Knowledge Representation and Reasoning Introduction and …ssll.cecs.anu.edu.au/files/slides/pagnucco.pdf · 2009-01-20 · Knowledge Representation and Reasoning Introduction and

Knowledge Representation andReasoning

Introduction and Motivation

Maurice PagnuccoSchool of Computer Sc. & Eng.University of New South Wales

NSW 2052, AUSTRALIA

[email protected]

NB: Many examples from: R. Brachman and H. J. Levesque,Knowledge Rep-resentation, Morgan Kaufmann, 2004.

LSS 2009, Thursday 5 February, 2009 KRR: Introduction 1

Knowledge Representation and Reasoning

� What is (the nature of) knowledge?

� How can we represent what we know?

� How can we use the representation to infer new knowledge?

� Reference: Ronald J. Brachman and Hector J. Levesque,KnowledgeRepresentation and Reasoning, Morgan Kaufmann Publishers, SanFrancisco, CA,, 2004.ISBN: 1-55860-932-6



� Foundations1 Introduction to KRR

2 Nonmonotonic Reasoning

3 Reasoning about Action

4 Belief Revision


Outline

� What is knowledge?

� Representation and Reasoning

� Why Knowledge?

� Why Representation?

� Why Knowledge Representation?

� Advantages of Knowledge Representation

� Forms of Knowledge Representation


What is Knowledge

� “John knows that . . . ”

◮ the ‘. . . ’ are replaced by aproposition

◮ proposition can be true/false

� Other types of knowledge:

◮ know how, know who, know what, know when, . . .

◮ sensorimotor: riding a bike

◮ affective: deep understanding

� Belief is similar but may not necessarily be true

� Note: we do not distinguish between knowledge and belief

� Main idea: take world to be one way and not another


Representation

� Symbols stand for things in the world

−→ first aid

−→ restaurant

“John” −→ John

“John loves Mary” −→ the proposition that John loves Mary

� Knowledge representation

◮ symbolic encoding of believed propositions


Reasoning

� Manipulation of symbols that encode propositions to producerepresentations of new propositions

� Analogy: arithmetic“1011” + “10” → “1101”

⇓ ⇓ ⇓

eleven two thirteen

“John is Mary’s father” −→ “John is an adult male”

⇓ ⇓


Why Knowledge?

� For systems that are reasonably complex it is often useful todescribethat system in terms ofbeliefs, goals, fears and intentions

◮ e.g., chess-playing program“because program believed that its queen was in danger but stillwanted to control centre of chess board”

◮ sometimes more useful than describing actual technique:“because evaluation using minimax procedure returned value of 7for this position”

� Intentional stance (Daniel Dennet)

� Howeveris KR just a convenient way of describing complex systems?

◮ anthropomorphising can be inappropriate

◮ . . . and misleading


Why Representation?

� Intentional stance says nothing about what is and is not representedsymbolically

� Knowledge Representation Hypothesis (Brian Smith)Any mechanically embodied intelligent process will becomprised of structural ingredients that a) we as externalobservers naturally take to represent a propositional accountof the knowledge that the overall process exhibits, and b)independent of such external semantic attribution, play aformal but causal and essential role in engendering thebehaviour that manifests that knowledge

� In other words, existence of structures that◮ can be interpreted propositionally

◮ determine how the system behaves� Knowledge-based system:a system designed in accordance with

these principles


Example

� Contrast:

printColour(snow) :- !, write("It’s white.").

printColour(grass) :- !, write("It’s green.").

printColour(sky) :- !, write("It’s yellow.").

printColour(X) :- write("Beats me.").

� with:

printColour(X) :- colour(X,Y), !,

write("It’s "), write(Y), write(".").

printColour(X) :- write("Beats me.").

colour(snow,white).

colour(sky,yellow).

colour(X,Y) :- madeof(X,Z), colour(Z,Y).

madeof(grass,vegetation).

colour(vegetation,green).


Example

� Both examples can be described intentionally

� Second example has a separate collection of symbolic structures (likethe KR hypothesis)

� It is a knowledge-based system


KR in AI

� Much of AI is concerned with building systems that are knowledge-based◮ natural language understanding

◮ planning and scheduling

◮ diagnosis

◮ “expert systems”

� some to a certain extent◮ game-playing

◮ vision

� some to a lesser extent◮ speech recognition

◮ motor control


Why KR?

� Why not “compile out” knowledge into specialised procedures?◮ distribute KB to procedures that need it

◮ usually achieves better performance

� Don’t think. Just do it!◮ riding a bike

◮ driving a car

◮ playing soccer

◮ playing chess?

◮ doing math?

◮ staying alive??

� Skills (Hubert Dreyfus)◮ novices think; experts react


Advantages of KR

� Knowledge-based system most suitable for open-ended tasks

� Good for

◮ explanation and justification

◮ “informability”: debugging the KB

◮ “extensibility”: new relations

◮ new applications

� Hallmark of a knowledge-based system: ability to be told facts aboutworld and adjust behaviour accordingly

� “Cognitive penetrability” (Zenon Plylyshyn)actions conditioned by what is currently believed (e.g., don’t leavethe room when alarm sounds if you believe alarm is being tested)


Advantages of Reasoning

� Want knowledge to affect action

◮ not do actionA if sentenceP is in KB

◮ but do actionA if world believed in satisifesP

� Difference

◮ P may not be explicitly represented

◮ need to apply what is known to particulars of given situation

� Patientx is allergic to medicationmAnybody allergic to medicationm is also allergic tom′

Is it ok to prescribem′ for x?

