Relevant Minimal Change in Belief Update

Transcription

1 Relevant Minimal Change in Belief Update Laurent Perrussel 1 Jerusa Marchi 2 Jean-Marc Thévenin 1 Dongmo Zhang 3 1 IRIT Université de Toulouse France 2 Universidade Federal de Santa Catarina Florianópolis Brazil 3 University of Western Sydney Penrith Australia Initially JELIA-12

2 Motivation (1/3) Belief Change (Revision and Update): Set of beliefs K. Goal: incorporating a new piece of information into K. Challenges: keeping consistency and entailing minimal change. E.g.: K = {p, q, p q} Revising by q: what beliefs should be removed? p and q, p q and q... Preferences should be set in order to tackle the choices.

3 Motivation (2/3) Preferences might lead to counter-intuitive changes: E.g.: remove all initial beliefs. Minimal change only beliefs relating to the new piece of information should be involved in belief revision. What means related to? Sharing common symbols. Example: revising K = {p, q, p q, r} by q should not change r, and possibly p. Relevant Revision operators: Initially proposed by Parikh. sub class of revision operators which takes cares of beliefs that do not use symbols used in the incoming information.

4 Motivation (3/3) Our goal: 1 Express Belief Update with the help of Prime Implicants. Prime implicant: specific notation which helps to represent beliefs in a minimal way. Focus on minimal set of literals entailing a belief set. Natural way to focus on relevant symbols (canonical DNF). 2 Extend the notion of Relevance to Belief Update.

5 Plan

6 Basic Definitions Logical language Focus on propositional beliefs. Literal L: p or p. Complementary literal L = p (respectively p) iff L = p (respectively p). Lang: function returning the set of symbols of a formula.

7 Playing with Implicants (1/3) Implicant and Prime Implicant Term D: conjunction of literals p q. Implicant of ψ: term entailing ψ p q = p q. Prime Implicant of ψ: minimal implicant p = p q.

8 Playing with Implicants (2/3) Implicant and Prime Implicant Implicants(ψ): set of all implicants of ψ Implicants(p q) = { p, p q...} PrimeImplicants(ψ): set of prime implicants of ψ PrimeImplicants(p q) PrimeImplicants(p q) = { p, q} viewed as terms = {{ p}, {q}} viewed as sets

9 Playing with Implicants (3/3) Adapting terms and implicants Set of all possible terms based on D: TermsBasedOn(D) = {D (D D ) D Implicants( )} Example: TermsBasedOn( p) = {{ p}, { p, q}, { p, q}, {p}, {p, q} } Set of all possible terms based on ψ: TermsBasedOn(ψ) = Dψ PrimeImplicants(ψ) TermsBasedOn(Dψ) Example: TermsBasedOn(p q) = {{ p}, {q}, { p, q}, { p, q}, {p}, {p, q}

10 Prime Implicant Based Belief Revision (1/3) Belief set: ψ; incoming information: µ. Revision: ψ µ. Based on Katzuno-Mendelzon model-based revision (preferences over models): [[ψ µ]] = min([[µ]], ψ ) Idem but choosing terms rather than models constraining preferences over terms (faithful assignment).

11 Prime Implicant Based Belief Revision (2/3) Faithful assignment: mapping ψ to a pre-order defined over TermsBasedOn(ψ) (C1-T) if D u, D v Implicants(ψ), then D u < ψ D v. (C2-T) if D u Implicants(ψ) and D v Implicants(ψ), then D u < ψ D v. (C3-T) if ψ ϕ, then ψ = ϕ. (C4-T) For all D u, D v Implicants(ψ), if (D u D v ) then D u ψ D v. preferences should not favor too specific terms.

12 Prime Implicant Based Belief Revision (3/3) Prime Implicant-based revision of ψ by µ: ψ PI µ. Operator: such that ψ PI µ = def min(termsbasedon(ψ, µ), ψ ) TermsBasedOn(ψ, µ) = {Dµ (Dψ Dµ) Dψ PrimeImplicants(ψ) and Dµ PrimeImplicants(µ)} Key properties: (R1) (R6) holds. Natural mapping to KM revision (model-based revision). More specific than KM revision (preferences are defined only over a subset of models). what is more specific? Focus on key symbols.

13 Relevant Revision (1/3) Key idea: Grounding relevance into the languages used for describing belief and incoming information. Only belief ϕ of ψ using symbols appearing in incoming information µ should change. Belief ϕ which do not use these symbols should not change. Parikh s postulate: (P) Let ψ = ϕ ϕ such that Lang(ϕ) Lang(ϕ ) =. If Lang(µ) Lang(ϕ), then ψ µ (ϕ µ) ϕ, where is the revision operator restricted to language Lang(ϕ). Postulate can be rephrased to avoid syntax dependence by using prime implicant representation.

14 Relevant Revision (2/3) Parikh s postulate considers a local revision operator. There should be only one version of that operator Suppose ψ = ϕ ϕ s.t. ϕ and ϕ use different symbols. Suppose a pre-order ϕ s.t. D ϕ D Adding prime implicants of ϕ to D and D should not changed the preferences: D {D ϕ } ψ D {D ϕ }. In other words: constraint should relate multiple faithful assignments. (CS-T) Let ψ ϕ ϕ such that Lang(PIϕ) Lang(PI ϕ ) =. For any D, D TermsBasedOn(ϕ): D ϕ D iff D D ϕ ψ D D ϕ such that D ϕ, D ϕ PI ϕ and D D ϕ, D D ϕ TermsBasedOn(ψ).

