Combining equilibrium logic and dynamic logic (an introduction and a - PowerPoint PPT Presentation

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Combining equilibrium logic and dynamic logic (an introduction and a very brief overview) Luis Fariñas del Cerro, Andreas Herzig and Ezgi Iraz Su University of Toulouse, IRIT-CNRS, France Toulouse, July 5, 2013 1 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Outline Introduction 1 HT logic and equilibrium logic 2 A dynamic extension of HT logic and of equilibrium logic 3 DL-PA: dynamic logic of propositional assignments 4 Relating D-HT and DL-PA 5 Conclusion 6 2 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Motivation motivation in a broad sense: recent studies reveals: Answer Set Programming (ASP): central to various approaches in non-monotonic reasoning equilibrium logic: semantical framework for ASP [Pearce, Lifschitz, . . . ] then, need for an extension of the language of ASP ... (with some supportive concepts like:) the representations of modalities actions ontologies updates main goal: beyond updates, adding other modalities to equilibrium logic (and, via that, to ASP) motivation, in particular, for this work: the update of answer set programs aim (here): manage atomic change of equi. models 3 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Outline Introduction 1 HT logic and equilibrium logic 2 A dynamic extension of HT logic and of equilibrium logic 3 DL-PA: dynamic logic of propositional assignments 4 Relating D-HT and DL-PA 5 Conclusion 6 4 / 22

� Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Here-and-there models a bit history: (idea [Gödel]) strength of “ → ” between material implication “ ⊃ ” and intuitionistic implication “ ⇒ ” � � here-and-there model (HT model) = H , T : H , T sets of propositional variables from P with H ⊆ T { p , q } ‘there’ ∅ ‘here’ truth conditions: H , T | = p iff p ∈ H H , T �| = ⊥ H , T | = ϕ ∧ ψ H , T | = ϕ and H , T | = ψ iff H , T | = ϕ ∨ ψ iff H , T | = ϕ or H , T | = ψ H , T | = ϕ → ψ H , T | = ϕ ⊃ ψ and T , T | = ϕ ⊃ ψ iff ( H , T ) is a HT model of ϕ iff H , T | = ϕ 5 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Equilibrium models Definition T ⊆ P is an equilibrium model of ϕ iff ( T , T ) is a HT model of ϕ ; 1 (minimality condition) no ( H , T ) with H ⊂ T is a HT model of ϕ . 2 example: T = ∅ is an equilibrium model of ¬ p = p → ⊥ : ∅ , ∅ | = p → ⊥ 1 min. cnd. always satisfied for T = ∅ . 2 actually the only one: e.g. for T = { q } we have ∅ , { q } | = p → ⊥ 3 p ∨ q has only 2 equi. models, namely T = { p } and T = { q } . ¬¬ p has no equilibrium model: { p } , { p } | = ¬¬ p 1 however, min. cnd. fails since ∅ , { p } | = ¬¬ p . 2 6 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Equilibrium models Definition T ⊆ P is an equilibrium model of ϕ iff ( T , T ) is a HT model of ϕ ; 1 (minimality condition) no ( H , T ) with H ⊂ T is a HT model of ϕ . 2 example: T = ∅ is an equilibrium model of ¬ p = p → ⊥ : ∅ , ∅ | = p → ⊥ 1 min. cnd. always satisfied for T = ∅ . 2 actually the only one: e.g. for T = { q } we have ∅ , { q } | = p → ⊥ 3 p ∨ q has only 2 equi. models, namely T = { p } and T = { q } . ¬¬ p has no equilibrium model: { p } , { p } | = ¬¬ p 1 however, min. cnd. fails since ∅ , { p } | = ¬¬ p . 2 6 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Equilibrium logic χ | = ϕ : logical consequence in HT models χ | ≈ ϕ : logical consequence in equilibrium models Definition ≈ ϕ iff for every equil. model T of χ , ( T , T ) is HT model of ϕ . χ | example: ⊤ | ≈ ¬ p and ¬ p → q | ≈ q 7 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Equilibrium logic χ | = ϕ : logical consequence in HT models χ | ≈ ϕ : logical consequence in equilibrium models Definition ≈ ϕ iff for every equil. model T of χ , ( T , T ) is HT model of ϕ . χ | example: ⊤ | ≈ ¬ p and ¬ p → q | ≈ q 7 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Outline Introduction 1 HT logic and equilibrium logic 2 A dynamic extension of HT logic and of equilibrium logic 3 DL-PA: dynamic logic of propositional assignments 4 Relating D-HT and DL-PA 5 Conclusion 6 8 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional The language L D-HT extension of L HT with dynamic modalities: (common to D-HT and dynamic equilibrium logic) : ϕ � p | ⊥ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | [ π ] ϕ | � π � ϕ L D-HT π � + p | − p | π ; π | π ∪ π | π ∗ | ϕ ? where p ranges over P . atomic programs: + p and − p minimally update an HT model ( H , T ) abbreviations: = ¬ ϕ ϕ → ⊥ = ⊤ ⊥ → ⊥ 9 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Dynamic here-and-there logic (1) notation: HT = { ( H , T ) : H ⊆ T ⊆ P } : the set of all HT models interpretation of formulas and programs in D - HT : for a formula ϕ , � ϕ � D-HT ⊆ HT . examples: �¬ p � D-HT = { ( H , T ) : p � T } 1 (and so p � H by the heredity constraint in HT logic, i.e., H ⊆ T ) � p ∨ ¬ p � D-HT = { ( H , T ) : p ∈ H or p � T } 2 (trivial, but important) �¬¬ p � D-HT = { ( H , T ) : p ∈ T } 3 (and therefore, upshot: � p � D-HT ⊂ �¬¬ p � D-HT ) for a program π , � π � D-HT is a relation on HT . 10 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Dynamic here-and-there logic (1) notation: HT = { ( H , T ) : H ⊆ T ⊆ P } : the set of all HT models interpretation of formulas and programs in D - HT : for a formula ϕ , � ϕ � D-HT ⊆ HT . examples: �¬ p � D-HT = { ( H , T ) : p � T } 1 (and so p � H by the heredity constraint in HT logic, i.e., H ⊆ T ) � p ∨ ¬ p � D-HT = { ( H , T ) : p ∈ H or p � T } 2 (trivial, but important) �¬¬ p � D-HT = { ( H , T ) : p ∈ T } 3 (and therefore, upshot: � p � D-HT ⊂ �¬¬ p � D-HT ) for a program π , � π � D-HT is a relation on HT . 10 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Dynamic here-and-there logic (1) notation: HT = { ( H , T ) : H ⊆ T ⊆ P } : the set of all HT models interpretation of formulas and programs in D - HT : for a formula ϕ , � ϕ � D-HT ⊆ HT . examples: �¬ p � D-HT = { ( H , T ) : p � T } 1 (and so p � H by the heredity constraint in HT logic, i.e., H ⊆ T ) � p ∨ ¬ p � D-HT = { ( H , T ) : p ∈ H or p � T } 2 (trivial, but important) �¬¬ p � D-HT = { ( H , T ) : p ∈ T } 3 (and therefore, upshot: � p � D-HT ⊂ �¬¬ p � D-HT ) for a program π , � π � D-HT is a relation on HT . 10 / 22

Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Dynamic here-and-there logic (1) notation: HT = { ( H , T ) : H ⊆ T ⊆ P } : the set of all HT models interpretation of formulas and programs in D - HT : for a formula ϕ , � ϕ � D-HT ⊆ HT . examples: �¬ p � D-HT = { ( H , T ) : p � T } 1 (and so p � H by the heredity constraint in HT logic, i.e., H ⊆ T ) � p ∨ ¬ p � D-HT = { ( H , T ) : p ∈ H or p � T } 2 (trivial, but important) �¬¬ p � D-HT = { ( H , T ) : p ∈ T } 3 (and therefore, upshot: � p � D-HT ⊂ �¬¬ p � D-HT ) for a program π , � π � D-HT is a relation on HT . 10 / 22

� � � � Introduction HT logic and equilibrium logic A dynamic extension of HT logic and of equilibrium logic DL-PA: dynamic logic of propositional Dynamic here-and-there logic (2) interpretation of atomic update operations: upgrade p: + p executable, viz. when p � H (‘here’) (ex: all, but black below) downgrade p: same for − p , viz. when p ∈ T (‘there’) (ex: all, but blue below) � � ∅ , { p , q } ���������� � �− p � D-HT � �− p � D-HT � � � � + r � D-HT � � � � � � � � � � � � { p } , { p , q } ∅ , { p , q , r } ∅ , { q } � + r � D-HT � � { r } , { p , q , r } remark: not allowed to apply − p to blue again, neither + r to green... Similarly, neither + p to black, nor − r to orange... 11 / 22