Model checking and validity in propositional Jonni Virtema and - PowerPoint PPT Presentation

Model checking and validity in propositional and modal inclusion logics Model checking and validity in propositional Jonni Virtema and modal inclusion logics Movativation & History Inclusion logics Team Semantics Jonni Virtema Complexity Results Proof ideas Hasselt University, Belgium jonni.virtema@gmail.com References Joint work with Lauri Hella 1 , Antti Kuusisto 2 , and Arne Meier 3 1 University of Tampere, Finland, 2 University of Bremen, Germany, 3 University of Hanover, Germany 23rd of August, 2017 – MFCS 2017 1/ 19

Model checking Core of Team Semantics and validity in propositional and modal inclusion logics Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Movativation & E.g., History ◮ a first-order assignment in first-order logic, Inclusion logics ◮ a propositional assignment in propositional logic, Team Semantics ◮ a possible world of a Kripke structure in modal logic. Complexity Results Proof ideas ◮ In team semantics sets of states of affairs are considered. References E.g., ◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic. ◮ These sets of things are called teams. 2/ 19

Model checking Core of Team Semantics and validity in propositional and modal inclusion logics Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Movativation & E.g., History ◮ a first-order assignment in first-order logic, Inclusion logics ◮ a propositional assignment in propositional logic, Team Semantics ◮ a possible world of a Kripke structure in modal logic. Complexity Results Proof ideas ◮ In team semantics sets of states of affairs are considered. References E.g., ◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic. ◮ These sets of things are called teams. 2/ 19

Model checking Team Semantics: Motivation and History and validity in propositional Logical modelling of uncertainty, imperfect information, and different notions of and modal inclusion logics dependence such as functional dependence and independence. Related to similar Jonni Virtema concepts in statistics, database theory etc. Movativation & History Historical development: Inclusion logics ◮ Branching quantifiers by Henkin 1959. � ∀ x ∃ y Team Semantics � ϕ ( x , y , x ′ , y ′ ) Complexity Results ∀ x ′ ∃ y ′ Proof ideas ◮ Independence-friendly logic by Hintikka and Sandu 1989. References ∀ x ∃ y ∀ x ′ ∃ y ′ / { x , y } ϕ ( x , y , x ′ , y ′ ) ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ Dependence logic and modal dependence logic by V¨ a¨ an¨ anen 2007. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Generalised atoms by Kuusisto (derived from generalised quantifiers). ◮ Multiteam and polyteam semantics by Hannula et al. 3/ 19

Model checking Team Semantics: Motivation and History and validity in propositional and modal Logical modelling of uncertainty, imperfect information, and different notions of inclusion logics dependence such as functional dependence and independence. Related to similar Jonni Virtema concepts in statistics, database theory etc. Movativation & History Historical development: Inclusion logics ◮ Branching quantifiers by Henkin 1959. Team Semantics Complexity Results ◮ Independence-friendly logic by Hintikka and Sandu 1989. Proof ideas ◮ Compositional semantics for independence-friendly logic by Hodges 1997. References (Origin of team semantics.) ◮ Dependence logic and modal dependence logic by V¨ a¨ an¨ anen 2007. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Generalised atoms by Kuusisto (derived from generalised quantifiers). ◮ Multiteam and polyteam semantics by Hannula et al. 3/ 19

Model checking Inclusion logics in first-order setting and validity in propositional and modal inclusion logics Jonni Virtema We study logics with inclusion dependencies: Movativation & For a set of first-order assignments X History Inclusion logics ∀ s ∈ X ∃ s ′ ∈ X : s ( � x ) = s ′ ( � X | = � x ⊆ � y iff y ) . Team Semantics Complexity Results Proof ideas In first-order setting FO( ⊆ ) has very interesting properties: References ◮ FO( ⊆ ) has the same expressive power as posGFP. ◮ FO( ⊆ ) with strict semantics has the same expressive power as ESO. ◮ Fragments of FO( ⊆ ) with strict semantics capture NTIME RAM ( n k ), fixed k . 4/ 19

Model checking Inclusion logics in first-order setting and validity in propositional and modal inclusion logics Jonni Virtema We study logics with inclusion dependencies: Movativation & For a set of first-order assignments X History Inclusion logics ∀ s ∈ X ∃ s ′ ∈ X : s ( � x ) = s ′ ( � X | = � x ⊆ � y iff y ) . Team Semantics Complexity Results Proof ideas In first-order setting FO( ⊆ ) has very interesting properties: References ◮ FO( ⊆ ) has the same expressive power as posGFP. ◮ FO( ⊆ ) with strict semantics has the same expressive power as ESO. ◮ Fragments of FO( ⊆ ) with strict semantics capture NTIME RAM ( n k ), fixed k . 4/ 19

