ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison An epistemic extension of equilibrium logic and its relation to Gelfond’s epistemic specifications Andreas Herzig University of Toulouse and CNRS, IRIT, France (joint work with Luis Fari˜ nas del Cerro and Ezgi Iraz Su; paper @ IJCAI 2015) Dagstuhl, May 2015 1 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Outline Answer-set programming and Gelfond’s epistemic 1 specifications 2 HT models and equilibrium models Epistemic HT models and equilibrium models 3 Comparison by examples 4 2 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Answer-set programming (ASP) logic program = set of rules rules of form Head ← Body Body may contain “ not ” ‘default negation’, ‘negation by failure’ = non-deducibility semantics: not every classical model of a program intended Π = { p ← not q } should have unique model { p } Π = { p ← not p } should have no model Π = { p ← p } should have unique model ∅ (all variables false) consensus only in the 90ies: answer set semantics fixed point definition: V is an answer set for Π iff V = reduct (Π , V ) nonmonotonic inference relation remarkably stable: 10+ different characterisations [Lifschitz ”Twelve Definfitions of a Stable Model”, ICLP 08] 3 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Answer-set programming (ASP) logic program = set of rules rules of form Head ← Body Body may contain “ not ” ‘default negation’, ‘negation by failure’ = non-deducibility semantics: not every classical model of a program intended Π = { p ← not q } should have unique model { p } Π = { p ← not p } should have no model Π = { p ← p } should have unique model ∅ (all variables false) consensus only in the 90ies: answer set semantics fixed point definition: V is an answer set for Π iff V = reduct (Π , V ) nonmonotonic inference relation remarkably stable: 10+ different characterisations [Lifschitz ”Twelve Definfitions of a Stable Model”, ICLP 08] 3 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison ASP lacks expressivity Example (scholarship eligibility program [Gelfond 94] ) eligible ← highGPA 1 eligible ← fairGPA , minority 2 eligible ← fairGPA , highGPA 3 interview ← not eligible , not eligible 4 highGPA or fairGPA ← 5 has the answer sets � AS (Π eligible ) = { highGPA , eligible } , � { fairGPA } Therefore: Π eligible | � interview ⇒ wanted: quantification over (possible) answer sets. . . 4 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison ASP lacks expressivity Example (scholarship eligibility program [Gelfond 94] ) eligible ← highGPA 1 eligible ← fairGPA , minority 2 eligible ← fairGPA , highGPA 3 interview ← not eligible , not eligible 4 highGPA or fairGPA ← 5 has the answer sets � AS (Π eligible ) = { highGPA , eligible } , � { fairGPA } Therefore: Π eligible | � interview ⇒ wanted: quantification over (possible) answer sets. . . 4 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Epistemic specifications [Gelfond 91, 94] Example (scholarship eligibility program, ES-version) eligible ← highGPA 1 eligible ← minority , fairGPA 2 eligible ← fairGPA , highGPA 3 interview ← not K eligible , not K eligible 4 highGPA or fairGPA ← 5 will have the answer sets � AS (Π K eligible ) = { highGPA , eligible , interview } , � { fairGPA , interview } Therefore: Π K eligible | ≈ interview 5 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Epistemic specifications [Gelfond 91, 94] Example (scholarship eligibility program, ES-version) eligible ← highGPA 1 eligible ← minority , fairGPA 2 eligible ← fairGPA , highGPA 3 interview ← not K eligible , not K eligible 4 highGPA or fairGPA ← 5 will have the answer sets � AS (Π K eligible ) = { highGPA , eligible , interview } , � { fairGPA , interview } Therefore: Π K eligible | ≈ interview 5 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Epistemic specifications: language rules of form Head ← Body Body may contain “ not ” Body may contain epistemic operators K q = “it is known that q ” M q = “ q may be believed” models: move from answer sets to world views = sets of answer sets 1 reduct Π W of an epistemic specification Π by a world view W 2 ⇒ eliminates the epistemic operators ⇒ procedural W is a world view of Π iff W = AS (Π W ) 3 ⇒ fixpoint equation ⇒ non-constructive 6 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Epistemic specifications: reduct definition [Kahl 14] Definition (reduct Π W of an epistemic specification Π by a world view W ) for each rule, literal in body: if true in W : if false in W : K l replace by l delete rule not K l replace by ⊤ replace by not l M l replace by ⊤ replace by not not l not M l replace by not l delete rule examples: { p ← K p } {∅} = ∅ { p ← K p } {{ p }} = { p ← p } { p ← M p } {∅} = { p ← not not p } { p ← M p } {{ p }} = { p ← ⊤} 7 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Outline Answer-set programming and Gelfond’s epistemic 1 specifications 2 HT models and equilibrium models Epistemic HT models and equilibrium models 3 Comparison by examples 4 8 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Which logical account of ASP? logic behind negation by failure? identify “ not ” with “ ¬ ” problem: classical negation doesn’t suit program { p ← not p } should have no model but equivalent to p in classical logic ⇒ logic of here-and-there (HT) minimisation of truth in HT models ⇒ equilibrium models 9 / 24
� � ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison The logic of here-and-there (HT) � • T H • simple Kripke models: only two possible worlds H (‘here’) and T (‘there’) accessibility relation is reflexive, and T is accessible from H idea: H = proved true, T = hypothesised, PVAR \ T = refuted intuitionistic heredity condition: H ⊆ T ⊆ PVAR language: connective ‘ → ’ stronger than material ‘ ⊃ ’: | = ¬ ϕ ↔ ( ϕ →⊥ ) | = ϕ → ¬¬ ϕ �| = ϕ ← ¬¬ ϕ �| = ϕ ∨ ¬ ϕ �| = ( ¬ ϕ → ϕ ) ↔ ϕ 10 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison The logic of here-and-there (HT) HT model = couple ( H , T ) such that H ⊆ T ⊆ PVAR H = T : ‘total model’ truth conditions: H , T | = p iff p ∈ H H , T | = ¬ ϕ iff T , T �| = ϕ H , T | = ϕ → ψ iff H , T | = ϕ ⊃ ψ and T , T | = ϕ ⊃ ψ (where ⊃ is material implication) Theorem (strong equivalence [Lifschitz et al. 01] ) AS (Π 1 ∪ Π) = AS (Π 2 ∪ Π) for every Π , iff | = HT Π 1 ↔ Π 2 11 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Equilibrium models = minimal HT models Definition ( T , T ) equilibrium model of ϕ iff T , T | = ϕ 1 H , T �| = ϕ for every H ⊂ T 2 Theorem ( [Pearce 96] ) equilibrium models of Π = answer sets of Π ⇒ t.b.d.: epistemic extension to capture epistemic specifications 12 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Outline Answer-set programming and Gelfond’s epistemic 1 specifications 2 HT models and equilibrium models Epistemic HT models and equilibrium models 3 Comparison by examples 4 13 / 24
ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison Epistemic equilibrium logic in a nutshell: combine HT and S5 1 semantics: sets of HT models intermediate modal logic (stronger than intuitionistic S5) maximise falsehood as in equilibrium logic: 2 ∅ | ≈ EE K ¬ p ≈ EE K ( p ∨ q ) p ∨ q | however: p ∨ q | � EE M p ∧ M q maximise ignorance: 3 p ∨ q | ≈ AEE M p ∧ M q ⇒ Autoepistemic [Moore 80] ⇒ “all-that-I-know” [Levesque 90] 14 / 24
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