� Usually need more than just DB-style retrieval of facts in KB


Forms of KRR

� Many forms of knowledge representation and reasoning have beenproposed and studied

� A large number of these are logic-based or closely related tologic

� In the following slides we will briefly list some approaches


Inheritance Networks

Clyde

gray

elephant

royal elephant

fat royal elephant


Frames

(Trip18

<IS-A Trip>

<firstStep TravelStep18-1>)

(TravelStep18-1

<IS-A TravelStep>

<beginDate 12/21/98>

<endDate 12/21/98>

<means>

<origin>

<destination Toronto>

<nextStep>

<previousStep>

<departureTime>

<arrivalTime>)


Frames

(Trip <totalCost

[IF-NEEDED

{let x $\leftarrow$ SELF.firstStep;

let result $\leftarrow$ 0;

repeat

{if x.nextStep then

{result $\leftarrow$ result + x.cost + x.destinationLodgingStay.c

x $\leftarrow$ x.nextStep}

else return result+x.cost}}]>)


Uncertainty

� Probabilistic vs. possibilistic

� Bayesian networks/belief networks

BurglaryP(Burglary)0.001

EarthquakeP(Earthquake)0.002

AlarmBurglaryTrueTrueFalseFalse

EarthquakeTrueFalseTrueFlase

P(Alarm)0.950.940.290.001

JohnCalls MaryCallsAlarmTrueFalse

P(JohnCalls)0.900.05

AlarmTrueFalse

P(MaryCalls)0.700.01

� Dempster-Shafer theory of evidence

� Fuzzy logic


Nonmonotonic Reasoning

� Classical logics obey property ofmonotonicity

If ∆ ⊆ Γ, thenCn(∆)⊆Cn(Γ)

� Nonmonotonic logics try to capture “commonsense” reasoning

� e.g., “birds usually fly”, “emus normally don’t fly”

� Next lecture will investigate some forms of nonmonotonic reasoning


Description Logic

� Object-centred representation and reasoning

� Concepts: types, categories

� Roles: properties, descriptions

RED-BORDEAUX-WINE↔

(AND WINE(FILLS color Red)(FILLS region Bordeaux)(FILLS sugar-content Dry))

� Form basis ofSemantic Web


Conclusion

� Tradeoff between expressiveness of knowledge representation andcomputational effort required to realise correct reasoning in thatrepresentation

� Less expressive KR often means more tractable reasoning

� Can also look to “approximative” inference

Outline Nonmonotonicity Closed World Assumption Predicate Completion Circumscription Default Logic Nonmonotonic Consequence

Knowledge Representation and ReasoningNonmonotonic Reasoning

Maurice Pagnucco

KRR Program, National ICT Australia andARC Centre of Excellence for Autonomous Systems

School of Computer Science and EngineeringThe University of New South Wales

Sydney, NSW, 2052

January 4, 2009

Maurice Pagnucco UNSW




Suppose you are told “Tweety is a bird”

What conclusions would you draw?

Now, consider being further informed that “Tweety is anemu”

What conclusions would you draw now? Do they differfrom the conclusions that you would draw without thisinformation? In what way(s)?

Nonmonotonic reasoning is an attempt to capture a form ofcommonsense reasoning




1 Nonmonotonicity

2 Closed World Assumption

3 Predicate Completion

4 Circumscription

5 Default Logic

6 Nonmonotonic ConsequenceKLM Systems





In classical logic the more facts (premises) we have, themore conclusions we can draw

This property is known as Monotonicity

If ∆ ⊆ Γ, then Cn(∆) ⊆ Cn(Γ)

(where Cn denotes classical consequence)

However, the previous example shows that we often do notreason in this manner

Might a nonmonotonic logic—one that does not satisfy theMonotonicity property—provide a more effective way ofreasoning?




Why Nonmonotonicity?

Problems with the classical approach to consequence

It is usually not possible to write down all we would like tosay about a domainInferences in classical logic simply make implicit knowledgeexplicit; we would also like to reason with tentativestatementsSometimes we would like to represent knowledge aboutsomething that is not entirely true or false; uncertainknowledge

Nonmonotonic reasoning is concerned with getting aroundthese shortcomings




Makinson’s Classification

Makinson has suggested the following classification ofnonmonotonic logics:

Additional background assumptions

Restricting the set of valuations

Additional rules

David Makinson, Bridges from Classical to NonmonotonicLogic, Texts in Computing, Volume 5, King’s CollegePublications, 2005.




Nonmonotonicity

Classical logic satisfies the following property

Monotonicity: If ∆ ⊆ Γ, then Cn(∆) ⊆ Cn(Γ)(equivalently, Γ ⊢ φ implies Γ ∪ ∆ ⊢ φ)

However, we often draw conclusions based on ‘what isnormally the case’ or ‘true by default’

More information can lead us to retract previousconclusionsWe shall adopt the following notation

⊢ classical consequence relation|∼ nonmonotonic consequence relation




Consequence Operation Cn

Other properties of consequence operation Cn:

Inclusion ∆ ⊆ Cn(∆)

Cumulative Transitivity ∆ ⊆ Γ ⊆ Cn(∆) implies Cn(Γ) ⊆ Cn(∆)

Compactness If φ ∈ Cn(∆) then there is a finite ∆′ ⊆ ∆ suchthat φ ∈ Cn(∆′)

Disjunction in the PremisesCn(∆ ∪ {a}) ∩ Cn(∆ ∪ {b}) ⊆ Cn(∆ ∪ {a ∨ b})

Note: ∆ ⊢ φ iff φ ∈ Cn(∆)alternatively: Cn(∆) = {φ : ∆ ⊢ φ}




Example

Suppose I tell you ‘Tweety is a bird’You might conclude ‘Tweety flies’I then tell you ‘Tweety is an emu’You conclude ‘Tweety does not fly’

bird(Tweety) |∼ flies(Tweety)bird(Tweety) ∧ emu(Tweety) |∼¬flies(Tweety)




The Closed World Assumption

A complete theory is one in which for every ground atom inthe language, either the atom or its negation appears in thetheoryThe closed world assumption (CWA) completes a base(non-closed) set of formulae by including the negation of aground atom whenever the atom does not follow from thebaseIn other words, if we have no evidence as to the truth of(ground atom) P, we assume that it is falseGiven a base set of formulae ∆ we first calculate theassumption set

¬P ∈ ∆asm iff for ground atom P, ∆ 6⊢ PCWA(∆) = Cn{∆ ∪ ∆asm}




Example

∆ = {P(a),P(b),P(a) → Q(a)}∆asm = {¬Q(b)}Theorem: The CWA applied to a consistent set of formulae ∆is inconsistent iff there are positive ground literals L1, . . . , Ln

such that ∆ |= L1 ∨ . . . ∨ Ln but ∆ 6|= Li for i = 1, . . . , n.

