Polyteam Semantics Jonni Virtema Backround Polyteam Semantics Team Semantics Axiomatisations in team semantics Polyteams and poly-dependence Jonni Virtema Axioms of poly-dependence Hasselt University, Belgium Poly-independence jonni.virtema@gmail.com Polyteam seamantics Joint work with Miika Hannula (University of Auckland) and Juha Kontinen (University of Helsinki) January 11, 2018 1/ 28
Polyteam Team Semantics: Motivation and History Semantics Logical modelling of uncertainty, imperfect information, and different notions of Jonni Virtema dependence, such as functional dependence and independence, from application Backround fields: statistics (probabilistic independence), database theory (database Team Semantics dependencies), social choice theory (arrows theorem), etc. Axiomatisations in team semantics Historical development: Polyteams and poly-dependence ◮ First-order logic and Skolem functions. Axioms of poly-dependence ◮ Branching quantifiers by Henkin 1959. Poly-independence ◮ Independence-friendly logic by Hintikka and Sandu 1989. Polyteam seamantics ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ Dependence logic 2007 and modal dependence logic 2008 by V¨ a¨ an¨ anen. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Approximate dependence by V¨ a¨ an¨ anen 2014 and multiteam semantics by 2/ 28 Durand et al. 2016.
Polyteam Team Semantics: Motivation and History Semantics Logical modelling of uncertainty, imperfect information, and different notions of Jonni Virtema dependence, such as functional dependence and independence, from application Backround fields: statistics (probabilistic independence), database theory (database Team Semantics dependencies), social choice theory (arrows theorem), etc. Axiomatisations in team semantics Historical development: Polyteams and poly-dependence ◮ First-order logic and Skolem functions. Axioms of poly-dependence ◮ Branching quantifiers by Henkin 1959. Poly-independence ◮ Independence-friendly logic by Hintikka and Sandu 1989. Polyteam seamantics ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ Dependence logic 2007 and modal dependence logic 2008 by V¨ a¨ an¨ anen. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Approximate dependence by V¨ a¨ an¨ anen 2014 and multiteam semantics by 2/ 28 Durand et al. 2016.
Polyteam First-Order Team Semantics (via database theoretic spectacles) Semantics Jonni Virtema Backround Team Semantics Axiomatisations in ◮ A team is a set of assignments that have a common domain of variables. team semantics ◮ A team can be seen as a database table. Polyteams and poly-dependence ◮ Variables correspond to attributes. Axioms of ◮ Assignments correspond to records. poly-dependence Poly-independence ◮ Dependency notions of database theory give rise to novel atomic formulae. Polyteam ◮ Functional dependence corresponds to dependence atoms =( x 1 , . . . , x n , y ). seamantics ◮ Inclusion dependence corresponds to inclusion atoms x ⊆ y . ◮ Embedded multivalued dependency gives rise to independence atoms y ⊥ x z . 3/ 28
Polyteam Dependence Logic Semantics Jonni Virtema Backround In FO, formulas are formed using connectives ∨ , ∧ , ¬ , and quantifiers ∃ and ∀ . Team Semantics Axiomatisations in Definition team semantics Polyteams and Dependence logic FO ( dep ) extends the syntax of FO by dependence atoms poly-dependence Axioms of poly-dependence =( x 1 , . . . , x n ) . Poly-independence Polyteam seamantics We consider also independence and inclusion atoms (and the corresponding logics) that replace dependence atoms respectively by y ⊥ x z and x ⊆ y . 4/ 28
Polyteam Assignments and Teams Semantics Jonni Virtema Backround Team Semantics Axiomatisations in The semantics of dependence logic is defined using the notion of a team. team semantics Polyteams and Teams: poly-dependence Axioms of Let A be a set and V = { x 1 , . . . , x k } a finite set of variables. A team X with poly-dependence domain V is a set of assignments Poly-independence Polyteam s : V → A . seamantics A is called the co-domain of X (the universe of a model). 5/ 28
Polyteam Interpretation of Dependence Atoms Semantics Jonni Virtema Backround Team Semantics Axiomatisations in team semantics Let A be a structure and X a team. Polyteams and poly-dependence = X =( x 1 , ..., x n ), if and only if, for all s , s ′ ∈ X : Axioms of A | poly-dependence Poly-independence � s ( x i ) = s ′ ( x i ) = ⇒ s ( x n ) = s ′ ( x n ) . Polyteam seamantics 0 < i < n 6/ 28
