Relevant Minimal Change in Belief Update Laurent Perrussel Jerusa - - PowerPoint PPT Presentation

Relevant Minimal Change in Belief Update

Laurent Perrussel✶ Jerusa Marchi✷ Jean-Marc Thévenin✶ Dongmo Zhang✸

✶IRIT – Université de Toulouse – France ✷Universidade Federal de Santa Catarina – Florianópolis – Brazil ✸University of Western Sydney – Penrith – Australia

Initially presented @ JELIA-12

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Motivation (1/3)

Belief Change (Revision and Update):

Set of beliefs ❑. Goal: incorporating a new piece of information into ❑. Challenges: keeping consistency and entailing minimal change.

E.g.: ❑ = {♣, q, ♣ → q} Revising by ¬q: ⇒ what beliefs should be removed? ♣ and q, ♣ → q and q... Preferences should be set in order to tackle the choices.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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

• f information should be involved in belief revision.

What means “related to”? Sharing common symbols. Example: revising ❑ = {♣, q, ♣ → q, r} by ¬q should not change r, and possibly ♣. 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.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Plan

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Basic Definitions

Logical language Focus on propositional beliefs. Literal ▲: ♣ or ¬♣. Complementary literal ▲ = ♣ (respectively ¬♣) iff ▲ = ¬♣ (respectively ♣). ▲❛♥❣: function returning the set of symbols of a formula.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

“Playing” with Implicants (1/3)

Implicant and Prime Implicant Term ❉: conjunction of literals ¬♣ ∧ q . Implicant of ψ: term entailing ψ ¬♣ ∧ q | = ♣ → q . Prime Implicant of ψ: minimal implicant ¬♣ | = ♣ → q .

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

“Playing” with Implicants (2/3)

Implicant and Prime Implicant ■♠♣❧✐❝❛♥ts(ψ): set of all implicants of ψ ■♠♣❧✐❝❛♥ts(♣ → q) = {¬♣, ¬♣ ∧ q...} Pr✐♠❡■♠♣❧✐❝❛♥ts(ψ): set of prime implicants of ψ Pr✐♠❡■♠♣❧✐❝❛♥ts(♣ → q) = {¬♣, q} viewed as terms Pr✐♠❡■♠♣❧✐❝❛♥ts(♣ → q) = {{¬♣}, {q}} viewed as sets

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

“Playing” with Implicants (3/3)

Adapting terms and implicants Set of all possible terms based on ❉: ❚❡r♠s❇❛s❡❞❖♥(❉) = {❉′ ∪ (❉ − ❉′)|❉′ ∈ ■♠♣❧✐❝❛♥ts(⊤)} Example:

❚❡r♠s❇❛s❡❞❖♥(¬♣) = {{¬♣}, {¬♣, q}, {¬♣, ¬q}, {♣}, {♣, q} · · · }

Set of all possible terms based on ψ: ❚❡r♠s❇❛s❡❞❖♥(ψ) =

• ❉ψ∈Pr✐♠❡■♠♣❧✐❝❛♥ts(ψ)

❚❡r♠s❇❛s❡❞❖♥(❉ψ) Example:

❚❡r♠s❇❛s❡❞❖♥(♣ → q) = {{¬♣}, {q}, {¬♣, q}, {¬♣, ¬q}, {♣}, {♣, q} · · ·

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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]] = ♠✐♥([[µ]], ψ) Idem but choosing terms rather than models ⇒ constraining preferences over terms (faithful assignment).

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Prime Implicant Based Belief Revision (2/3)

Faithful assignment: mapping ψ to a pre-order defined over ❚❡r♠s❇❛s❡❞❖♥(ψ) (C1-T) if ❉✉, ❉✈ ∈ ■♠♣❧✐❝❛♥ts(ψ), then ❉✉ <ψ ❉✈. (C2-T) if ❉✉ ∈ ■♠♣❧✐❝❛♥ts(ψ) and ❉✈ ∈ ■♠♣❧✐❝❛♥ts(ψ), then ❉✉ <ψ ❉✈. (C3-T) if ψ ≡ ϕ, then ψ=ϕ. (C4-T) For all ❉✉, ❉✈ ∈ ■♠♣❧✐❝❛♥ts(ψ), if (❉✉ ⊆ ❉✈) then ❉✉ ∼ψ ❉✈. ⇒ preferences should not favor too specific terms.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Prime Implicant Based Belief Revision (3/3)

Prime Implicant-based revision of ψ by µ: ψ ◦P■ µ. Operator: ψ ◦P■ µ =❞❡❢

• ♠✐♥(❚❡r♠s❇❛s❡❞❖♥(ψ, µ), ψ)

such that ❚❡r♠s❇❛s❡❞❖♥(ψ, µ) = {❉µ ∪ (❉ψ − ❉µ)| ❉ψ ∈ Pr✐♠❡■♠♣❧✐❝❛♥ts(ψ) and ❉µ ∈ Pr✐♠❡■♠♣❧✐❝❛♥ts(µ)} Key properties: (R1)–(R6) holds. Natural mapping to KM revision (model-based revision). More specific than KM revision (preferences are defined

• nly over a subset of models).