Note that in the example above we limited our attention tothe object constants that appeared in ∆ however thelanguage could contain other constants. This is known asthe Domain Closure Assumption (DCA)Another common assumption is the Unique-NamesAssumption (UNA).

If two ground terms can’t be proved equal,assume that they are not.




Predicate Completion

Idea: The only objects that satisfy a predicate are those thatmust

For example, suppose we have P(a). Can view this as∀x . x = a → P(x)

the if-half of a definition

Can add the only if part:∀x . P(x) → x = a

Giving:∀x . P(x) ↔ x = a





Definition: A clause is solitary in a predicate P ifwhenever the clause contains a postive instance of P, itcontains only one instance of P.

For example, Q(a) ∨ P(a) ∨ ¬P(b) is not solitary in PQ(a) ∨ R(a) ∨ P(b) is solitary in P

Completion of a predicate is only defined for sets ofclauses solitary in that predicate





Each clause can be written:∀y . Q1 ∧ . . . ∧ Qm → P(t) (P not contained in Qi )∀y . ∀x . (x = t) ∧ Q1 ∧ . . . ∧ Qm → P(x)∀x .(∀y . (x = t) ∧ Q1 ∧ . . . ∧ Qm → P(x)) (normal form ofclause)Doing this to every clause gives us a set of clauses of theform:∀x . E1 → P(x). . .∀x . En → P(x)Grouping these together we get:∀x . E1 ∨ . . . ∨ En → P(x)Completion becomes: ∀x . P(x) ↔ E1 ∨ . . . ∨ En

and we can add this to the original set of formulaeMaurice Pagnucco UNSW



Example

Suppose ∆ = {∀x . Emu(x) → Bird(x),Bird(Tweety),¬Emu(Tweety)}

We can write this as∀x . (Emu(x) ∨ x = Tweety) → Bird(x)

Predicate completion of P in ∆ becomes∆ ∪ {∀x . Bird(x) → Emu(x) ∨ x = Tweety}




Circumscription

Idea: Make extension of predicate as small as possible

Example:

∀x .Bird(x) ∧ ¬Ab(x) → Flies(x)Bird(Tweety), Bird(Sam), Tweety 6= Sam,¬Flies(Sam)

Want to be able to conclude Flies(Tweety) but¬Flies(Sam)

Accept interpretations where Ab predicate is as “small” aspossible

That is, we minimise abnormality




Circumscription

Given interpretations I1 = 〈D, I1〉, I2 = 〈D, I2〉, I1 ≤ I2 iff forevery predicate P ∈ P, I1[P] ⊆ I2[P].

Γ |=circ φ iff for every interpretation I such that I |= Γ, eitherI |= φ or there is a I′ < I and I′ |= Γ.

φ is true in all minimal models

Now consider∀x .Bird(x) ∧ ¬Ab(x) → Flies(x)∀x .Emu(x) → Bird(x) ∧ ¬Flies(x)Bird(Tweety)




Reiter’s Default Logic (1980)Add default rules of the form α:β

γ

“If α can be proven and consistent to assume β, thenconclude γ”

Often consider normal default rules α:ββ

Example: bird(x):flies(x)flies(x)

Default theory 〈D, W 〉D – set of defaults; W – set of facts

Extension of default theory contains as many defaultconclusions as possible and must be consistent (and isclosed under classical consequence Cn)Concluding whether formula φ follows from 〈D, W 〉

Sceptical inference: φ occurs in every extension of 〈D, W 〉Credulous inference: φ occurs in some extension of 〈D, W 〉




Examples

W = {}; D = { :p¬p} – no extensions

W = {p ∨ r}; D = {p:qq ,

r :qq } – one extension {p ∨ r}

W = {p ∨ q}; D = { :¬p¬p ,

:¬q¬q } – two extensions

{¬p, p ∨ q}, {¬q, p ∨ q}

W = {emu(Tweety), ∀x .emu(x) → bird(x)};D = {bird(x):flies(x)

flies(x) } – one extension

What if we add emu(x):¬flies(x)¬flies(x) ?

Poole (1988) achieves a similar effect (but not quite asgeneral) by changing the way the underlying logic is usedrather than introducing a new element into the syntax




Default Theories—Properties

Observation: Every normal default theory (default rules are allnormal) has an extensionObservation: If a normal default theory has severalextensions, they are mutually inconsistentObservation: A default theory has an inconsistent extension iffD is inconsistentTheorem: (Semi-monotonicity)Given two normal default theories 〈D, W 〉 and 〈D′, W 〉 suchthat D ⊆ D′ then, for any extension E(D, W ) there is anextension E(D′, W ) where E(D, W ) ⊆ E(D′, W )(The addition of normal default rules does not lead to theretraction of consequences.)




Nonmonotonic Consequence

Abstract study and analysis of nonmonotonic consequencerelation |∼ in terms of general properties Kraus, Lehmannand Magidor (1991)

Some common properties include:Supraclassicality If φ ⊢ ψ, then φ |∼ψ

Left Logical Equivalence If ⊢ φ↔ ψ and φ |∼χ, then ψ |∼χ

Right Weakening If ⊢ ψ → χ and φ |∼ψ, then φ |∼χ

And If φ |∼ψ and φ |∼χ, then φ |∼ψ ∧ χ

Plus many more!




KLM Systems

KLM Systems

Kraus, Lehman and Magidor (1991) study various classesof nonmonotonic consequence relations

Cumulative

Cumulative-Loop Preferential

Cumulative-Monotonic

Monotonic

⊇

This has been extended since. A good reference for thisline of work is Schlechta (1997)




KLM Systems

Summary

Nonmonotonic reasoning attempts to capture a form ofcommonsense reasoning

Nonmonotonic reasoning often deals with inferencesbased on defaults or ‘what is usually the case’

Belief change and nonmonotonic reasoning: two sides ofthe same coin?

