Belief Revision

Lecture 2: Beyond AGM

Gregory Wheeler

greg@di.fct.unl.pt

Outline

• Withdrawal vs. Contraction

• Belief Bases

• Updates

• Choice

• Uncertainty

Belief Bases

In AGM, a belief state is modeled as a deductively closed theory. Thus, there is no distinction between ‘basic beliefs’ and ‘derived beliefs’.

Thus, retracting a formula does not entail retracting all formulas derived from .

Belief Bases

Example:

Let B = {(Match Campfire), Match}.

Let K = Cn(B).

Notice: Campfire (K - Match)

This is an example of the coherence approach to theory change, where the maximum amount of information is preserved.

Belief Bases

A belief base is a finite set of non-deductively closed formulas.

B1 = {, }

B2 = {, }

Note that B1 B2, but Cn(B1) = Cn(B2).

Belief Bases

A Foundationalist approach to theory change maintains that changes are only made to belief bases, not to deductively closed theories.

B1 = {, } (B1 -- )

B2 = {, } (B2 -- )

Note: Base revisions and contractions have properties different than AGM style operators.

Updates vs. Revision

The difference lies in the source of incorrect beliefs

Belief revision assumes that the world is unchanging: an agent’s change in belief occurs when that agent discovers something new about the static world or discovers an error in his beliefs about the static world.

Belief updates involve a change in belief prompted by actions in a dynamic world. This process behaves differently than revision.

Updates vs. Revision

Suppose:I believe that Reema lives in Rochester.

Types of changes:1. I learn that she has just moved to Cairo.

(Update)

2. I learn that I have been mistaken and that she lives in Cairo.

(Revision)

Changes in the World: The idea

• Interpretation of a belief set B– the set of possible worlds where B is true

• notification of some change in the actual world• The agent’s description of the possible states of

affairs must be modified accordingly:– Our description of the actual world is typically incomplete, which

means that there are several states of affairs (possible worlds) that are consistent with what we believe. Hence, an update must ensure that the changes are made true in the “candidate worlds” that survive the update.

Selection Functions

• Interpretation of a belief base B:

B = {w : w is a possible world, and w B}

= [B]

Selection Functions

• Interpretation of a belief base B:

B = {w : w is a possible world, and w B}

= [B]• Update of a world w [B] by

= the possible worlds w close to where is true

= w [] (selection function) []

w

Selection Function Semantics

Update B by = [B ] = w [B]w []

[B] [B ]

w

v

w []

v []

Updates vs. Conditionals

• Update Problem:– current belief base B

– input – a belief

• Ternary Relation R(B, , ); algebraically:1. R(B, , ) iff B (B updated by entails )

B “B has been updated by ” used in databases

2. R(B, , ) iff B (B entails if updated by )

“if one updated with then follows” (counterfactuals)

The Ramsey Test

B – iff B – If two people are arguing ‘If p will q’ and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q.

-Frank Ramsey 1931

This is how to evaluate a conditional: First, add the antecedent (hypothetically) to your stock of beliefs; second, make whatever adjustments are required to maintain consistency (without modifying the hypothetical belief in the antecedent); finally, consider whether or not the consequent is true. -Robert Stalnaker 1968

Gärdenfors Impossibility Theorem

Theorem (Gärdenfors 1978): Suppose is a metalanguage belief change operator, mapping

bases and target formula to new bases;

• L is a logic for an object language and satisfy the Ramsey test:

B – iff –L B

• the logic is ‘rich enough’;

Then, does not satisfy the AGM postulates.

Gärdenfors Impossibility Theorem

Remarks:• The result does not depend upon –, and the

theorem holds even if – is substructural or non-monotonic (Makinson 1990).

• Similar to Gödel’s incompleteness result, no object language predicate can express consistency.

• Result does not hold for updates (Grahne, KR 1991).

Language vs. Metalanguage

• Traditionally metalanguage operation

• single model

• non-constructive postulates “If o is consistent, then y -d o is consistent”,…

– object language operator• class of models

• axiomatics (constructive

Language vs. Metalanguage

• Now:– two dyadic modal operators in the object

language:• update operator:

• conditional operator:

Logics based on the Ramsey Test

• classical logic, plus

• two inference rules, “the Ramsey Rules”

B B B B

where ‘B’ is the conjunction of the elements (wffs) in the belief base.

Updates and conditionals as inverse modal operators

(Ryan and Schobbens, JoLLI 1997)

Switching Lemma (de Rijke, Venema, Roorda, Dunn)

• The conversion axioms

B (B )

( ) can be derived from the Ramsey rules

• …and vice versa

(cf. tense logics, dynamic logic with inverse modality, program logics.)