⇒ what is more specific? Focus on key symbols.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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 ▲❛♥❣(ϕ) ∩ ▲❛♥❣(ϕ′) = ∅. If ▲❛♥❣(µ) ⊆ ▲❛♥❣(ϕ), then ψ ◦ µ ≡ (ϕ ◦′ µ) ∧ ϕ′, where ◦′ is the revision operator restricted to language ▲❛♥❣(ϕ).

Postulate can be rephrased to avoid syntax dependence by using prime implicant representation.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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. ❉ ϕ ❉′ Adding prime implicants of ϕ′ to ❉ and ❉′ should not changed the preferences: ❉ ∪ {❉ϕ′} ψ ❉′ ∪ {❉′

ϕ′}.

In other words: constraint should relate multiple faithful assignments.

(CS-T) Let ψ ≡ ϕ ∧ ϕ′ such that ▲❛♥❣(P■ϕ) ∩ ▲❛♥❣(P■ϕ′) = ∅. For any ❉, ❉′ ∈ ❚❡r♠s❇❛s❡❞❖♥(ϕ): ❉ ϕ ❉′ iff ❉ ∪ ❉ϕ′ ψ ❉′ ∪ ❉′

ϕ′ such that ❉ϕ′, ❉′ ϕ′ ∈ P■ϕ′ and

❉ ∪ ❉ϕ′, ❉′ ∪ ❉′

ϕ′ ∈ ❚❡r♠s❇❛s❡❞❖♥(ψ).

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Relevant Revision (3/3)

Main result:

1

Relevance satisfaction: If faithful assignment satisfies constraint (CS-T) then ◦P■ 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.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Belief Update: quick summary

Change in a dynamic context Change is performed world by world: [[ψ ⋄ µ]] =

• ✇∈[[ψ]]

♠✐♥([[µ]], ✇) Faithful assignment: mapping worlds and partial pre-orders. KM postulates: axiomatic definition of updates.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Relevance Criterion for Belief Update (1/3)

“Naive” Parikh’s postulate for Update:

(P-U) Let ψ = ϕ ∧ ϕ′ such that ▲❛♥❣(ϕ) ∩ ▲❛♥❣(ϕ′) = ∅. If ▲❛♥❣(µ) ⊆ ▲❛♥❣(ϕ), then ψ ⋄ µ ≡ (ϕ ⋄′ µ) ∧ ϕ′, where ⋄′ is the update operator restricted to language ▲❛♥❣(ϕ).

... and go?

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Relevance Criterion for Belief Update (2/3)

Consider: ψ = (♣✷ ∧ ♣✸ ∧ ♣✺) ∨ (♣✹ ∧ ♣✺) and µ = (♣✶ ∧ ♣✷ ∧ ¬♣✸) ∨ (¬♣✶ ∧ ¬♣✷ ∧ ¬♣✸) Update using Forbus:

[[ψ ⋄ µ]] = {{♣✶, ♣✷, ¬♣✸, ♣✹, ♣✺}, {¬♣✶, ¬♣✷, ¬♣✸, ♣✹, ♣✺} {♣✶, ♣✷, ¬♣✸, ¬♣✹, ♣✺}, {¬♣✶, ¬♣✷, ¬♣✸, ¬♣✹, ♣✺}} that corresponds to the following implicants: ψ ⋄ µ = (♣✶ ∧ ♣✷ ∧ ¬♣✸ ∧ ♣✺) ∨ (¬♣✶ ∧ ¬♣✷ ∧ ¬♣✸ ∧ ♣✺)

µ focuses on symbols ♣✶, ♣✷, ♣✸. ⇒ ♣✺ is preserved ⇒ But ♣✹ is not! ... and ♣✹ is not related to µ.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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 = (♣✷ ∧ ♣✸ ∧ ♣✺) ∨ (♣✹ ∧ ♣✺) and µ = (♣✶ ∧ ♣✷ ∧ ¬♣✸) ∨ (¬♣✶ ∧ ¬♣✷ ∧ ¬♣✸). At first change should consider ♣✷ ∧ ♣✸ ∧ ♣✺ and second ♣✹ ∧ ♣✺. In each implicant change, keep as much as possible of the implicant.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Prime Implicant based Belief Update (1/2)

