Decidability of predicate logics with team semantics Jonni Virtema Decidability of predicate logics with team semantics Backround Team semantics Dependence logic Jonni Virtema Validity for D2 University of Helsinki, Finland jonni.virtema@gmail.com Joint work with Juha Kontinen and Antti Kuusisto MFCS 2016 23rd of August, 2016 1/ 13
Decidability of Core of Team Semantics predicate logics with team semantics Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Backround Team semantics E.g., Dependence logic ◮ a first-order assignment in first-order logic, Validity for D2 ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic. ◮ In team semantics sets of states of affairs are considered. 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/ 13
Decidability of Core of Team Semantics predicate logics with team semantics Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Backround Team semantics E.g., Dependence logic ◮ a first-order assignment in first-order logic, Validity for D2 ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic. ◮ In team semantics sets of states of affairs are considered. 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/ 13
Decidability of Core of Team Semantics predicate logics with team semantics Jonni Virtema ◮ In most studied logics formulae are evaluated in a single state of affairs. Backround Team semantics E.g., Dependence logic ◮ a first-order assignment in first-order logic, Validity for D2 ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic. ◮ In team semantics sets of states of affairs are considered. 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/ 13
Decidability of Team Semantics: Motivation and History predicate logics with team semantics Logical modelling of uncertainty, imperfect information, and different notions of Jonni Virtema dependence such as functional dependence and independence. Related to similar Backround concepts in statistics, database theory etc. Team semantics Historical development: Dependence logic Validity for D2 ◮ First-order logic and Skolem functions. ◮ Branching quantifiers by Henkin 1959. ◮ Independence-friendly logic by Hintikka and Sandu 1989. ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ 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. ◮ Generalized atoms by Kuusisto (derived from generalised quantifiers). 3/ 13
Decidability of Team Semantics: Motivation and History predicate logics with team semantics Logical modelling of uncertainty, imperfect information, and different notions of Jonni Virtema dependence such as functional dependence and independence. Related to similar Backround concepts in statistics, database theory etc. Team semantics Historical development: Dependence logic Validity for D2 ◮ First-order logic and Skolem functions. ◮ Branching quantifiers by Henkin 1959. ◮ Independence-friendly logic by Hintikka and Sandu 1989. ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ 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. ◮ Generalized atoms by Kuusisto (derived from generalised quantifiers). 3/ 13
Decidability of First-order logic predicate logics with team semantics Jonni Virtema Grammar of first-order logic FO in negation normal form: Backround Team semantics ϕ ::= x = y | ¬ ( x = y ) | R ( � x ) | ¬ R ( � x ) | ( ϕ ∨ ϕ ) | ( ϕ ∧ ϕ ) | ∃ x ϕ ( x ) | ∀ x ϕ ( x ) Dependence logic Validity for D2 A team of an FO -structure A is any set X of assignments s : VAR → A with a common domain VAR of FO variables. We want to define team semantics for FO s.t. we have the following property ( flattness ): If ϕ is an FO -formula, A a first-order structure, and X a set of assignments: A | = X ϕ ⇐ ⇒ ∀ s ∈ X : A , s | = ϕ. 4/ 13
Decidability of First-order logic predicate logics with team semantics Jonni Virtema Grammar of first-order logic FO in negation normal form: Backround Team semantics ϕ ::= x = y | ¬ ( x = y ) | R ( � x ) | ¬ R ( � x ) | ( ϕ ∨ ϕ ) | ( ϕ ∧ ϕ ) | ∃ x ϕ ( x ) | ∀ x ϕ ( x ) Dependence logic Validity for D2 A team of an FO -structure A is any set X of assignments s : VAR → A with a common domain VAR of FO variables. We want to define team semantics for FO s.t. we have the following property ( flattness ): If ϕ is an FO -formula, A a first-order structure, and X a set of assignments: A | = X ϕ ⇐ ⇒ ∀ s ∈ X : A , s | = ϕ. 4/ 13
Decidability of Team semantics for first-order logic predicate logics with team semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team semantics Dependence logic x ) ∈ R A A , s | = R ( � x ) ⇔ s ( � Validity for D2 x ) �∈ R A A , s | = ¬ R ( � ⇔ s ( � x ) A , s | = ϕ ∧ ψ ⇔ A , s | = ϕ and A , s | = ψ A , s | = ϕ ∨ ψ ⇔ A , s | = ϕ or A , s | = ψ A , s | = ∀ x ϕ ⇔ A , s ( a / x ) | = ϕ for all a ∈ A A , s | = ∃ x ϕ ⇔ A , s ( a / x ) | = ϕ for some a ∈ A 5/ 13
Decidability of Team semantics for first-order logic predicate logics with team semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team semantics Dependence logic x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � Validity for D2 x ) �∈ R A A | = X R ( � ⇔ ∀ s ∈ X : s ( � x ) A , s | = ϕ ∧ ψ ⇔ A , s | = ϕ and A , s | = ψ A , s | = ϕ ∨ ψ ⇔ A , s | = ϕ or A , s | = ψ A , s | = ∀ x ϕ ⇔ A , s ( a / x ) | = ϕ for all a ∈ A A , s | = ∃ x ϕ ⇔ A , s ( a / x ) | = ϕ for some a ∈ A 5/ 13
Decidability of Team semantics for first-order logic predicate logics with team semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team semantics Dependence logic x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � Validity for D2 x ) �∈ R A A | = X R ( � ⇔ ∀ s ∈ X : s ( � x ) A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ A , s | = ϕ ∨ ψ ⇔ A , s | = ϕ or A , s | = ψ A , s | = ∀ x ϕ ⇔ A , s ( a / x ) | = ϕ for all a ∈ A A , s | = ∃ x ϕ ⇔ A , s ( a / x ) | = ϕ for some a ∈ A 5/ 13
Decidability of Team semantics for first-order logic predicate logics with team semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team semantics Dependence logic x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � Validity for D2 x ) �∈ R A A | = X R ( � ⇔ ∀ s ∈ X : s ( � x ) A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ A | = X ϕ ∨ ψ ⇔ A | = Y ϕ and A | = Z ψ for some Y ∪ Z = X A , s | = ∀ x ϕ ⇔ A , s ( a / x ) | = ϕ for all a ∈ A A , s | = ∃ x ϕ ⇔ A , s ( a / x ) | = ϕ for some a ∈ A 5/ 13
Decidability of Team semantics for first-order logic predicate logics with team semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team semantics Dependence logic x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � Validity for D2 x ) �∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ A | = X ϕ ∨ ψ ⇔ A | = Y ϕ and A | = Z ψ for some Y ∪ Z = X A | = X ∀ x ϕ ⇔ A | = X [ A / x ] ϕ A , s | = ∃ x ϕ ⇔ A , s ( a / x ) | = ϕ for some a ∈ A 5/ 13
Decidability of Team semantics for first-order logic predicate logics with team semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team semantics Dependence logic x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � Validity for D2 x ) �∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ A | = X ϕ ∨ ψ ⇔ A | = Y ϕ and A | = Z ψ for some Y ∪ Z = X A | = X ∀ x ϕ ⇔ A | = X [ A / x ] ϕ A | = X ∃ x ϕ ⇔ A | = X [ F / x ] ϕ for some F : X → P ( A ) \ ∅ 5/ 13
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