belief revision bernhard nebel, stefan w ol , and marco...

Principles of Knowledge Representation and ReasoningBelief Revision

Bernhard Nebel, Stefan Wolfl, and Marco Ragni

Albert-Ludwigs-Universitat Freiburg

June 14, 2010

Nebel, Wolfl, Ragni (Uni Freiburg) KRR June 14, 2010 1 / 21

Principles of Knowledge Representation and ReasoningJune 14, 2010 — Belief Revision

Belief RevisionChange vs. RevisionChange OperatorsAGM PostulatesBase RevisionPrioritiesRevision vs. NMRConclusions

Literature


Belief Revision Change vs. Revision

Belief Change

I A dual approach to nonmonotonic reasoning is belief change.

I We start with some belief state K . When new information arrives, wechange the belief state in order to accommodate the new information.

I In the general case, the changed belief state may not be a superset ofthe original belief state.

I Contrary to nonmonotonic reasoning, here we deal with temporalnonmonotonicity, i.e., the nonmonotonic evolution of a knowledgebase or belief state over time.



Two Scenarios

I We have a theory about the world, and the new information is meantto correct our theory . . .

belief revision: change your belief state minimally in order toaccommodate the new information

I We have a correct theory about the current state of the world, andthe new information is meant to record a change in the world . . .

belief update: incorporate the change by assuming that the world haschanged minimally



Update and Revision are Different

Assume the new information is consistent with our old beliefs.

I In case of belief revision, we would like to add the new informationmonotonically to our old beliefs.

I For belief update this is not necessarily the case.

◦ Assume we know that the door is open or the window is open.◦ Assume we learn that the world has changed and the door is now

closed.I In this case, we do not want to add this information monotonically to

our theory, since we would be forced to conclude that the window isopen.


Belief Revision Change Operators

Belief Change Operations

General assumption:

I A belief state is modeled by a deductively closed theory, i.e.,K = Cn(K ) with Cn the consequence operator

I L: logical language (propositional logic)

I ThL: set of deductively closed theories (or belief sets) over L

Belief change operations

Monotonic addition: +: ThL × L → ThL

K + ψ = Cn(K ∪ {ψ})Revision: u : ThL × L → ThL

Reasonable revision operations?AGM Revision Postulates (Alchourron, Gardenfors, Makinson)


Belief Revision AGM Postulates

AGM Postulates:Constraining the Space of Revision Operations

AGM postulates:

(u1) K u ϕ ∈ ThL;

(u2) ϕ ∈ K u ϕ;

(u3) K u ϕ ⊆ K + ϕ;

(u4) If ¬ϕ 6∈ K , then K + ϕ ⊆ K u ϕ;

(u5) K u ϕ = Cn(⊥) only if ` ¬ϕ;

(u6) If ` ϕ↔ ψ then K u ϕ = K u ψ;

(u7) K u (ϕ ∧ ψ) ⊆ (K u ϕ) + ψ;

(u8) If ¬ψ 6∈ K u ϕ,then (K u ϕ) + ψ ⊆ K u (ϕ ∧ ψ).



Canonical Revision Operations?

I The postulates constrain the space to fully rational revisionoperations, but do not pick a single one.

I Revision operations are closed under intersection, so should we choosethe minimum?

I NO!

This is full meet revision, which is known to be useless sinceK u ϕ = Cn(ϕ) for all ϕ that are inconsistent with K .

I What other ways are there to generate a reasonable revisionoperation?



Belief Revision Schemes

I Preference information (what to keep and what to give up)

I . . . may be different for different K ’s, but independent from the newinformation ϕ

compose revision operation pointwise for each K

I In general, a belief revision scheme (BRS) is a “recipe” for deriving arevision operation – restricted to a particular set K – from

◦ the belief set and◦ preference information over this belief set



Examples

Partial meet revision (AGM): Preference information is given by a selectionfunction γ over the set of maximal subtheories consistent with the newinformation:

K u ϕdef=(⋂

γ(K ↓ ¬ϕ))

+ ϕ,

where K + ϕ = Cn(K ∪ {ϕ}).

Cut revision (GM): Preference information is given by a complete preorder� over all ψ ∈ K :

K u ϕdef= {ψ ∈ K | ¬ϕ ≺ ψ}+ ϕ.

Provided � satisfies a number of axioms (epistemic entrenchment), cutrevisions correspond to fully rational revision operations.



Revision – Viewed Computationally

I We don’t want to deal with deductively closed theories . . .

I Consider belief bases (finite sets of propositions) to represent beliefsets.

I We don’t want to specify an arbitrary amount of preferenceinformation . . .

I A theory K over the propositional logic L with n propositional atomscan have as much as

I 22n

different propositions,I 2n different models.

I Consider ways of specifying preference information in a concise way,i.e., polynomial in the size of the belief base.


Belief Revision Base Revision

Base Revision Schemes

I Start with a finite belief base A and preference information over theelements of A . . .