Model checking Inclusion logics in propositional setting and validity in propositional and modal inclusion logics Jonni Virtema ϕ, � For a set of propositional assignments X and � ψ ∈ PL Movativation & ∀ s ∈ X ∃ s ′ ∈ X : s ( � History ϕ ⊆ � ϕ ) = s ′ ( � X | = � ψ iff ψ ) . Inclusion logics Team Semantics In propositional setting PL( ⊆ ) and ML( ⊆ ) have interesting properties: Complexity Results ◮ PL( ⊆ ) definable classes of propositional teams are exactly those C s.t. Proof ideas ◮ ∅ ∈ C and References ◮ C is union closed ( X ∈ C , Y ∈ C ⇒ X ∪ Y ∈ C ). ◮ ML( ⊆ ) definable classes of Kripke models with teams are those C s.t. ◮ (K , ∅ ) ∈ C , for every K, ◮ C is union closed ((K , X ) ∈ C , (K , Y ) ∈ C ⇒ (K , X ∪ Y ) ∈ C ), ◮ C is closed under team k -bisimulation for some k . 5/ 19

Model checking Inclusion logics in propositional setting and validity in propositional and modal inclusion logics Jonni Virtema ϕ, � For a set of propositional assignments X and � ψ ∈ PL Movativation & ∀ s ∈ X ∃ s ′ ∈ X : s ( � History ϕ ⊆ � ϕ ) = s ′ ( � X | = � ψ iff ψ ) . Inclusion logics Team Semantics In propositional setting PL( ⊆ ) and ML( ⊆ ) have interesting properties: Complexity Results ◮ PL( ⊆ ) definable classes of propositional teams are exactly those C s.t. Proof ideas ◮ ∅ ∈ C and References ◮ C is union closed ( X ∈ C , Y ∈ C ⇒ X ∪ Y ∈ C ). ◮ ML( ⊆ ) definable classes of Kripke models with teams are those C s.t. ◮ (K , ∅ ) ∈ C , for every K, ◮ C is union closed ((K , X ) ∈ C , (K , Y ) ∈ C ⇒ (K , X ∪ Y ) ∈ C ), ◮ C is closed under team k -bisimulation for some k . 5/ 19

Model checking Inclusion logics in propositional setting and validity in propositional and modal inclusion logics Jonni Virtema ϕ, � For a set of propositional assignments X and � ψ ∈ PL Movativation & ∀ s ∈ X ∃ s ′ ∈ X : s ( � History ϕ ⊆ � ϕ ) = s ′ ( � X | = � ψ iff ψ ) . Inclusion logics Team Semantics In propositional setting PL( ⊆ ) and ML( ⊆ ) have interesting properties: Complexity Results ◮ PL( ⊆ ) definable classes of propositional teams are exactly those C s.t. Proof ideas ◮ ∅ ∈ C and References ◮ C is union closed ( X ∈ C , Y ∈ C ⇒ X ∪ Y ∈ C ). ◮ ML( ⊆ ) definable classes of Kripke models with teams are those C s.t. ◮ (K , ∅ ) ∈ C , for every K, ◮ C is union closed ((K , X ) ∈ C , (K , Y ) ∈ C ⇒ (K , X ∪ Y ) ∈ C ), ◮ C is closed under team k -bisimulation for some k . 5/ 19

Model checking Propositional team semantics and validity in propositional and modal Syntax of propositional logic: inclusion logics Jonni Virtema ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) Movativation & History Semantics via propositional assignments: Inclusion logics Team Semantics p q r Complexity Results s | = q ∧ r 0 1 1 s Proof ideas References Team semantics / semantics via sets of assignments: p q r s 0 1 1 { s , t , u } | = q , { s , t } | = p ∨ r 1 1 0 t u 0 1 0 6/ 19

Model checking Team semantics and validity in propositional and modal inclusion logics Jonni Virtema Movativation & History Inclusion logics We want that for each formula ϕ of propositional logic and for each team X Team Semantics Complexity Results X | = ϕ iff ∀ s ∈ X : s | = ϕ. Proof ideas References 7/ 19