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 of 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 : Express Belief Update with the help of Prime Implicants. 1 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). Extend the notion of Relevance to Belief Update. 2 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 ) = ❚❡r♠s❇❛s❡❞❖♥ ( ❉ ψ ) ❉ ψ ∈ Pr✐♠❡■♠♣❧✐❝❛♥ts ( 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 only 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: Relevance satisfaction: If faithful assignment satisfies 1 constraint (CS-T) then ◦ P■ satisfies postulate (P) (Parikh’s relevance postulate). Dalal is then relevant since (CS-T) holds for Dalal. 2 Immediate question : Is it the same with belief update? Naive translation of Parikh for Update. 1 Reprashing with prime implicants. 2 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 ]] = ♠✐♥ ([[ µ ]] , � ✇ ) ✇ ∈ [[ ψ ]] 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
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