Can introduce abstract study of nonmonotonicconsequence relations in same way as we study classicalconsequence relations

Similar links exist with conditionals

One area where nonmonotonic reasoning is important isreasoning about action (dynamic systems)



Outline RAA Sit Calc Frame Problem Summary

Knowledge Representation and ReasoningReasoning About Actions

Maurice Pagnucco



Sydney, NSW, 2052

January 4, 2009




NB: Many examples from: R. J. Brachman and H. J. Levesque,Knowledge Representation, Morgan Kaufmann, 2004.




1 RAA

2 Sit Calc

3 Frame Problem

4 Summary




Reasoning About Actions

One method to reason about action is to simply change theagent’s knowledge base

Erase some sentence(s) that should no longer be true andadd sentences that will now be true (i.e., after performingaction)

However, we can only answer questions about the currentstate

It will not be possible to reason about past or future states

On the other hand, if all we want to do is reason aboutwhich actions to perform, this may be a viable approach(may return to this later)




Modelling Domains and Actions

Aspects we need to consider:The state of the worldActions that change state of the world and what changesthey effectConstraints on legal scenarios (won’t deal much with thesein this lecture)Can you think of anything else?




Situation Calculus

The situation calculus is a way of describing change infirst-order logic

In simple terms it may be viewed as a dialect of FOLTerms

actionssituations

Fluents—predicates or functions whose values may vary




State of the WorldMethod 1:

on(C, A, S1)on(A, Table, S1)on(B, Table, S1)clear(B, S1)clear(C, S1)

Note: we reify states (i.e., make them entities in ourformalisation)Another common way using the situation calculus is asfollowsMethod 2:

holds(on(C, A), S1)holds(on(A, Table), S1)holds(on(B, Table), S1)holds(clear(B), S1)holds(clear(C), S1)




Actions

Actions are namedput(x , y) — put object x on top of object ymove(x , y , z) — move block x from y to zclear(x) — clear x




Situations

Situation — a snapshot of the world at a particular point intime

Alternate view — world historiesS0/init — initial situation (no actions have been performed)do(a, s) — situation resulting from performing action a insituation s

For example, do(put(A, B), do(put(B, C), S0))Situation resulting from putting block B on block C in theinitial situation and then placing block A on block B




Preconditions

Predicates and functions whose values may vary fromsituation to situation

For example,¬Broken(x , s) ∧ Broken(x , do(drop(r , x), s))

Special predicate Poss(a, s) denotes that action a may beperformed in state s

For example, Poss(pickup(r , x), s) ≡∀z¬Holding(r , z, s) ∧ ¬Heavy(x) ∧ NextTo(r , x , s)




Effects

Actions can have positive effectsFragile(x) ⊃ Broken(x , do(drop(r , x), s))

and negative effects¬Broken(x , do(repair(r , x), s))




Domain ConstraintsAlso known as state constraintsTrue at all (legal) states even though they involvestate-dependent relations

x is on the table iff it is not on top of another block

on(x , Table, s) ≡ ¬∃y(on(x , y , s) ∧ y 6= Table)

x is clear iff there is no block on top of it

clear(x , s) ≡ ¬∃y on(y , x , s)

If y is a block and there is another block on it, then y is not clear

on(x , y , s) ∧ ¬(y = Table) → ¬clear(y , s)

etc.Maurice Pagnucco UNSW



The Frame ProblemAction descriptions are not complete:

They describe what changes BUT do not specify whatstays the same!

The (famous) Frame Problem:

The problem of characterising those aspects ofthe state description that are not changed by anaction

One solution — Frame AxiomsMoving an object does not change its colourColour(x , c, s) ⊃ Colour(x , c, do(put(x , y), s))Fragile things do not break¬Broken(x , s) ∧ (x 6= y ∨ ¬Fragile(x)) ⊃¬Broken(x , do(drop(r , y), s)Since actions often leave most fluents unchanged, manyframe axioms may be required




Qualification Problem

What are the ramifications (direct and indirect effects) ofperforming an action

¬clear(b, do(move(c, a, b), S0))

Recent approaches have investigated the use of explicitnotions of causality in an attempt to solve this problemefficiently

What qualifications (preconditions) do we require inspecifying actions and their effects

Trying to specify exactly under which conditions an actionhas a particular effect is very difficult (in principle, the list ofpreconditions can be vast)




What counts as a solution to the frame problem?

Once we have described the actions of a system, we wouldlike a systematic method for automatically generatingframe axioms

Preferably, the representation should be conciseReasons:

Require frame axioms for reasoningThey are not entailed by other axiomsReduce possiblity of errors in determining frame axiomsCan easily update frame axioms if additional effects arespecified




Projection

Determining what is true in the situation resulting from theperforming of a sequence of actions a1, . . . , an

Suppose we gather all the axioms above in a sentence F .To determine whether a formula φ is true after performingthe sequence of actions a1, . . . , an, we need to determine

Γ |= φ(do(an, do(an−1, . . . , do(a1, S0) . . .)))




Legality

However, we don’t know whether the sequence of actionsa1, . . . , an can be performedA situation is legal iff:

Legal(S0) — it is the initial situationLegal(do(a, s)) ≡ Legal(s) ∧ Poss(a, s) — it results fromperforming the action in a legal situation where itsprecondition is satisified

Adding these axioms to Γ, we can determine whether asequence of actions can be performed by showing thatthey lead to a legal situation

Γ |= Legal(do(an, do(an−1, . . . , do(a1, S0) . . .)))




