Extending answer set programming using generalized possibilistic - PowerPoint PPT Presentation

Extending answer set programming using generalized possibilistic logic Steven Schockaert (joint work with Didier Dubois and Henri Prade) School of Computer Science & Informatics Cardi ff University, Cardi ff , UK schockaerts1@cardi ff .ac.uk http://users.cs.cf.ac.uk/S.Schockaert

1. Possibilistic logic 2. Generalized possibilistic logic 3. Answer set programming 4. Extending answer set programming

Possibilistic logic: syntax Knowledge bases in possibilistic logic are sets of weighted formulas of the form: ( p ∧ ( ¬ q → r ) , 0 . 7) propositional formula certainty degree, taken from Λ = { 0 , 1 k , 2 k , ..., 1 }

Possibilistic logic: syntax Knowledge bases in possibilistic logic are sets of weighted formulas of the form: ( p ∧ ( ¬ q → r ) , 0 . 7) K = { ( loc ( IJCAI2015 , Argentina ) , 1) , ( loc ( IJCAI2015 , Argentina ) → loc ( IJCAI2015 , SouthAmerica ) , 1) , ( loc ( IJCAI2016 , NewYork ) → loc ( IJCAI2016 , NorthAmerica ) , 1) , ( loc ( IJCAI2015 , SouthAmerica ) → ¬ loc ( IJCAI2016 , SouthAmerica ) , 0 . 75) , ( loc ( IJCAI2015 , SouthAmerica ) → ¬ loc ( IJCAI2016 , NorthAmerica ) , 0 . 5) } reflects my degree of surprise if I found out that the formula were false

Possibilistic logic: syntax Certainty degrees are interpreted qualitatively, i.e. possibilistic logic theories can be seen as stratified classical logic theories → ¬ loc ( IJCAI2015 , Argentina ) loc ( IJCAI2015 , Argentina ) → loc ( IJCAI2015 , SouthAmerica ) loc ( IJCAI2016 , NewYork ) → loc ( IJCAI2016 , NorthAmerica ) loc ( IJCAI2015 , SouthAmerica ) → ¬ loc ( IJCAI2016 , SouthAmerica ) loc ( IJCAI2015 , SouthAmerica ) → ¬ loc ( IJCAI2016 , NorthAmerica ) Using numbers makes it easier to describe the semantics and to formulate inference rules

Possibilistic logic: semantics At the semantic level, the role of models is taken up by possibility distributions , which assign to every propositional interpretation (or possible world) a degree of possibility, e.g. π ( ω 1 ) = 0 π ( ω 2 ) = 0 . 4 π ( ω 3 ) = 0 . 7 π ( ω 4 ) = 1 Intuitively, a model corresponds to the epistemic state of an agent, encoded as the weighted set of worlds it considers possible

Possibilistic logic: semantics If π represents the epistemic state of an agent, then that agent considers a formula α possible to the degree that some model of α is considered possible { | | } ¬ − Π ( α ) = Π ( { ω | ω | = α } ) = max { π ( ω ) | ω | = α } possibility measure

Possibilistic logic: semantics If π represents the epistemic state of an agent, then that agent considers a formula α possible to the degree that some model of α is considered possible { | | } ¬ − Π ( α ) = Π ( { ω | ω | = α } ) = max { π ( ω ) | ω | = α } The agent considers the formula necessary to the degree that all counter-models of α are impossible N ( α ) = N ( { ω | ω | = α } ) = min { 1 � π ( ω ) | ω 6 | = α } = 1 � Π ( ¬ α ) necessity measure

Possibilistic logic: semantics A possibility distribution π satisfies a formula ( α , λ ) iff N( α ) ≥ λ , with N the necessity measure induced by π A possibility distribution π is called a model of a set of formulas K iff π satisfies all formulas in K

