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On quantified propositional logics and the exponential time hierarchy On quantified propositional logics Jonni Virtema and the exponential time hierarchy Team Semantics Quantified propositional logic Dependency atoms Jonni Virtema Expressive Power Complexity University of Helsinki, Finland jonni.virtema@gmail.com Exponential hierarchy Joint work with Miika Hannula, Juha Kontinen, and Martin L¨ uck GandALF 2016 15th of September, 2016 1/ 16

On quantified Core of Team Semantics propositional logics and the exponential time hierarchy Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Team Semantics E.g., Quantified ◮ a first-order assignment in first-order logic, propositional logic ◮ a propositional assignment in propositional logic, Dependency atoms ◮ a possible world of a Kripke structure in modal logic. Expressive Power Complexity ◮ In team semantics sets of states of affairs are considered. Exponential E.g., hierarchy ◮ 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/ 16

On quantified Core of Team Semantics propositional logics and the exponential time hierarchy Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Team Semantics E.g., Quantified ◮ a first-order assignment in first-order logic, propositional logic ◮ a propositional assignment in propositional logic, Dependency atoms ◮ a possible world of a Kripke structure in modal logic. Expressive Power Complexity ◮ In team semantics sets of states of affairs are considered. Exponential E.g., hierarchy ◮ 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/ 16

On quantified Core of Team Semantics propositional logics and the exponential time hierarchy Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Team Semantics E.g., Quantified ◮ a first-order assignment in first-order logic, propositional logic ◮ a propositional assignment in propositional logic, Dependency atoms ◮ a possible world of a Kripke structure in modal logic. Expressive Power Complexity ◮ In team semantics sets of states of affairs are considered. Exponential E.g., hierarchy ◮ 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/ 16

On quantified Team Semantics: Motivation and History propositional logics and the exponential time Logical modelling of uncertainty, imperfect information, and different notions of hierarchy dependence such as functional dependence and independence. Related to similar Jonni Virtema concepts in statistics, database theory etc. Team Semantics Quantified Historical development: propositional logic ◮ Branching quantifiers by Henkin 1959. Dependency atoms Expressive Power ◮ Independence-friendly logic by Hintikka and Sandu 1989. Complexity ◮ Compositional semantics for independence-friendly logic by Hodges 1997. Exponential hierarchy (Origin of team semantics.) ◮ Dependence logic by V¨ a¨ an¨ anen 2007. ◮ Modal dependence logic by V¨ a¨ an¨ anen 2008. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Generalized atoms by Kuusisto (derived from generalised quantifiers). 3/ 16

On quantified Team Semantics: Motivation and History propositional logics and the exponential time Logical modelling of uncertainty, imperfect information, and different notions of hierarchy dependence such as functional dependence and independence. Related to similar Jonni Virtema concepts in statistics, database theory etc. Team Semantics Quantified Historical development: propositional logic ◮ Branching quantifiers by Henkin 1959. Dependency atoms Expressive Power ◮ Independence-friendly logic by Hintikka and Sandu 1989. Complexity ◮ Compositional semantics for independence-friendly logic by Hodges 1997. Exponential hierarchy (Origin of team semantics.) ◮ Dependence logic by V¨ a¨ an¨ anen 2007. ◮ Modal dependence logic by V¨ a¨ an¨ anen 2008. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Generalized atoms by Kuusisto (derived from generalised quantifiers). 3/ 16

On quantified Quantified propositional logic propositional logics and the exponential time hierarchy Jonni Virtema Grammar of quantified propositional logic QPL (or QBF) in negation normal form: Team Semantics ϕ ::= p | ¬ p | ( ϕ ∨ ϕ ) | ( ϕ ∧ ϕ ) | ∃ p ϕ | ∀ p ϕ. Quantified propositional logic A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Dependency atoms domain. Expressive Power Complexity We want to define team semantics for QPL s.t. we have the following property Exponential hierarchy ( flattness ): If ϕ is an QPL -formula and X a set of propositional assignments: X | = ϕ ⇐ ⇒ ∀ s ∈ X : s | = ϕ. 4/ 16