Normal form for effect axiomsGiven positive effect axioms for fluent Broken:Fragile(x) ⊃ Broken(x , do(drop(r , x), s))NextTo(b, x , s) ⊃ Broken(x , do(explode(b), s))

Rewrite them:∃r{a = drop(r , x) ∧ Fragile(x)}∨∃b{a = explode(b) ∧ NextTo(b, x , s)} ⊃

Broken(x , do(a, s))

Negative effect axiom:¬Broken(x , do(repair(r , x), s))

Rewrite as:∃r{a = repair(r , x)} ⊃ ¬Broken(x , do(a, s))

These formulae or of the form:PF (x1, . . . , xn, a, s) ⊃ F (x1, . . . , xn, do(a, s))NF (x1, . . . , xn, a, s) ⊃ ¬F (x1, . . . , xn, do(a, s))




Explanation Closure

Assumption: The previous two formulae characterise theonly way in which a fluent may changeExplanation Closure Axioms

¬F (x, s) ∧ F (x, do(a, s)) ⊃ PF (x, a, s)F (x, s) ∧ ¬F (x, do(a, s)) ⊃ NF (x, a, s)

Disguised frame axioms:¬F (x, s) ∧ ¬PF (x, a, s) ⊃ ¬F (x, do(a, s))F (x, s) ∧ ¬NF (x, a, s) ⊃ ¬F (x, do(a, s))




Successor State Axioms

Additional axioms:Integrity of effect axioms¬∃x, a, s PF (x, a, s) ∧ NF (x, a, s)Unique names for actionsA(x1, . . . , xn) = A(y1, . . . , yn) ⊃ (x1 = y1) ∧ . . . ∧ (xn = yn)A(x1, . . . , xn) 6= B(y1, . . . , ym) for distinct A and B

Together, axioms on last three slides equivalent tosuccessor state axiom for F :F (x, do(a, s)) ≡ PF (x, a, s) ∨ (F (x, s) ∧ ¬NF (x, a, s))

Broken(x , do(a, s)) ≡∃r{a = drop(r , x) ∧ Fragile(x)}∨∃b{a = explode(b) ∧ NextTo(b, x , s)}∨Broken(x , s) ∧ ¬∃r{a = repair(r , x)}




What we cannot do

Explicit time

Exogenous actions

Concurrent actions

Continuous actions

Complex actions

. . .




SummaryReasoning about actions is a very interesting area ofartificial intelligence and often makes use of nonmonotonicreasoning techniques

We have seen that a number of challenging problems arisethat we must deal with in order to reason effectively

One of the problems, however, is the possible proliferationof axioms

The search continues for a concise solution to the frameproblem (and associated problems)

Other formalisms include the event calculus, A languages,features and fluents, fluent calculus

Current research: causal approaches, cognitive robotics,planning (an area in its own right)



Outline Belief Change AGM Constructions Summary

Knowledge Representation and ReasoningBelief Revision

Maurice Pagnucco



Sydney, NSW, 2052

January 4, 2009




Introduction to AGM Approach

��




Belief Change — An example

(Gärdenfors & Rott 1995)

BeliefsThe bird caught in the trap is a swanThe bird caught in the trap comes from SwedenSweden is part of EuropeAll European swans are white

Consequences

The bird caught in the trap is white

New informationThe bird caught in the trap is black

Which sentence(s) would you give up?Maurice Pagnucco UNSW



Belief Change—An example

Logical considerations alone are not sufficient to answer thisquestion.

One possibility

The bird caught in the trap is a swanThe bird caught in the trap comes from SwedenSweden is part of EuropeAll European swans, except for some of theSwedish, are whiteThe bird caught in the trap is black




1 Belief Change

2 AGMBelief ExpansionBelief Contraction

3 ConstructionsMaximal Non-implying SubsetsEpistemic EntrenchmentSystems of Spheres

4 Summary




Belief Change—The General Idea

Dynamics of epistemic states

(old)

epistemic

state

input

state

epistemic

(new)

epistemic

However, we are not interested in all such ways ofchanging beliefs but only those that follow certainprinciples (i.e., rational belief change)




Rationality Criteria

Principle of Categorial Matching The representation of a belief stateafter change should be of the same format as thatprior to change

Consistency Beliefs in belief state should be consistent

Deductive Closure If the beliefs in a belief state logically entail asentence φ, then φ should be included in the state

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Principle of Informational Economy The amount of information lostduring change should be kept to a minimum

Preference Beliefs considered more important or entrenchedshould be retained in favour of less important ones.

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AGM Framework

(Alchourrón, Gärdenfors and Makinson 1985)While there are many frameworks for belief change weconcentrate on the AGM here as it is very common in theliterature

Epistemic states: belief sets (closed under Cn)

Epistemic input: formulaOperators: belief expansion, belief contraction, beliefrevision

Guided by principles of rationality (e.g., minimal change)Characterised by postulates for rational belief change(delineating a class of functions with desirable properties)Constructions: Selection functions, systems of spheres,epistemic entrenchment




Epistemic Attitudes

An agent can have three attitudes towards a formula φ:φ ∈ K agent believes φ

¬φ ∈ K agent disbelieves φφ, ¬φ 6∈ K agent is indifferent towards φ

In other models of belief change there may be a muchlarger number of attitudes (for example, suppose we attachprobabilities to beliefs)




AGM Belief Change Operations

belief expansion (+) epistemic input added to the current beliefstate without removal of any existing beliefs

belief contraction ( .−) beliefs removed from the current belief

state in order to effect removal of the epistemicinput

belief revision (∗) epistemic input is incorporated into thecurrent belief state but some existing beliefs mayalso need to be removed to maintain consistency

Belief change function +,.−, ∗ : K × L → K

(where L is the object language and K is the set of all beliefsets).




Belief Expansion

Belief Expansion

Want to add a belief(s) without giving anything up

(K+1) For any sentence α and any belief set K ,K + α is a belief set (closure)

(K+2) α ∈ K + α (success)(K+3) K ⊆ K + α (inclusion)(K+4) If α ∈ K , then K + α = K (vacuity)(K+5) If K ⊆ H, then K + α ⊆ H + α (monotonicity)(K+6) For all belief sets K and sentences α, K + α is the

smallest belief set satisfying (K+1) – (K+5) (minimality)

Theorem: The expansion function + satisfies (K + 1) – (K + 6) iffK + α = Cn(K ∪ {α}).