Polyteam Interpreting Inclusion and Independence Atoms Semantics Jonni Virtema Backround Team Semantics Inclusion atoms: Axiomatisations in = X x ⊆ y , if and only if, for all s ∈ X there exists s ′ ∈ X s.t. s ( x ) = s ′ ( y ). team semantics A | Polyteams and poly-dependence Independence atoms: = X y ⊥ x z , iff, for all s , s ′ ∈ X : if s ( x ) = s ′ ( x ) then there exists s ′′ ∈ X such Axioms of A | poly-dependence that Poly-independence Polyteam ◮ s ′′ ( x ) = s ( x ), seamantics ◮ s ′′ ( y ) = s ( y ), ◮ s ′′ ( z ) = s ′ ( z ). 7/ 28
Polyteam Examples of teams Semantics Jonni Virtema We may think of the variables x i as attributes of a database such as Backround x 0 = SALARY and x 2 = JOB TITLE . Team Semantics Axiomatisations in team semantics . . . x 0 x n Polyteams and . . . s 0 a 0 , m a n , m poly-dependence . Axioms of poly-dependence . Poly-independence . Polyteam seamantics . . . s m a 0 , m a n , m Then dependence atom =( x 2 , x 0 ) expresses the functional dependence JOB TITLE → SALARY . 8/ 28
Polyteam Team semantics for first-order logic Semantics Jonni Virtema Backround Recall that a team is a set of first-order assignments with a common domain. Team Semantics Axiomatisations in team semantics Polyteams and x ) ∈ R A A , s | = R ( � x ) ⇔ s ( � poly-dependence Axioms of x ) �∈ R A A , s | = ¬ R ( � ⇔ s ( � x ) poly-dependence A , s | = ϕ ∧ ψ ⇔ A , s | = ϕ and A , s | = ψ Poly-independence Polyteam A , s | = ϕ ∨ ψ ⇔ A , s | = ϕ or A , s | = ψ seamantics 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 9/ 28
Polyteam Team semantics for first-order logic Semantics Jonni Virtema Backround Recall that a team is a set of first-order assignments with a common domain. Team Semantics Axiomatisations in team semantics Polyteams and x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � poly-dependence Axioms of x ) �∈ R A A | = X R ( � ⇔ ∀ s ∈ X : s ( � x ) poly-dependence A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ Poly-independence Polyteam A , s | = ϕ ∨ ψ ⇔ A , s | = ϕ or A , s | = ψ seamantics 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 9/ 28
Polyteam Team semantics for first-order logic Semantics Jonni Virtema Backround Recall that a team is a set of first-order assignments with a common domain. Team Semantics Axiomatisations in team semantics Polyteams and x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � poly-dependence Axioms of x ) �∈ R A A | = X R ( � ⇔ ∀ s ∈ X : s ( � x ) poly-dependence A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ Poly-independence Polyteam A | = X ϕ ∨ ψ ⇔ A | = Y ϕ and A | = Z ψ for some Y ∪ Z = X seamantics 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 9/ 28
Polyteam Team semantics for first-order logic Semantics Jonni Virtema Backround Recall that a team is a set of first-order assignments with a common domain. Team Semantics Axiomatisations in team semantics Polyteams and x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � poly-dependence x ) �∈ R A Axioms of A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � poly-dependence A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ Poly-independence Polyteam A | = X ϕ ∨ ψ ⇔ A | = Y ϕ and A | = Z ψ for some Y ∪ Z = X seamantics A | = X ∀ x ϕ ⇔ A | = X [ A / x ] ϕ A | = X ∃ x ϕ ⇔ A | = X [ F / x ] ϕ for some F : X → P ( A ) \ ∅ 9/ 28
Polyteam Team semantics for first-order logic Semantics Jonni Virtema Recall that a team is a set of first-order assignments with a common domain. Backround Team Semantics Axiomatisations in x ) ∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � team semantics Polyteams and x ) �∈ R A A | = X R ( � x ) ⇔ ∀ s ∈ X : s ( � poly-dependence A | = X ϕ ∧ ψ ⇔ A | = X ϕ and A | = X ψ Axioms of poly-dependence A | = X ϕ ∨ ψ ⇔ A | = Y ϕ and A | = Z ψ for some Y ∪ Z = X Poly-independence A | = X ∀ x ϕ ⇔ A | Polyteam = X [ A / x ] ϕ seamantics A | = X ∃ x ϕ ⇔ A | = X [ F / x ] ϕ for some F : X → P ( A ) \ ∅ For every FO -formula ϕ the following holds: A | = X ϕ ⇐ ⇒ ∀ s ∈ X : A , s | = ϕ. 9/ 28
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