On quantified Quantified propositional logic propositional logics and the exponential time hierarchy Jonni Virtema Grammar of quantified propositional logic QPL (or QBF) in negation normal form: Team Semantics ϕ ::= p | ¬ p | ( ϕ ∨ ϕ ) | ( ϕ ∧ ϕ ) | ∃ p ϕ | ∀ p ϕ. Quantified propositional logic A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Dependency atoms domain. Expressive Power Complexity We want to define team semantics for QPL s.t. we have the following property Exponential hierarchy ( flattness ): If ϕ is an QPL -formula and X a set of propositional assignments: X | = ϕ ⇐ ⇒ ∀ s ∈ X : s | = ϕ. 4/ 16

On quantified Team Semantics for Propositional Logics propositional logics and the exponential time hierarchy Jonni Virtema A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Team Semantics domain. Quantified propositional logic s | ⇔ = p s ( p ) = 1 Dependency atoms s | = ¬ p ⇔ s ( p ) = 0 Expressive Power Complexity s | = ϕ ∧ ψ ⇔ s | = ϕ and s | = ψ Exponential s | = ϕ ∨ ψ ⇔ s | = ϕ or s | = ψ hierarchy s | = ∃ p ϕ ⇔ s ( b / p ) | = ϕ for some b ∈ { 0 , 1 } s | = ∀ p ϕ ⇔ s ( b / p ) | = ϕ for all b ∈ { 0 , 1 } 5/ 16

On quantified Team Semantics for Propositional Logics propositional logics and the exponential time hierarchy Jonni Virtema A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Team Semantics domain. Quantified propositional logic X | ⇔ ∀ s ∈ X : s ( p ) = 1 = p Dependency atoms s | = ¬ p ⇔ s ( p ) = 0 Expressive Power Complexity s | = ϕ ∧ ψ ⇔ s | = ϕ and s | = ψ Exponential s | = ϕ ∨ ψ ⇔ s | = ϕ or s | = ψ hierarchy s | = ∃ p ϕ ⇔ s ( b / p ) | = ϕ for some b ∈ { 0 , 1 } s | = ∀ p ϕ ⇔ s ( b / p ) | = ϕ for all b ∈ { 0 , 1 } 5/ 16

On quantified Team Semantics for Propositional Logics propositional logics and the exponential time hierarchy Jonni Virtema A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Team Semantics domain. Quantified propositional logic X | ⇔ ∀ s ∈ X : s ( p ) = 1 = p Dependency atoms X | = ¬ p ⇔ ∀ s ∈ X : s ( p ) = 0 Expressive Power Complexity s | = ϕ ∧ ψ ⇔ s | = ϕ and s | = ψ Exponential s | = ϕ ∨ ψ ⇔ s | = ϕ or s | = ψ hierarchy s | = ∃ p ϕ ⇔ s ( b / p ) | = ϕ for some b ∈ { 0 , 1 } s | = ∀ p ϕ ⇔ s ( b / p ) | = ϕ for all b ∈ { 0 , 1 } 5/ 16

On quantified Team Semantics for Propositional Logics propositional logics and the exponential time hierarchy Jonni Virtema A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Team Semantics domain. Quantified propositional logic X | ⇔ ∀ s ∈ X : s ( p ) = 1 = p Dependency atoms X | = ¬ p ⇔ ∀ s ∈ X : s ( p ) = 0 Expressive Power Complexity X | = ϕ ∧ ψ ⇔ X | = ϕ and X | = ψ Exponential s | = ϕ ∨ ψ ⇔ s | = ϕ or s | = ψ hierarchy s | = ∃ p ϕ ⇔ s ( b / p ) | = ϕ for some b ∈ { 0 , 1 } s | = ∀ p ϕ ⇔ s ( b / p ) | = ϕ for all b ∈ { 0 , 1 } 5/ 16

On quantified Team Semantics for Propositional Logics propositional logics and the exponential time hierarchy Jonni Virtema A propositional team is a set of assigments s : PROP → { 0 , 1 } with the same Team Semantics domain. Quantified propositional logic X | ⇔ ∀ s ∈ X : s ( p ) = 1 = p Dependency atoms X | = ¬ p ⇔ ∀ s ∈ X : s ( p ) = 0 Expressive Power Complexity X | = ϕ ∧ ψ ⇔ X | = ϕ and X | = ψ Exponential X | = ϕ ∨ ψ ⇔ Y | = ϕ and Z | = ψ for some Y ∪ Z = X hierarchy s | = ∃ p ϕ ⇔ s ( b / p ) | = ϕ for some b ∈ { 0 , 1 } s | = ∀ p ϕ ⇔ s ( b / p ) | = ϕ for all b ∈ { 0 , 1 } 5/ 16