There is only one AGM expansion function—classical consequence Cn

This is not the case for AGM contraction and revision




Belief Contraction

Contraction Example

Suppose K contains the closure of the following formulae:

rain → wet_grassrainshop_openshop_open → light_on

Consider possibilities for K .−wet_grass:

rain → wet_grass rain shop_openshop_open shop_open shop_open → light_onshop_open → light_on shop_open → light_on




Belief Contraction

Belief Contraction

Want to give up a belief or suspend judgement; do not want to add anybeliefs

(K .−1) For any sentence φ and any belief set K ,K .−φ is a belief set (closure)

(K .−2) K .−φ ⊆ K (inclusion)(K .−3) If φ 6∈ K , then K .−φ = K (vacuity)(K .−4) If 6⊢ φ then φ 6∈ K .−φ (success)(K .−5) If φ ∈ K , K ⊆ (K .−φ) + φ (recovery)(K .−6) If ⊢ φ↔ ψ, then K .−φ = K .−ψ (preservation)(K .−7) K .−φ ∩ K .−ψ ⊆ K .−(φ ∧ ψ) (conj. overlap)(K .−8) If φ 6∈ K .−(φ ∧ ψ), then K .−(φ ∧ ψ) ⊆ K .−φ (conj. inclusion)

In particular, note the difference between

K .−(φ ∧ ψ): need only give up either φ or ψ

K .−(φ ∨ ψ): must give up both φ and ψ




Belief Contraction

Additional Properties

The following properties follow from the AGM postulates forbelief contraction.

1 If φ ∈ K , then (K .−φ) + φ ⊆ K

2 K .−φ = K ∩ (K .

−φ) + ¬φ

3 K .−φ ∩ Cn({φ}) ⊆ K .

−(φ ∧ ψ)

4 Either K .−(φ ∧ ψ) ⊆ K .

−φ or K .−(φ ∧ ψ) ⊆ K .

−ψ

5 Either K .−(φ ∧ ψ) = K .

−φ or K .−(φ ∧ ψ) = K .

−ψ orK .−(φ ∧ ψ) = K .

−φ ∩ K .−ψ

6 If ψ → φ ∈ K .−φ and φ→ ψ ∈ K .

−ψ, then K .−φ = K .

−ψ




Belief Contraction

Digression—Commensurability Thesis (Levi 1991)

“Given an initial state of full belief K1 and another state of full belief K2, thereis always a sequence of expansions and contractions, beginning with K1,remaining within the state of potential states of full belief and terminating withK2.”

Levi Identity: K ∗ φ = (K .−¬φ) + φ

• Given a contraction function we can construct a(associated) revision function: contract anything that wouldcause the addition of φ to lead to inconsistency and thenexpand by φ

Harper Identity: K .−φ = K ∩ K ∗ ¬φ

• Given a revision function we can construct a (associated)contraction function: revise by ¬φ (which would remove φ if itwere currently believed so as to have a consistent revision)and keep those beliefs in K that were maintained by thisrevision




Constructing Contraction Functions

AGM contraction function is simply a mapping from a beliefset and a formula to a new belief set that satisfies certainrestrictions

How would we go about “constructing” such a functionespecially if we wanted to implement one in a computerprogram?

Storing all the possible mappings is out of the question!

There are a number of constructions for contractionfunctions that we shall investigate

The first idea is to consider removing just enough formulaefrom K so that it no longer implies φ




Maximal Non-implying Subsets


Definition: K ′ is a maximal subset of K that fails to imply φ (aφ-remainder) iff

(i) K ′ ⊆ K

(ii) φ 6∈ K ′

(iii) for any ψ ∈ K such that ψ 6∈ K ′, ψ → φ ∈ K ′

We denote by K⊥φ the set of all such maximal non-implyingsubsets.Definition: A selection function γ : 2K → K is a function suchthat for any K ∈ K and φ ∈ L, ∅ 6= γ(K⊥φ) ⊆ K⊥φ wheneverK⊥φ 6= ∅ and K otherwise. If γ always returns a singletonwhenever K⊥φ 6= ∅, then γ is referred to as an opinionatedselection function.





Maxichoice Contraction

Idea: select the “best” element from K⊥α (minimal change).Definition: Let γ be an opinionated selection function. Amaxichoice contraction function over K may be defined asfollows

(Def Max) K .−φ =

{

γ(K⊥φ) whenever K⊥φ 6= ∅K otherwise

Theorem: If .− is a maxichoice contraction function over K ,

then it satisfies (K .−1) – (K .

−6).Theorem: If a revision function ∗ is obtained from a maxichoicecontraction function .

− via the Levi Identity, then for any φ suchthat ¬φ ∈ K , K ∗ φ is complete.• Maxichoice doesn’t remove enough





Full Meet Contraction

Idea: All or nothing!Definition: A full meet contraction over K may be defined asfollows

(Def Full) K .−φ =

{⋂

(K⊥φ) whenever K⊥φ 6= ∅K otherwise

Theorem: Any full meet contraction function satisfies (K .−1) –

(K .−6)

Theorem: If a revision function ∗ is obtained from a full meetcontraction function .

− via the Levi Identity, then for any φ suchthat ¬φ ∈ K , K ∗ φ = Cn(φ).• Full meet removes too much





Partial Meet Contraction

Idea: Compromise!Definition: Let γ be a selection function. A partial meetcontraction over K may be defined as follows

(Def Partial) K .−φ =

{⋂

γ(K⊥φ) whenever K⊥φ 6= ∅K otherwise

Theorem: For every belief set K , .− is a partial meet

contraction function iff .− satisfies (K .

−1) – (K .−6).





Selection Functions — more details

We can define a selection function as follows and then applyrestrictions to see what properties resultMarking-off identity ≤γ(K⊥φ) = {K ′ ∈ K⊥φ : K ′′ ≤ K ′ for all K ′′ ∈ K⊥φ}Definition: γ is a transitively relational iff it can be defined via amarking-off identity ≤ which is transitive.Theorem:For every belief set K , .

