtering with Logical Formulae

• Belief state S represented by • Actions: Filter[a](t) logical resultst+1 of

t Action-Definition(a)t,t+1

• Observations: Filter[o]() = o• t+1 = Filter[o](Filter[a](t))

AAtt v B v Btt BBttCCt+1t+1 AAt+1t+1 v (B v (Bt+1 t+1 CCt+1t+1))

Filtering with Logical Formulae

• Belief state S represented by • Actions: Filter[a](t) logical resultst+1 of

t Action-Definition(a)t,t+1

• Observations: Filter[o]() = o• t+1 = Filter[o](Filter[a](t))

• Theorem: formula filtering implements possible-worlds semantics

Contents

• Tracking and filtering logical knowledge

• Foundations for efficient filtering

• Compact representation indefinitely

• Possible projects

Distribution Properties

Filter[a](Filter[a](Filter[a]()

• Filtering a DNF belief state by factoring

Distribution Properties

Filter[a](Filter[a](Filter[a]()

Filter[a](Filter[a](Filter[a]()

Filter[a](Filter[a](Filter[a](TRUE)

• Filtering a DNF belief state by factoring

Distribution for Some Actions

Filter[a](Filter[a](Filter[a]() Filter[a](Filter[a](Filter[a]() Filter[a](Filter[a](Filter[a](TRUE)

• Filter literals in the belief-state formula separately, and combine the results

• STRIPS ActionsSTRIPS Actions• 1:1 Actions1:1 Actions

Actions that map states 1:1• Examples:

– flip(light) but not turn-on(light)– increase(speed,+10) but not set(speed,50)– pickUp(X,Y) but not

pickUp(X)

• Most actions are 1:1 in proper formulation

Actions that map states 1:1• Reason for distribution over

Filter[a](Filter[a](Filter[a](Filter[a](Filter[a](Filter[a]())

Filter[a](Filter[a](Filter[a](Filter[a](Filter[a](Filter[a]())

1:11:1

Non-1:1Non-1:1

STRIPS Actions• Possibly nondeterministic effects

• No conditions on effects

• Example: turn-on(light)

• Used extensively in planning

Distribution for Some Actions

Filter[a](Filter[a](Filter[a]() Filter[a](Filter[a](Filter[a]() Filter[a](Filter[a](Filter[a](TRUE)

• Filter literals in the belief-state formula separately, and combine the results

• STRIPS ActionsSTRIPS Actions• 1:1 Actions1:1 Actions

Example: Filtering a Literal• Initial knowledge:

in(broom,closet)

• Apply fetch(broom,closet)

Preconds: in(broom,closet) locked(closet)

Effects: has(broom) in(broom,closet)

• Resulting knowledge:

has(broom) in(broom,closet)

locked(closet)

Example: Filtering a Formula• Initial knowledge:

in(broom,closet) locked(closet)

• Apply fetch(broom,closet)

Preconds: in(broom,closet) locked(closet)

Effects: has(broom) in(broom,closet)

• Resulting knowledge:

has(broom) in(broom,closet) locked(closet)

Filtering a Single Literal• Closed-form solution:

Filter[a](literal) = (Eff1 Effu) B(a) literal╞ Pre1 Preu

– a has effect rules (and frame rules) a causes Effi if Prei

• Eff1 Effu - effects of action a• Pre1 Preu - preconditions of action a

– Roughly, B(a) Filter[a](TRUE)

Filtering a Literal

• Filter[a](literal) = (Eff1 Effu) B(a) literal╞ Pre1 Preu

• Belief state (: locked(closet)• Action (a): fetch(broom,closet) with

“fetch(X,Y) causes has(X) in(X,Y) if locked(Y) in(X,Y)”

• Belief state after a:Filter[a](locked(closet)

locked(closet) in(broom,closet)

Filtering a Literal

• Filter[a](literal) = (Eff1 Effu) B(a) literal╞ Pre1 Preu

• Action (a): fetch(broom,closet) with “fetch(X,Y) causes has(X) in(X,Y)

if locked(Y) in(X,Y)”• Belief state after a:Filter[a](locked(closet)locked(closet) in(broom,closet)• Reason: locked(closet)╞ locked(closet) in(broom,closet)) in(broom,closet)

Algorithm for Permutation Actions

• Belief state (: locked(closet) (in(broom,closet) in(broom,shed))

• Action (a): fetch(broom,closet) with “fetch(X,Y) causes

has(X) in(X,Y) if locked(Y) in(X,Y)”

• Resulting belief state:

Filter[a](Filter[a](locked(closet)

Filter[a](in(broom,closet)Filter[a](in(broom,shed)))

• Filter[a](locked(closet)locked(closet) in(broom,closet)

Algorithm for Permutation Actions• Belief state (:

locked(closet) (in(broom,closet) in(broom,shed))

• Filter[a](locked(closet)locked(closet) in(broom,closet)

• Filter[a](in(broom,closet)locked(closet)

in(broom,closet) has(broom)) in(broom,closet)

