Polyteam Semantics Jonni Virtema Backround Polyteam Semantics Team Semantics Axiomatizations in team semantics Polyteams and Jonni Virtema poly-dependence Axioms of University of Helsinki, Finland poly-dependence jonni.virtema@gmail.com Poly-independence Polyteam seamantics Joint work with Juha Kontinen (University of Helsinki) and Miika Hannula (University of Auckland) Expressivity of polyteam logics GaLoP 2017 22nd of April, 2017 1/ 22
Polyteam Team Semantics: Motivation and History Semantics Jonni Virtema Logical modelling of uncertainty, imperfect information, and di ff erent notions of dependence such as functional dependence and independence. Related to similar Backround concepts in statistics, database theory etc. Team Semantics Axiomatizations in team semantics Historical development: Polyteams and I First-order logic and Skolem functions. poly-dependence Axioms of I Branching quantifiers by Henkin 1959. poly-dependence I Independence-friendly logic by Hintikka and Sandu 1989. Poly-independence Polyteam I Compositional semantics for independence-friendly logic by Hodges 1997. seamantics (Origin of team semantics.) Expressivity of polyteam logics I Dependence logic 2007 and modal dependence logic 2008 by V¨ a¨ an¨ anen. I Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. I Approximate dependence by V¨ a¨ an¨ anen 2014 and multiteam semantics by Durand et al. 2016. 2/ 22
Polyteam Team Semantics: Motivation and History Semantics Jonni Virtema Logical modelling of uncertainty, imperfect information, and di ff erent notions of dependence such as functional dependence and independence. Related to similar Backround concepts in statistics, database theory etc. Team Semantics Axiomatizations in team semantics Historical development: Polyteams and I First-order logic and Skolem functions. poly-dependence Axioms of I Branching quantifiers by Henkin 1959. poly-dependence I Independence-friendly logic by Hintikka and Sandu 1989. Poly-independence Polyteam I Compositional semantics for independence-friendly logic by Hodges 1997. seamantics (Origin of team semantics.) Expressivity of polyteam logics I Dependence logic 2007 and modal dependence logic 2008 by V¨ a¨ an¨ anen. I Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. I Approximate dependence by V¨ a¨ an¨ anen 2014 and multiteam semantics by Durand et al. 2016. 2/ 22
Polyteam First-Order Team Semantics (via database theoretic spectacles) Semantics Jonni Virtema Backround Team Semantics Axiomatizations in I A team is a set of assignments that have a common domain of variables. team semantics I A team is a database table. Polyteams and poly-dependence I Variables correspond to attributes. Axioms of I Assignments correspond to records. poly-dependence Poly-independence I Dependency notions of database theory give rise to novel atomic formulae. Polyteam I Functional dependence gives rise to dependence atoms =( x 1 , . . . , x n ). seamantics I Inclusion dependence gives rise to inclusion atoms x ✓ y . Expressivity of I Embedded multivalued dependency gives rise to independence atoms y ? x z . polyteam logics 3/ 22
Polyteam Dependence Logic Semantics Jonni Virtema Backround In FO, formulas are formed using connectives _ , ^ , ¬ , and quantifiers 9 and 8 . Team Semantics Axiomatizations 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 Expressivity of logics) that replace dependence atoms respectively by polyteam logics y ? x z and x ✓ y . 4/ 22
Polyteam Assignments and Teams Semantics Jonni Virtema Backround Team Semantics Axiomatizations 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 set of variables. A team X with poly-dependence domain V is a set of assignments Poly-independence Polyteam s : V ! A . seamantics Expressivity of polyteam logics A is called the co-domain of X (the universe of a model). 5/ 22
Polyteam Interpretation of Dependence Atoms Semantics Jonni Virtema Backround Team Semantics Axiomatizations in Let A be a structure and X a team with co-domain Dom ( A ) and domain V s.t. team semantics { x 1 , ..., x n } ✓ V . Polyteams and poly-dependence Axioms of = X =( x 1 , ..., x n ), if and only if, for all s , s 0 2 X : poly-dependence A | Poly-independence Polyteam ^ s ( x i ) = s 0 ( x i ) = ) s ( x n ) = s 0 ( x n ) . seamantics 0 < i < n Expressivity of polyteam logics 6/ 22