− is a transitively relational partial meetcontraction function iff .

− satisfies (K .−1) – (K .

−8).





Recovery

The opposite half of (K .−5) follows from (K .

−1) – (K .−4) giving

the following propertyIf φ ∈ K , then K = (K .

−φ) + φ

Counterexample?: (Hansson 1991)

George is a murderer (m)George is a law breaker (b)George is a tax evader (t)K = Cn({m} ∪ {b})m 6∈ K .

−b¬t 6∈ K ⊇ K .

−bK ⊆ Cn(K .

−b ∪ {b}) ⊆ Cn(K .−b ∪ {t})

m ∈ Cn(K .−b ∪ {t})





Withdrawals (Makinson 1986)

Idea: Let’s consider functions that don’t satisfy (Recovery)Definition: A function .

−is a withdrawal function iff it satisfies postulates(K .

−1) – (K .−4) and (K .

−6) for contraction over K .Definition: Two withdrawal functions − and .

− are revision equivalent iff theygenerate the same revision function via the Levi Identity.Theorem: Let K be any belief set. Then for each withdrawal operation − onK , there is a unique contraction function .

−on K that is revision equivalent to− and this .

− is the greatest element of [−].

In other words, withdrawal functions can be partitioned into equivalence

classes where the revision function associated with each member of a class

behaves the same. The maximal element of each class (the one removing

fewest beliefs) is an AGM contraction function (and there is only one per

class).





Belief Revision

Want to incorporate a belief in a consistent fashion

(K*1) For any sentence φ and any belief set K ,K ∗ φ is a belief set (closure)

(K*2) φ ∈ K ∗ φ (success)(K*3) K ∗ φ ⊆ K + φ (inclusion)(K*4) If ¬φ 6∈ K , then K + φ ⊆ K ∗ φ (preservation)(K*5) K ∗ φ = K⊥ if and only if ⊢ ¬φ (vacuity)(K*6) If ⊢ φ↔ ψ, then K ∗ φ = K ∗ ψ (extensionality)(K*7) K ∗ φ ∧ ψ ⊆ (K ∗ φ) + ψ (super expansion)(K*8) If ¬ψ 6∈ K ∗ φ, then (K ∗ φ) + ψ ⊆ K ∗ (φ ∧ ψ) (sub expansion)






1 If φ ∈ K , then K ∗ φ = K

2 K ∗ φ = (K ∩ K ∗ φ) + φ

3 K ∗ φ = K ∗ ψ if and only if ψ ∈ K ∗ φ and φ ∈ K ∗ ψ

4 K ∗ φ ∩ K ∗ ψ ⊆ K ∗ (φ ∨ ψ)

5 If ¬ψ 6∈ K ∗ (φ ∨ ψ), then K ∗ (φ ∨ ψ) ⊆ K ∗ ψ

6 K ∗ (φ ∨ ψ) = K ∗ φ or K ∗ (φ ∨ ψ) = K ∗ ψ orK ∗ (φ ∨ ψ) = K ∗ φ ∩ K ∗ ψ




Epistemic Entrenchment


Ordering over formulae in L

Certain beliefs about the world are more important thanothers when planning future actions, etc.

φ ≤ ψ: ψ is at least as epistemically entrenched as φ

In contraction, sentences in K with lower entrenchmentgiven up

Tautologies maximally entrenched, non-beliefs minimallyentrenched

• But what does an epistemic entrenchment relation looklike?






(EE1) If φ ≤ ψ and ψ ≤ γ then φ ≤ γ (transitivity)(EE2) If {φ} ⊢ ψ then φ ≤ ψ (dominance)(EE3) For any φ and ψ, φ ≤ φ ∧ ψ or ψ ≤ φ ∧ ψ (conjunctiveness)(EE4) When K 6= K⊥, φ ∈ K iff φ ≤ ψ for all ψ (minimality)(EE5) If φ ≤ ψ for all φ then ⊢ ψ (maximality)

• Essentially we have a series of “ranks” or levels containingformulae of equal entrenchment. Moreover, the tautologies aremaximally entrenched (we cannot give these up!) andnon-beliefs are minimally entrenched (we don’t care aboutthese!)






1 φ ≤ ψ or ψ ≤ φ (connectedness)

2 If ψ ∧ χ ≤ φ, then φ ≤ φ or χ ≤ φ

3 φ < ψ iff φ ∧ ψ < ψ

4 If χ ≤ φ and χ ≤ ψ, then χ ≤ φ ∧ ψ

5 If φ ≤ ψ, then φ ≤ φ ∧ ψ

6 φ ∧ ψ = min(φ, ψ)

7 φ ∨ ψ ≥ max(φ, ψ)

(NB: φ < ψ ≡ ψ 6≤ φ)





Belief Change via Entrenchment

TY

X ^ Y

X

X v Y?

φ:

φ not in K X v Y?





Belief Change via Entrenchment

(Gärdenfors and Makinson 1988)(C ≤) φ ≤ ψ iff φ 6∈ K .

−(φ ∧ ψ) or ⊢ φ ∧ ψ• Prefer ψ to φ if we would give up φ when given a choice between giving upφ or ψ or if it’s not possible to give up either formula(C .

−) ψ ∈ K .−φ iff ψ ∈ K and either φ < φ ∨ ψ or ⊢ φ

• Retain belief if it was originally believed and there is “independentevidence” for maintaining it or if it is not possible to remove φ(C∗) ψ ∈ K ∗ φ iff either ¬φ ≤ ¬φ ∨ ψ or ⊢ ¬φTheorem: If an ordering ≤ satisfies (EE1) – (EE5), then the contractionfunction which is uniquely determined by (C .

−) satisfies (K .−1) – (K .

−8) aswell as condition (C ≤).Theorem: If a contraction function .

− satisfies (K .−1) – (K .

−8), then the

ordering that is uniquely determined by (C ≤) satisfies (EE1) – (EE8) as well

as condition (C .−).





