Knowledge Representation and Reasoning Introduction and Motivation Maurice Pagnucco School 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, Knowledge Representation and Reasoning, Morgan Kaufmann Publishers, San Francisco, CA,, 2004. ISBN: 1-55860-932-6
35
Embed
Knowledge Representation and Reasoning Introduction and …ssll.cecs.anu.edu.au/files/slides/pagnucco.pdf · 2009-01-20 · Knowledge Representation and Reasoning Introduction and
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Knowledge Representation andReasoning
Introduction and Motivation
Maurice PagnuccoSchool of Computer Sc. & Eng.University of New South Wales
� 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
� 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
� 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)
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
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}
∆ = {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.
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 〉
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
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 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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
Knowledge Representation and ReasoningReasoning About Actions
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
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
NB: Many examples from: R. J. Brachman and H. J. Levesque,Knowledge Representation, Morgan Kaufmann, 2004.
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
1 RAA
2 Sit Calc
3 Frame Problem
4 Summary
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem 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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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?
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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:
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
Preconditions
Predicates and functions whose values may vary fromsituation to situation
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
Effects
Actions can have positive effectsFragile(x) ⊃ Broken(x , do(drop(r , x), s))
and negative effects¬Broken(x , do(repair(r , x), s))
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline RAA Sit Calc Frame Problem Summary
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
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
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
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
1 Belief Change
2 AGMBelief ExpansionBelief Contraction
3 ConstructionsMaximal Non-implying SubsetsEpistemic EntrenchmentSystems of Spheres
4 Summary
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions 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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
9
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
;
Sta
tic
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.
9
>
>
>
>
>
=
>
>
>
>
>
;
Dyn
amic
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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).
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Belief Contraction
Contraction Example
Suppose K contains the closure of the following formulae:
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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.
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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).
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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).
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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})
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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).
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Maximal Non-implying Subsets
Additional Properties
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Epistemic Entrenchment
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
• But what does an epistemic entrenchment relation looklike?
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Epistemic Entrenchment
Epistemic Entrenchment
(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!)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Epistemic Entrenchment
Additional Properties
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Epistemic Entrenchment
Belief Change via Entrenchment
TY
X ^ Y
X
X v Y?
φ:
φ not in K X v Y?
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Epistemic Entrenchment
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 .−).
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Epistemic Entrenchment
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 ]
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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.
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Systems of Spheres
AGM Revision
[K]
ML
[φ]
c ( )
f ( )
S
S
φ
φ
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Systems of Spheres
AGM Partial Meet Contraction
[K]
ML
[ φ]¬
c ( )
f ( )
S
S
φ ¬
φ ¬
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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 ]
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Systems of Spheres
AGM Maxichoice Contraction
[K]
ML
[ φ]¬
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
Systems of Spheres
AGM Full Meet Contraction
[K]
ML
[ φ]¬
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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)
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Maurice Pagnucco UNSW
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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
Knowledge Representation and Reasoning
Outline Belief Change AGM Constructions Summary
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