• Filter[a](in(broom,shed)in(broom,shed)

• Filter[a](= locked(closet) in(broom,closet)

has(broom) in(broom,shed))

Summary: Efficient Update

• Fast exact update with any observation formulae, if one of the following:– STRIPS action (possibly nondeterministic)– Action is a 1:1 mapping between states– Belief states include all their prime implicates

Talk Outline

• Tracking and filtering knowledge

• Tractability results

• Compact representation over time

• Discussion & Future work

Tractability and Representation Size

• Theorem1: Every propositional repn. of the belief state grows exponentially for some systems, even when initial belief state is compactly represented (follows from [Boppana & Sipser ’90])

1 Rough statement. Complete one in [A. & Russell ’03].

Example: A Cleaning Robot• Initial Knowledge:

in(broom,closet) in(broom,shed) • Apply action fetch(broom,closet)• Resulting knowledge (has(broom) locked(closet) in(broom,closet)) (has(broom) locked(closet) in(broom,closet)) (has(broom) in(broom,shed))

• Reason for space explosion: uncertainty of action’s success and preconditions applied

Small formulaSmall formula Big formulaBig formula

Compact & Tractable Cases• Compact belief state representation

– STRIPS actions with belief state in k-CNF– 1:1 actions with belief state in k-CNF …

• Observations in 2-CNF

• Theorem: Filtering with STRIPS actions– k-CNF k-CNF – time ~ O(|| 2#rules(a))

• Corollary: Filtering with STRIPS actions keeps belief state in O(nk) size (k fixed).

STRIPS-Filter: ExperimentsAverage time per step

Filtering step

Filt

er ti

me

(m.s

ec)

~270 features

~240 features

~210 features

~180 features~150 features

STRIPS-Filter: ExperimentsAverage space per step

Filtering step

Filt

er s

pace

(lit

eral

s) ~210 features

~185 features~160 features~135 features~110 features

Intuition for More Results

• Filtering with deterministic action a is equivalent to filtering with actions a1 (“a succeeds”) or a2 (“a fails”) successfully, – a1,a2 STRIPS with known success/failure

Filter[a](φ) Filter[a1](φ) v Filter[a2](φ)

• STRIPS with known success/failure:

Filter[a](l1lu) = (l1lu) B(a) or B(a)

Recent Results: (unpublished) #1

• Compact representation indefinitely for STRIPS, if failure leaves features unchanged, and effects are 2-clauses

a causes  (f v g) & (g v -h)  if  x & y

• Starting from belief state with r clauses we get at most max(r,n) clauses indefinitely, if effects are conjunction of at most two clauses

Recent Results: (unpublished) #2

• Compact representation indefinitely for STRIPS, if failure has nondeterministic effect on affected features

a causes  f & g  if  x & y

a causes (f v -f) & (g v -g) if (-x v -y)

• Belief state in k-CNF maintained indefinitely, if effects in k1-CNF, preconditions in k2-DNF, k=k1+k2

Related Work

• Stochastic filtering– [Kalman ’60], [Doucet et-al. ’00], [Dean &

Kanazawa ’88], [Boyen & Koller ’98], …

• Action theories and semantics– [Gelfond & Lifschitz ’97], [Baral & Son ’01],

[Doherty et-al. ’98], …

• Computation of progression– [Winslett ’90], [del Val ’92], [Lin & Reiter ’97],

[Simon & del Val ’01], …

Possible Projects

• More families of actions/observations– Stochastic conditions on observations– Different data structures (BDDs? Horn?)

• Compact and efficient stochastic filtering

• Relational / first-order filtering

• Dynamic observation models, filtering in expanding worlds

• Logical Filtering of numerical variables

More Projects

• Filtering for Kriegspiel (partially observable chess)

• Autonomous exploration of uncharted domains

• Smart agents in rich environments

THE END

Example: Explosion of Space• Initial Knowledge:

in(broom,closet) in(broom,shed)

• Apply action fetch(broom,closet)• Resulting knowledge (has(broom) locked(closet) in(broom,closet)) (has(broom) locked(closet) in(broom,closet)) (has(broom) in(broom,shed))

Tractability Problem

• Formula filtering is NP-hard in general

Actions: Filter[a](t)

Cn(t (Precond(a)t Effect(a)t+1)

Frame-Axioms(a))

Cn(•) = Logical consequences of •

• Specific cases?

• Approximation?