Derivable principles for updates -1

• B1 B2 (syntax-independence)

B1 B2

• B1 B2 (monotony)

B1 B2

• (B1 B2 ) (B1 B2) (updating disjunctions)

(inconsistency preservation)

Derivable principles for updates -2

• Example: proof of 1. by classical logic

2. by the Ramsey Rules

Derivable principles for updates -3

• Example: proof of B1 B2 .

B1 B2 1. B1 B2 hypothesis

2. B1 from 1, with new B2

3. B1 from 2, by the Ramsey rules

B2

4. B2 from 3

5. B1 by a proof similar to 1-4.

Derivable principles for conditionals

(Wansing, JLC 1994)

1. 1 2 . 2. 1 2 .

1 2 1 2

3. (1 2) ( 1 2)

4. T

1-4 are axioms of semi-normal conditional logic Ck (Chellas, JPL 1975)

Recovering normal conditional logics

CK = (Conditional, Kripke)

=

Selection function semantics:

M = W, ·, V such that:

• W (set of possible worlds)

• · : W x 2 2 (selection function)

• V(w) ATM for all w W. (truth assignment)

[ ] = {w W : w · [] []}

Thm (Chellas, 1975): sound, complete, finite model prop., decidable.

€

Ck + φ1 ↔ φ2

φ1[]→ψ ↔ φ2 []→ψ

Conversely

• Given:– a conditional logic L

– its tense extension L, by the Ramsey rules

• then we may ask which properties does L inherit?

(completeness, decidability, finite model, correspondence)

• ALL if L is one of the standard systems

• Correspondence only in the general case (Wolter, NDJFL 1998)

Normal Update Logics

UCK = (Update Conditional, Kripke)

= CK + the Ramsey Rules

=

=

Thm (Herzg 1998): sound and complete for selection function semantics

€

classical logic + Ramsey rules + φ1 ↔ φ2

φ1[]→ψ ↔ φ2[]→ψ

€

classical logic + Ramsey rules + φ1 ↔ φ2

ψ ←◊ φ1 ↔ ψ ←◊ φ2

Four desiderata for belief update

• syntax independence

if 1 2 then B 1 B 2

if B1 B2 then B1 B2 • success

B – • “maximum” consistency

B is consistent iff both B and are consistent.• minimal change

B is “as close to B as possible”

(Note: the last 3 require extensions of UCK.)

UCK and update desiderata

• success

– axiomatically: UCK + (B ) – semantically: w · [] []

• consistency

– axiomatically:

– semantically: w · [] , for all w W, for all consistent formula of classical logic.

€

UCK + B ←◊ φ →⊥

B →⊥

UCK and update desiderata

• minimum change – axiomatically: difficult to express if B is inconsistent

– semantically: minimal change “defined away” by selection function

• weaker forms of minimal change

minimality ~ orderings€

UCK + B →φ

B ←◊ φ ↔ B (idempotence)

Cumulative update logics

Minimal change: orderings of closeness associated to worlds.

To update by o, go to worlds where o is true and stop--these are the closest worlds.

[]

w · []w

Cumulative update logics - semantics• M = W, {≤w : w W}, V such that

– W a set of possible worlds– ≤w partial pre-order (transitive, reflexive relation) on W

u ≤w v = “u is at least as close to w as v is”

• Sw = {u W : there is v W such that u ≤w v}

= accessible worlds

• w · [] = the -worlds closest to w

= min≤w([] Sw)

= {u [] Sw : v [] Sw, if v ≤w u then v ≤w u}

• truth conditions as for UCKif W is infinite: Limit Assumption entails that [] implies w · [] .

Cumulative conditional logics -axiomatics

CL = (Conditional, Lewis)

= CK

+ + ((1 ) (2 )) ((12) )

+ ((1 2) (2 1))

((1 ) (2 ))

(Lewis 1973; Burgess 1981)

Cumulative conditional logics -axiomatics

UCL = (Update, Conditional, Lewis)

= UCK + CL

= UCK

+ (B ) + (B (1 2)) (B 1) (B 2)

€

+(B ←◊ φ1) → φ2,(B ←◊ φ2) → φ1

(B ←◊ φ1) ↔ (B ←◊ φ2)

UCL: derivable principles

• cumulativity

€

• (φ1[]→ φ2) → ((φ1[]→ψ ) ↔ ((φ1∧φ2)[]→ψ ))

€

• B ←◊ φ1 → φ2

(B ←◊ φ1) ↔ (B ←◊ (φ1∧φ2))

KM theory

(Katsuno and Mendelzon, 1991)

A KM-model is a UCL-model M = W, {≤w : w W}, V s.t.:

• W = 2ATM the set of all logically possible worlds= all truth assignments

• ≤w is a ‘faithful’ partial order:

– w ≤w u for all u

– if u ≤w w then u = w.