15 Relevant Revision (3/3) Main result: 1 Relevance satisfaction: If faithful assignment satisfies constraint (CS-T) then PI satisfies postulate (P) (Parikh s relevance postulate). 2 Dalal is then relevant since (CS-T) holds for Dalal. Immediate question: Is it the same with belief update? 1 Naive translation of Parikh for Update. 2 Reprashing with prime implicants.

16 Belief Update: quick summary Change in a dynamic context Change is performed world by world: [[ψ µ]] = w [[ψ]] min([[µ]], w ) Faithful assignment: mapping worlds and partial pre-orders. KM postulates: axiomatic definition of updates.

17 Relevance Criterion for Belief Update (1/3) Naive Parikh s postulate for Update: (P-U) Let ψ = ϕ ϕ such that Lang(ϕ) Lang(ϕ ) =. If Lang(µ) Lang(ϕ), then ψ µ (ϕ µ) ϕ, where is the update operator restricted to language Lang(ϕ).... and go?

18 Relevance Criterion for Belief Update (2/3) Consider: ψ = (p 2 p 3 p 5 ) (p 4 p 5 ) and µ = (p 1 p 2 p 3 ) ( p 1 p 2 p 3 ) Update using Forbus: [[ψ µ]] = {{p 1, p 2, p 3, p 4, p 5 }, { p 1, p 2, p 3, p 4, p 5 } {p 1, p 2, p 3, p 4, p 5 }, { p 1, p 2, p 3, p 4, p 5 }} that corresponds to the following implicants: ψ µ = (p 1 p 2 p 3 p 5 ) ( p 1 p 2 p 3 p 5 ) µ focuses on symbols p 1, p 2, p 3. p 5 is preserved But p 4 is not!... and p 4 is not related to µ.

19 Relevance Criterion for Belief Update (3/3) Relevance enforces focus on most important literals Prime implicants represent: 1 relevant literals 2 alternatives for entailing ψ (ie. uncertainty) Change should be performed implicant by implicant rather than world by world. Consider again ψ = (p 2 p 3 p 5 ) (p 4 p 5 ) and µ = (p 1 p 2 p 3 ) ( p 1 p 2 p 3 ). At first change should consider p 2 p 3 p 5 and second p 4 p 5. In each implicant change, keep as much as possible of the implicant.

20 Prime Implicant based Belief Update (1/2) New Update operator: PI. 2 new postulates (U7-T) If PrimeImplicants(ψ) = {Dψ} then (ψ PI µ 1 ) (ψ PI µ 2 ) implies ψ PI (µ 1 µ 2 ). (U9-T) If PrimeImplicants(ψ) = {Dψ} and PrimeImplicants(µ) = {Dµ} then ψ PI µ = Dµ (Dψ Dµ). (U7-T) combined with (U8): (ψ 1 ψ 2 ) PI µ (ψ 1 PI µ) (ψ 2 PI µ) entails that update is performed implicant by implicant and (U9-T) focuses on relevant literals.

21 Prime Implicant based Belief Update (2/2) New Update operator: PI. Faithful assignment defined over TermsBasedOn(D): (CU1-T) For all D, D Implicants( ), if D TermsBasedOn(D), D TermsBasedOn(D) then D < D D (C4U-T) For all D, D TermsBasedOn(D), if D D then D D D. Definition of PI : ψ PI µ = def Dψ PIψ min(termsbasedon(dψ, µ), D ψ ) KM (U1)-(U6) + (U8) hold and also (U7-T) & (U9-T)

22 Relevant Belief Update (1/3) Second reading of Parikh s postulate for Update: (P-UT) Let ψ = ϕ ϕ such that Lang(ϕ) Lang(ϕ ) =. If Lang(µ) Lang(ϕ), then ψ PI µ (ϕ µ) PI ϕ, where is the PI PI update operator restricted to language Lang(ϕ). Remaining task: how to translate this postulate in terms of constraint(s) on faithful assignment

23 Relevant Belief Update (2/3) Constraining faithul assignment (constraint (CUS-T)): Suppose D D 1 D 2 and D 1 Suppose two terms D, D TermsBasedOn(D 1, µ) s.t. D D 1 D. Relevance states that adding D 2 to D and D should not switch the preferences related to D and D (if D 2 is expressed with symbols that differ from the symbols of D 1 ) D D 2 D D D 2

24 Relevant Belief Update (3/3) Main results: Faithful Assign. + Constraint (CUS-T) IP + Postulate (PU-T) Forbus is not relevant. Forbus IP is relevant.

25 Illustration: revisiting the example PIψ = (p 2 p 3 p 5 ) (p 4 p 5 ) PIµ = (p 1 p 2 p 3 ) ( p 1 p 2 p 3 ). Faithful assignment with pre-orders D ψ : {p 1, p 2, p 3, p 5 } < {p2,p 3,p 5 } { p 1, p 2, p 3, p 5 } {p 1, p 2, p 3, p 4, p 5 } {p4,p 5 } { p 1, p 2, p 3, p 4, p 5 } Let Fo be the PI update operator: PI ψ Fo PI µ = (p 1 p 2 p 3 p 5 ) (p 1 p 2 p 3 p 4 p 5 ) ( p 1 p 2 p 3 p 4 p 5 ) Remind: ψ Fo µ = (p 1 p 2 p 3 p 5 ) ( p 1 p 2 p 3 p 5 )

26 Conclusion Main Results Extension of the notion of relevance to belief update New definition of update: implicant by implicant Parikh-like postulate for relevant update Sound and complete definition: Postulates + Faith. Assign. Future Work Link with classical belief belief update. Lower and upper bounds for belief change.