Possibilistic logic: semantics A possibility distribution π satisfies a formula ( α , λ ) iff N( α ) ≥ λ , with N the necessity measure induced by π A possibility distribution π is called a model of a set of formulas K iff π satisfies all formulas in K K = { ( a, 0 . 8) , ( a ! b, 1) } π ( { a, b } ) = 1 π ( { a } ) = 0 π ( { b } ) = 0 . 2 π ( {} ) = 0 . 2

Possibilistic logic: semantics A possibility distribution π satisfies a formula ( α , λ ) iff N( α ) ≥ λ , with N the necessity measure induced by π A possibility distribution π is called a model of a set of formulas K iff π satisfies all formulas in K K = { ( a, 0 . 8) , ( a ! b, 1) } N ( a ) = 1 � Π ( ¬ a ) = 1 � max( π ( { b } ) , π ( {} )) = 0 . 8 π ( { a, b } ) = 1 π ( { a } ) = 0 N ( a ! b ) = 1 � Π ( a ^ ¬ b ) π ( { b } ) = 0 . 2 = 1 � max( π ( { a } )) π ( {} ) = 0 . 2 = 1

Possibilistic logic: semantics π 1 is less specific than π 2 if ⌃ ω . π 1 ( ω ) ⇥ π 2 ( ω ) ⌃ ω . π ( ω ) = 1 Completely uninformative: every world remains possible π ( ω 0 ) = 1 and ⌃ ω ⇧ = ω 0 . π ( ω ) = 0 Maximally informative: exactly one world is considered possible

Possibilistic logic: semantics π 1 is less specific than π 2 if ⌃ ω . π 1 ( ω ) ⇥ π 2 ( ω ) ⌃ ω . π ( ω ) = 1 Completely uninformative: every world remains possible π ( ω 0 ) = 1 and ⌃ ω ⇧ = ω 0 . π ( ω ) = 0 Maximally informative: exactly one world is considered possible Every consistent set of formulas K in possibilistic logic has a unique least specific model π K

Possibilistic logic: inference inference rules if ( α , λ ) 2 K then K ` ( α , λ ) ( if α ⌘ β and K ` ( α , λ ) then K ` ( β , λ ) ( if λ 1 � λ 2 and K ` ( α , λ 1 ) then K ` ( α , λ 2 ) ( if K ` ( α _ β , λ 1 ) and K ` ( ¬ α _ γ , λ 2 ) then K ` ( β _ γ , min( λ 1 , λ 2 )) possibilistic logic is based on a weakest link idea

Possibilistic logic: inference inference rules if ( α , λ ) 2 K then K ` ( α , λ ) ( if α ⌘ β and K ` ( α , λ ) then K ` ( β , λ ) ( if λ 1 � λ 2 and K ` ( α , λ 1 ) then K ` ( α , λ 2 ) ( if K ` ( α _ β , λ 1 ) and K ` ( ¬ α _ γ , λ 2 ) then K ` ( β _ γ , min( λ 1 , λ 2 )) soundness and completeness The following statements are equivalent: 1. K ` ( α , λ ) can be derived from (1)–(4). 2. Every model π of K is a model of ( α , λ ). 3. The least specific model π K of K is a model of ( α , λ ).

Possibilistic logic: applications Possibilistic logic is closely related to AGM belief revision and the rational closure of default rules K = { ( loc ( IJCAI2015 , Argentina ) , 1) , ( loc ( IJCAI2015 , Argentina ) → loc ( IJCAI2015 , SouthAmerica ) , 1) , ( loc ( IJCAI2016 , NewYork ) → loc ( IJCAI2016 , NorthAmerica ) , 1) , ( loc ( IJCAI2015 , SouthAmerica ) → ¬ loc ( IJCAI2016 , SouthAmerica ) , 0 . 75) , ( loc ( IJCAI2015 , SouthAmerica ) → ¬ loc ( IJCAI2016 , NorthAmerica ) , 0 . 5) } + ( loc ( IJCAI2016 , NewYork ) , 1)