New Update operator: ⋄P■. 2 new postulates (U7-T) If Pr✐♠❡■♠♣❧✐❝❛♥ts(ψ) = {❉ψ} then (ψ ⋄P■ µ✶) ∧ (ψ ⋄P■ µ✷) implies ψ ⋄P■ (µ✶ ∨ µ✷). (U9-T) If Pr✐♠❡■♠♣❧✐❝❛♥ts(ψ) = {❉ψ} and Pr✐♠❡■♠♣❧✐❝❛♥ts(µ) = {❉µ} then ψ ⋄P■ µ = ❉µ ∪ (❉ψ − ❉µ). (U7-T) combined with (U8): (ψ✶ ∨ ψ✷) ⋄P■ µ≡ (ψ✶ ⋄P■ µ) ∨ (ψ✷ ⋄P■ µ) entails that update is performed implicant by implicant and (U9-T) focuses on relevant literals.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Prime Implicant based Belief Update (2/2)

New Update operator: ⋄P■. Faithful assignment defined over ❚❡r♠s❇❛s❡❞❖♥(❉): (CU1-T) For all ❉′, ❉′′ ∈ ■♠♣❧✐❝❛♥ts(⊤), if ❉′ ∈ ❚❡r♠s❇❛s❡❞❖♥(❉), ❉′′ ∈ ❚❡r♠s❇❛s❡❞❖♥(❉) then ❉′ <❉ ❉′′ (C4U-T) For all ❉′, ❉′′ ∈ ❚❡r♠s❇❛s❡❞❖♥(❉), if ❉′ ⊆ ❉′′ then ❉′ ∼❉ ❉′′. Definition of ⋄P■: ψ ⋄P■ µ =❞❡❢

• ❉ψ∈P■ψ

♠✐♥(❚❡r♠s❇❛s❡❞❖♥(❉ψ, µ), ❉ψ) KM (U1)-(U6) + (U8) hold and also (U7-T) & (U9-T)

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Relevant Belief Update (1/3)

“Second reading” of Parikh’s postulate for Update: (P-UT) Let ψ = ϕ ∧ ϕ′ such that ▲❛♥❣(ϕ) ∩ ▲❛♥❣(ϕ′) = ∅. If ▲❛♥❣(µ) ⊆ ▲❛♥❣(ϕ), then ψ ⋄P■ µ ≡ (ϕ ⋄′

P■ µ) ∧ ϕ′, where ⋄′ P■ is the PI

update operator restricted to language ▲❛♥❣(ϕ). Remaining task: how to translate this postulate in terms of constraint(s) on faithful assignment

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Relevant Belief Update (2/3)

Constraining faithul assignment (constraint (CUS-T)): Suppose ❉ ≡ ❉✶ ∧ ❉✷ and ❉✶ Suppose two terms ❉′, ❉′′ ∈ ❚❡r♠s❇❛s❡❞❖♥(❉✶, µ) s.t. ❉′ ❉✶ ❉′′. Relevance states that adding ❉✷ to ❉′ and ❉′′ should not switch the preferences related to ❉′ and ❉′′ (if ❉✷ is expressed with symbols that differ from the symbols of ❉✶) ❉′ ∪ ❉✷ ❉ ❉′′ ∪ ❉✷

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Relevant Belief Update (3/3)

Main results: Faithful Assign. + Constraint (CUS-T) ⇐ ⇒ ⋄■P + Postulate (PU-T) Forbus ⋄ is not relevant. Forbus ⋄■P is relevant.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

Illustration: revisiting the example

P■ψ = (♣✷ ∧ ♣✸ ∧ ♣✺) ∨ (♣✹ ∧ ♣✺) P■µ = (♣✶ ∧ ♣✷ ∧ ¬♣✸) ∨ (¬♣✶ ∧ ¬♣✷ ∧ ¬♣✸). Faithful assignment with pre-orders ❉ψ: {♣✶, ♣✷, ¬♣✸, ♣✺} <{♣✷,♣✸,♣✺} {¬♣✶, ¬♣✷, ¬♣✸, ♣✺} {♣✶, ♣✷, ¬♣✸, ♣✹, ♣✺} {♣✹,♣✺} {¬♣✶, ¬♣✷, ¬♣✸, ♣✹, ♣✺} Let ⋄❋♦

P■ be the PI update operator:

ψ ⋄❋♦

P■ µ =

(♣✶ ∧ ♣✷ ∧ ¬♣✸ ∧ ♣✺)∨ (♣✶ ∧ ♣✷ ∧ ¬♣✸ ∧ ♣✹ ∧ ♣✺)∨ (¬♣✶ ∧ ¬♣✷ ∧ ¬♣✸ ∧ ♣✹ ∧ ♣✺) Remind: ψ ⋄❋♦ µ = (♣✶ ∧ ♣✷ ∧ ¬♣✸ ∧ ♣✺) ∨ (¬♣✶ ∧ ¬♣✷ ∧ ¬♣✸ ∧ ♣✺)

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update

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.

• L. Perrussel, J. Marchi, J.M. Thévenin, D. Zhang

Relevant Minimal Change in Belief Update