Polyteam Interpreting Inclusion and Independence Atoms Semantics Jonni Virtema Backround Team Semantics Inclusion atoms: Axiomatizations in = X x ✓ y , if and only if, for all s 2 X there exists s 0 2 X s.t. s ( x ) = s 0 ( y ). team semantics A | Polyteams and poly-dependence Independence atoms: = X y ? x z , i ff , for all s , s 0 2 X : if s ( x ) = s 0 ( x ) then there exists s 00 2 X such Axioms of A | poly-dependence that Poly-independence Polyteam I s 00 ( x ) = s ( x ), seamantics I s 00 ( y ) = s ( y ), Expressivity of polyteam logics I s 00 ( z ) = s 0 ( z ). 7/ 22
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 = ID NUMBER . Team Semantics Axiomatizations 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 Expressivity of polyteam logics Then dependence atom =( x 2 , x 0 ) expresses the functional dependence ID NUMBER ! SALARY . 8/ 22
Polyteam Expressive Power Semantics Dependence logic defines all downward closed ESO properties of teams. Jonni Virtema Backround Theorem (Kontinen, V¨ a¨ an¨ anen 2009) Team Semantics For every sentence ψ 2 ESO [ τ [ { R } ] , in which R appears only negatively, there Axiomatizations in team semantics is φ ( y 1 , . . . , y k ) 2 FO ( dep )[ τ ] s.t. for all A and X 6 = ; with domain { y 1 , . . . , y k } Polyteams and poly-dependence A | = X φ ( ) ( A , R := X ( y )) | = ψ . Axioms of poly-dependence Poly-independence Independence logic defines all ESO properties of teams. Polyteam seamantics Expressivity of Theorem (Galliani 2012) polyteam logics For every sentence ψ 2 ESO [ τ [ { R } ] there is φ ( y 1 , . . . , y k ) 2 FO ( ? )[ τ ] s.t. for all A and X 6 = ; with domain { y 1 , . . . , y k } : = X φ ( ) ( A , R := X ( y )) | A | = ψ . 9/ 22
Polyteam Amstrong’s Axioms for Functional Dependence Semantics Jonni Virtema Backround This inference system consists of only three rules which we depict below using Team Semantics the standard notation for functional dependencies, i.e., X ! Y denotes that an Axiomatizations in team semantics attribute set X functionally determines another attribute set Y . Polyteams and poly-dependence Definition (Armstrong 1974) Axioms of poly-dependence I Reflexivity: If Y ✓ X , then X ! Y . Poly-independence Polyteam I Augmentation: if X ! Y , then XZ ! YZ seamantics I Transitivity: if X ! Y and Y ! Z , then X ! Z . Expressivity of polyteam logics The same axiomatization works for dependence atoms =( x , y ) when we add some rules that permutes and adds/removes duplicates to/from x . 10/ 22
Polyteam Amstrong’s Axioms for Functional Dependence Semantics Jonni Virtema Backround This inference system consists of only three rules which we depict below using Team Semantics the standard notation for functional dependencies, i.e., X ! Y denotes that an Axiomatizations in team semantics attribute set X functionally determines another attribute set Y . Polyteams and poly-dependence Definition (Armstrong 1974) Axioms of poly-dependence I Reflexivity: If Y ✓ X , then X ! Y . Poly-independence Polyteam I Augmentation: if X ! Y , then XZ ! YZ seamantics I Transitivity: if X ! Y and Y ! Z , then X ! Z . Expressivity of polyteam logics The same axiomatization works for dependence atoms =( x , y ) when we add some rules that permutes and adds/removes duplicates to/from x . 10/ 22
Polyteam Axioms for Pure (Marginal) Independence Semantics Jonni Virtema Backround Team Semantics For x ? y , where x and y have no variables in common, a complete Axiomatizations in axiomatization is given by the following Independence Axioms : team semantics Polyteams and 1. Permutation and redundancy as before. poly-dependence 2. x ?; (Empty Set Rule). Axioms of poly-dependence 3. If x ? y , then y ? x (Symmetry Rule). Poly-independence 4. If x ? yz , then x ? y (Weakening Rule) Polyteam seamantics 5. If x ? y and xy ? z , then x ? yz (Exchange Rule). Expressivity of polyteam logics This axiomatization due to Geiger, Paz, and Pearl (1991) for marginal independence X ? ? Y between two sets of random variables. 11/ 22
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