Possibilistic logic: applications Possibilistic logic is closely related to AGM belief revision and the rational closure of default rules K = { ( loc ( IJCAI2015 , Argentina ) , 1) , ( loc ( IJCAI2016 , NewYork ) , 1) , ( loc ( IJCAI2015 , Argentina ) ! loc ( IJCAI2015 , SouthAmerica ) , 1) , ( loc ( IJCAI2016 , NewYork ) ! loc ( IJCAI2016 , NorthAmerica ) , 1) , ( loc ( IJCAI2015 , SouthAmerica ) ! ¬ loc ( IJCAI2016 , SouthAmerica ) , 0 . 75) , ( loc ( IJCAI2015 , SouthAmerica ) ! ¬ loc ( IJCAI2016 , NorthAmerica ) , 0 . 5) }

Possibilistic logic: applications Possibilistic logic is closely related to AGM belief revision and the rational closure of default rules K = { ( loc ( IJCAI2015 , Argentina ) , 1) , ( loc ( IJCAI2016 , NewYork ) , 1) , ( loc ( IJCAI2015 , Argentina ) ! loc ( IJCAI2015 , SouthAmerica ) , 1) , ( loc ( IJCAI2016 , NewYork ) ! loc ( IJCAI2016 , NorthAmerica ) , 1) , ( loc ( IJCAI2015 , SouthAmerica ) ! ¬ loc ( IJCAI2016 , SouthAmerica ) , 0 . 75) , ( loc ( IJCAI2015 , SouthAmerica ) ! ¬ loc ( IJCAI2016 , NorthAmerica ) , 0 . 5) }

1. Possibilistic logic 2. Generalized possibilistic logic 3. Answer set programming 4. Extending answer set programming

Generalized possibilistic logic In possibilistic logic , a formula ( α , λ ) corresponds to the constraint N( α ) ≥ λ at the semantic level, and a knowledge base corresponds to a conjunction of such constraints Π ( α ) ≥ λ

Generalized possibilistic logic In possibilistic logic , a formula ( α , λ ) corresponds to the constraint N( α ) ≥ λ at the semantic level, and a knowledge base corresponds to a conjunction of such constraints In generalized possibilistic logic (GPL), arbitrary propositional combinations of such formulas are allowed To emphasize the view of possibilistic logic as a modal logic , we use the following notations ) means that alternative notation for ( α , λ ) whereas N λ ( α ) with ailable beliefs suggests that abbreviation for ¬ N 1 − λ + 1 while Π λ ( α ) with k ( ¬ α ) ( Π ( ) ⌅ corresponds to the constraint Π ( α ) ≥ λ

Generalized possibilistic logic Syntax � ⇥ N 0 . 4 ( a ∧ ¬ b ) ∨ Π 0 . 3 ( a → ( b ∨ c )) → N 0 . 7 ( b ) Note: every propositional atom is encapsulated by a modality Note: no nesting of modalities

Generalized possibilistic logic Syntax � ⇥ N 0 . 4 ( a ∧ ¬ b ) ∨ Π 0 . 3 ( a → ( b ∨ c )) → N 0 . 7 ( b ) Note: every propositional atom is encapsulated by a modality Note: no nesting of modalities � ⇥ Semantics � ⇥ N ( a ∧ ¬ b ) < 0 . 4 ∧ Π ( a → ( b ∨ c )) < 0 . 3 ∨ N ( b ) ≥ 0 . 7 Note: models are possibility distributions, which are interpreted as epistemic states Axiomatization Weighted version of the axiomatization of the modal logic KD without introspection (KR 2012)

possibilistic logic GPL express propositional express lower bounds on formulas: combinations of lower the necessity of a bounds on the necessity of propositional formula a propositional formula models: weighted epistemic weighted epistemic states states minimally specific unique 0, 1 or more models: useful for reasoning useful for reasoning about the consequences about the revealed beliefs of one’s own beliefs of another agent