(hence: min≤w (W) = {w})

• V(w) = w(every w is a set of atoms)

KM theory

• The formula ‘B ’ denotes the replacement belief base resulting from an update of B by .

• The formula here denotes a new fact observed by an agent in response to a change in the world.

KM Update Postulates

(U1) (B ) .

Updating a belief base by a target formula yields a replacement belief base that implies the target formula.

(U2) If B then (B ) B.

If a belief base implies a target formula , then updating by yields a replacement belief base that is logically equivalent to the original belief base. (problematic)

KM Update Postulates

(U3) If B and are consistent, then (B ) is consistent.

Updating a consistent belief base by a consistent target formula yields a consistent replacement belief base.

(U4) If , B1 B2, then (B ) (B ).

Logically equivalent belief bases updated by logically equivalent target formulas yield logically equivalent replacement belief bases.

KM Update Postulates

(U4.1) If B1 B2, then (B1 ) (B2 ).

(U4.1) If , then (B ) (B ).

Postulates of syntax independence: U4.1 and U4.2 together hold that if logically equivalent belief bases are updated by logically equivalent target formula yield logically equivalent replacement belief bases.

KM Update Postulates

(U5) ((B ) ) (B ( )).

The conjunction and the updated belief base B by a target formula implies the update of B by the conjunction .

(U6) If (B 1) 2 and (B 2) 1, then

(B 1) (B 2)

If an update by , implies and an update by implies , then an update by is equivalent to an update by .

KM Update Postulates

(U7) If B is complete, then

((B 1) (B 2)) (B (1 2)).

Updating by the target formula 1 and updating by the target formula 2 entails an update by the target formula

(1 2). This requires a finite language.

(U8) (B1 B2) (B1 ) (B2 ).

Updating one or another belief base by a target formula is logically equivalent to updating one belief base by a target formula or updating the other by the target formula.

KM theory

Remarks:Success (U1); syntax independence (U4.1) + (U4.2); consistency ‘as much as possible’ (U3).

Minimal change? Not always satisfied.

KM theory

• a faithful partial pre-ordering for maximal change:

≤w = {w,u : w W}

• consequence:

min≤w([]) = {w}, if w []; otherwise [].

• the maximal change operation

satisfies the KM postulates.

€

B ←◊ φ ↔B if B |− φ

φ if B does not entail φ

⎧ ⎨ ⎩

References -1

• John Burgess, “Quick completeness proofs for some logics of conditionals”, Notre Dame Journal of Formal Logic, 22:76-84, 1981

• Brian Chellas, “Basic conditional logics”, Journal of Philosophical Logic, 4:133-153, 1973.

• Peter Gärdenfors, “Conditionals and changes of belief”, in I. H\Niiniluoto and R. Tuomela (eds.), The Logic and Epistemology of Scientific Change, vol. 30, Acta Philosophica Fennica, 1978.

• G. Grahne, “Updates and counterfactuals”, Proc. KR 1991.

• Andreas Herzig, “Logics for belief base updating”, in Handbook of defeasible reasoning and uncertainty management, vol. 3. D. Gabbay et. al. (eds.), Kluwer Academic Publishers, 1998.

References -2

• Hirofumi Katsuno and Alberto O. Mendelzon, “On the difference between updating a knowledge base and revising it”, in Belief Revision, P. Gärdenfors (ed.), Cambridge University Press, 1992. (First version appearing in Proc. KR 1991).A.

• A.M Keller and M. Winslett, “On the use of an extended relational model to handle changing incomplete information”, in IEEE Transactions on Software Engineering, 1985.

• David Lewis, Counterfactuals. Basil Blackwell, Oxford 1973.

• David Makinson, “The Gärdenfors Impossibility Theorem in non-monotonic contexts”, Studia Logica, 49:1-6, 1992.

• Frank P. Ramsey, Collected Essays of F. P. Ramsey, Oxford Press.

• Robert Stalknaker, “A theory of conditionals”, Studies in Logical Theory. Blackwell, Oxford, 1968.

References -3

• H. Wansing, “Sequent Calculi for Normal Modal Propositional Logics” , Journal of Logic and Computation, 4:125-42, 1994.

• F. Wolter, “A counterexample in tense logic”, Notre Dame J. of Formal Logic, 37(2), 1997. (Special issue on combining logics).