I We want to generate a revision operation (restricted to Cn(A))

I Assume a partitioning of A into n priority classes A1, . . . ,An such thatthe elements of Ai are more important or relevant than those of Aj

for j < i

I Equivalently, consider a complete preorder E over A comparingpriorities (epistemic relevance)

I Define a (base) revision scheme that keeps as many of the morerelevant propositions as possible

⇒ Base revision schemes generate revision operations in the same way asordinary schemes do.


Belief Revision Priorities

Example: Prioritized Meet-Base Revision

Let (A ⇓ ¬ϕ) be the maximal subsets of A that are consistent with ϕ andmaximize relevant propositions.

step 1step 2

step 4step 3

maximal and consistent

Level n−1Level n

Level n−2Level n−3

Level 1

U Cn(Xi)

step n



Prioritized Meet-Base Revision

Prioritized Meet-Base Revision (PMBR):

A⊕ ϕ def=( ⋂B∈(A⇓¬ϕ)

Cn(B))

+ ϕ.

Define a revision operation u on Cn(A) (that depends on A and thepriority information) by

Cn(A)u ϕdef= A⊕ ϕ.



Properties of PMBRsI Generates partial meet revision, but does not satisfy (u8) in general.I Deciding whether A⊕ ϕ ` ψ is Πp

2-complete, even for one priorityclass.

I A revised base can be represented by

A⊕ ϕ = Cn((∨

(A ⇓ ¬ϕ))∧ ϕ).

I A revised base can become exponentially large:

A = {p1, . . . , pm, q1, . . . , qm}

ϕ =m∧i=1

(pi ↔ ¬qi )

(A ⇓ ϕ) has size exponential in |A|.I Worse, in some cases there exists no concise representation of the

revised base (provided the polynomial hierarchy does not collapse[Cadoli et al 94]).


Belief Revision Revision vs. NMR

Revision vs. Nonmonotonic Reasoning

Belief Revision and Nonmonotonic Reasoning seem to be of differentnature, but there exists a tight connection:

I Given K and a revision operation u,a nonmonotonic consequence relation can be defined as follows:ϕ |∼ ψ iff ψ ∈ K u ϕ.

In this case,

I the rationality postulates correspond to principles of NMR (such ascautious monotonicity, etc.);

I in the case of prerequisite-free, normal defaults D, the cautionsconclusions from (W ,D) are simply D ⊕W with one priority level;

I a similar relationship holds between Brewka’s level default theoriesand PMBRs.


Belief Revision Revision vs. NMR

NMR Principles and Rationality Postulates

(u2) ϕ ∈ K u ϕ;

I Reflexivity

(u3) K u ϕ ⊆ K + ϕ;

I Supraclassicality

(u6) If ` ϕ↔ ψ then K u ϕ = K u ψ;

I Left Logical Equivalence

(u8) If ¬ψ 6∈ K u ϕ,then (K u ϕ) + ψ ⊆ K u (ϕ ∧ ψ);

I Rational Monotonicity


Belief Revision Conclusions

Conclusions from the Correspondence

I NMR can be thought of as the other side of the same coin.

I NMR (at least for default logic) is as hard as belief revision.

I Representing the conclusions from a propositional default theory usingclassical propositional logic cannot be done in polynomial space,provided the polynomial hierarchy does not collapse.

I In other words, nonmonotonic logics can be thought of representing(some) information in a denser way than classical logic, and with thatcome higher computational costs.


Belief Revision Conclusions

Outlook & Summary

I While NMR and Belief Revision seem to be the two sides of the samecoin, there are notable pragmatic differences:

I Belief revision seems to require that we can easily represent thechanged belief base, while for NMR it makes sense to use denserepresentations.

I A similar argument could be made for the computational complexity.

I NMR and Belief Revision can be thought of as qualitative ways ofdealing with uncertainty in a purely logical setting.

I There exists a strong correspondence between NMR and BeliefRevision.

I Both are computationally expensive and representational problematic.

I There are cases, though, that are tractable and practical.


Literature

Literature I

D. Makinson.How to give it up: A survey of some formal aspects of theory change.Synthese, 62:347–363, 1985.Very good introduction to the topic.

B. Nebel.Belief revision and default reasoning: Syntax-based approaches.In KR-91, pages 417–428, Cambridge, MA, Apr. 1991.

C. E. Alchourron, P. Gardenfors, and D. Makinson.On the logic of theory change: Partial meet contraction and revision functions.Journal of Symbolic Logic, 50(2):510–530, June 1985.Introduces the so-called AGM approaches: Characterizing belief revision operations by postulates.

P. Gardenfors.Knowledge in Flux—Modeling the Dynamics of Epistemic States.MIT Press, Cambridge, MA, 1988.


Literature

Literature II

B. Nebel.How hard is it to revise a belief base?In D. Dubois and H. Prade (eds.), Handbook of Defeasible Reasoning andUncertainty Management Systems, Vol. 3: Belief Change,Kluwer Academic, Dordrecht, The Netherlands, 1998, 77-145.

P. Gardenfors.Belief revision and nonmonotonic logic: Two sides of the same coin?ECAI-90, 768-773.

H. Rott.Change, choice and inference: A study of belief revision and nonmonotonicreasoning,Clarendon, Oxford, 2001.


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