Grove’s Spheres

Ordering over “possible worlds” (maximally consistent setsof formulae)

Motivated by Lewis’ sphere semantics for counterfactuals

System of spheres: sets of possible worlds nested onewithin the other

The set of all worlds ML is the outermost (largest) sphere

[K ] is the set of worlds consistent with K ; these worldsform the innermost (smallest) sphere

System of spheres centred on [K ]




Systems of Spheres

Systems of Spheres

Definition: Let S be any collection of subsets of ML. We call Sa system of spheres, centred on X ⊆ ML, if it satisfies thefollowing conditions:

(S1) S is totally ordered by ⊆; that is, if U, V ∈ S, thenU ⊆ V or V ⊆ U

(S2) X is the ⊆-minimum of S

(S3) ML is the ⊆-maximum of S

(S4) If φ ∈ L and 6⊢ ¬φ, then there is a smallest spherein S intersecting [φ] (i.e., there is a sphere U ∈ Ssuch that U ∩ [φ] 6= ∅, and V ∩ [φ] 6= ∅ impliesU ⊆ V for all V ∈ S)(Lewis’ Limit Assumption)




Systems of Spheres

Belief Change via SOSs

cS(φ) — the smallest sphere intersecting [φ]fS(φ) = cS(φ) ∩ [φ] — innermost φ-worldsTheorem: Let S be any system of spheres in ML centred on[K ] for some theory K ∈ K. If one defines, for any φ ∈ L, K ∗ φto be th(fS(φ)), then the axioms (K*1) – (K*8) are satisfied.Theorem: Let ∗ : K× L → K be any function satisfying axioms(K*1) – (K*8). Then for any (fixed) theory K there is a system ofspheres on ML, S say, centred on [K ] and satisfyingK ∗ φ = th(fS(φ)) for all φ ∈ L.




Systems of Spheres

AGM Revision

[K]

ML

[φ]

c ( )

f ( )

S

S

φ

φ




Systems of Spheres

AGM Partial Meet Contraction

[K]

ML

[ φ]¬

c ( )

f ( )

S

S

φ ¬

φ ¬




Systems of Spheres

Properties of th (Grove 1988)

th : 2ML → K

(i) th([K ]) = K for all belief sets (i.e., theories) K ifthe underlying logic is compact

(ii) th(X ) 6= K⊥ if and only if X is nonempty

(iii) For any sentence φ ∈ L and X ⊆ ML,th(X ∩ [φ]) = Cn(th(X ) ∪ {φ})

(iv) For X , X ′ ⊆ ML, if X ⊆ X ′, then th(X ′) ⊆ th(X )

(v) For K , K ′ ∈ K, if K ⊆ K ′, then [K ′] ⊆ [K ]




Systems of Spheres

AGM Maxichoice Contraction

[K]

ML

[ φ]¬




Systems of Spheres

AGM Full Meet Contraction

[K]

ML

[ φ]¬




Systems of Spheres

SOS ⇔ EE

(Gärdenfors 1988)Can translate back and forth from a Systems of Spheres S andan epistemic entrenchment relation ≤ using the followingcondition:

φ ≤ ψ iff cS(¬φ) ⊆ cS(¬ψ)




Summary

AGM framework characterised in terms of (intuitive)postulates and constructionsOperations: expansion, contraction and revision (we shalllook at this operation in the next lecture)Belief contraction and revision can be related in terms ofthe Levi and Harper identities

Essentialy this means that we only need expansion plusone of contraction or revision

Operators characterised in terms of postulates andconstructions: maximal non-implying subsets, epistemicentrenchment, systems of spheresSome postulates—such as Recovery—are open toquestion




Belief Change and NMRCan define φ |∼ Kψ iff ψ ∈ K ∗ φ (cf. Ramsey test)Translation of AGM revision postulates:(K*1) = If φ |∼ψi for all ψi ∈ K and K ⊢ χ, then φ |∼χ

(Closure)(K*2) = φ |∼φ (Reflexivity)(K*3) = If φ |∼ψ, then ⊤ |∼φ→ ψ (Weak Conditionalisation)(K*4) = If ⊤ |6∼¬φ and ⊤ |∼φ→ ψ, then φ |∼ψ (WeakRational Monotony)(K*5) = If φ |∼⊥, then φ ⊢ ⊥ (Consistency Preservation)(K*6) = Left Logical Equivalence(K*7) = If φ ∧ ψ |∼χ, then φ |∼ψ → χ (Conditionalisation)(K*8) = If φ |6∼¬ψ and φ |∼χ, then φ ∧ ψ |∼χ (RationalMonotony)Note that |∼ is with respect to a particular KAll these postulates are known in the nonmonotonicconsequence literatureMaurice Pagnucco UNSW



Belief Change and Defaults

Can view normal defaults α:ββ

as supernormal defaults⊤:α→βα→β

Can encode such defaults in epistemic entrenchment asα→ ¬β < α→ β

‘Given information α, strictly prefer β to ¬β’

bird(Tweety) → ¬flies(Tweety) < bird(Tweety) →flies(Tweety)

It is consistent to havebird(Tweety) ∧ emu(Tweety) → flies(Tweety) <

bird(Tweety) ∧ emu(Tweety) → ¬flies(Tweety)




The big picture!

* -.

SOS

<

|~

Rational choice

Conditionals

...

Harper Identity

Levi Identity

(C*) (C-).




Areas of Study in Belief Change

Belief bases and computational approaches

Coherence (how do beliefs ‘cohere’)

Contraction proposals

Iterated revision

Relationships between belief change, nonmonotonicreasoning, conditionals, rational choice, . . .

Non-prioritised belief change

Abductive belief change

Reasoning about action and belief change

Dealing with uncertainty (Spohn, Bayesian networks, . . . )



Documents

Knowledge Representation and Reasoning Introduction and …ssll.cecs.anu.edu.au/files/slides/pagnucco.pdf · 2009-01-20 · Knowledge Representation and Reasoning Introduction and