Example: A Cleaning Robot• Initial Knowledge:

?• Apply action

fetch(broom,closet)• Resulting knowledge

in(broom,closet)• Reason:

– If initially in(broom,closet), then still in(broom,closet)– If initially in(broom,closet), then now in(broom,closet)

Filtering Beliefs

• Filtering: Update knowledge of the world after actions and observations

• Stochastic filtering examples:– Dynamic Bayes Nets (DBNs): factored

representation– Kalman Filter: Gaussian belief state and linear

transition model

s1 s4s3s2World state

Agents Acting in The World

• Agents in partially observable domains– Cognitive, medical assistants– Cleaning, gardening robots– Space robots (exploration, repair, assist)– Game-playing/companion agents

• Knowledgeable agents– Use knowledge to decide on actions– Update knowledge about the world

Example: A Cleaning Robot

• Decides to clean the current room

• Knows the broom is in the closet

• Fetches the broom from the closet

• Now knows that the broom is in its hand and not in the closet

Filtering a Single Literal• Closed-form solution:

Filter[a](literal) = (Eff1 Effu) B(a) literal╞ Pre1 Preu

– a has effect rules (and frame rules) a causes Effi if Prei

• Eff1 Effu - effects of action a• Pre1 Preu - preconditions of action a

– Roughly, B(a) Filter[a](TRUE)

Permutation Actions• Actions that permute the states:

– flip(light) but not turn-on(light)– increase(speed,+10) but not set(speed,50)– pickUp(X,Y) but not

pickUp(X)

Results: Tractable Cases

• Filtering a single literal

• Permutation actions

• STRIPS actions

• Prime-implicate representation of belief state

Filtering Logical Formulae:STRIPS-Filter

• If every executed action was possible to execute (or we observed an error), and actions do not have conditional effects (but may have nondeterministic effects), and the belief state representation in PI-CNF, thenFilter[a]( Ci ) = Filter[a](Ci)

Summary: Tractable Cases

• Fast approximate update: propositional belief state represented in NNF,CNF,DNF

• Fast exact update (if one of the following):– Action is a 1:1 mapping between states– STRIPS action (unconditional, nondeterministic

effects of actions; observations distinguish success from failure of action)

– Belief states include all their prime implicates– Any observations

Sources of Difficulty for Compact Representation

• For action a with effect rule

a causes Eff if Pre– We always know after the action that

Eff Pre– If we know (Pre p), then after the action we

know

Eff p

How Is the State Kept Compact?

• STRIPS (nondeterministic) actions– We always know that the precondition held (or we got

a signal that the action failed)– There are no conditional effects

• Permutation actions– We restrict the preconditions and effects, e.g.,

• All rules of the form a causes l1 if l2, or

• One of the preconditions is always satisfied, or …

• Observations (and obs. model): in 2-CNF

STRIPS-Filter: Experimental Results

[A. & Russell ’03]

Tractability and Representation Size

• Theorem1: Every propositional repn. of the belief state grows exponentially for some systems, even when initial belief state is compactly represented (follows from [Boppana & Sipser ’90])

• However, special cases can can be computed efficiently and represented compactly

• s in 2-CNF1 Rough statement. Complete one in [A. & Russell ’03].

STRIPS-Filter: Experimental Results

[A. & Russell ’03]

Applications

• Tractable filtering and tracking of the world in high-dimensional domains with many objects, locations and relationships

• Learn effects and preconditions of actions in partially-observable domains

• Autonomous exploration of uncharted domains

Related Work

• Stochastic filtering– [Kalman ’60], [Blackman & Popoli ’99],

[Doucet et-al. ’00], …

• Action theories and semantics– [Gelfond & Lifschitz ’97], [Baral & Son ’01],

[Doherty et-al. ’98], …

• Computation of progression– [Winslett ’90], [del Val ’92], [Lin & Reiter ’97],

[Simon & del Val ’01], …

THE END

Filtering STRIPS Actions

• STRIPS:– Action was executed or we observed an error,– No conditional effects, and– Possibly nondeterministic effects

Logical Filtering: Progress Outlook

• 18 months: relational filtering, learning actions in partially-observable domains

• 36 months: dynamic observation models, Horn belief states, filtering in expanding worlds, autonomous agents in games

• 54 months: first-order filtering, factored belief states, continuous time, autonomous exploration of uncharted domains

Today

1. Probabilistic graphical models

2. Treewidth methods:1. Variable elimination

2. Clique tree algorithm

3. Applications du jour: Sensor Networks

Contents

1. Probabilistic graphical models

2. Exact inference and treewidth:1. Variable elimination

2. Junction trees

3. Applications du jour: Sensor Networks

Application: Planning

• General-purpose planning problem:– Given:

• Domain features (fluents)• Action descriptions: effects, preconditions• Initial state• Goal condition

– Find:• Sequence of actions that is guaranteed to achieve

the goal starting from the initial state

Application: Planning with partitions

PartPlan Algorithm• Start with a tree-structured

partition graph• Identify goal partition• Direct edges toward goal• In each partition

– Generate all plans possible with depth d and width k

– Pass messages toward goal

Factored Planning: Analysis

• Planner is sound and complete

• Running time for finding plans of width w with m partitions of treewidth k is O(mw22w+2k)

• Factoring can be done in polynomial time

• Goal can be distributed over partitions by adding at most 2 features per partition

Next Time

• Probabilistic Graphical Models:– Directed models: Bayesian Networks– Undirected models: Markov Fields

• Requires prior knowledge of:– Treewidth and graph algorithms– Probability theory