Inclusion and exclusion atoms in team semantics Pietro Galliani - PowerPoint PPT Presentation

Non-Functional Dependencies Semantics Expressivity Definability in I/E Logic Inclusion and exclusion atoms in team semantics Pietro Galliani Institute for Logic, Language and Computation Universiteit van Amsterdam Finite Model Theory Seminar Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Expressivity Definability in I/E Logic Outline Non-Functional Dependencies 1 Independence Atoms Inclusion and Exclusion Atoms Semantics 2 Strict and Lax Operators Game Theoretic Semantics Expressivity 3 Exclusion Logic Inclusion/Exclusion Logic Definability in I/E Logic 4 Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Outline Non-Functional Dependencies 1 Independence Atoms Inclusion and Exclusion Atoms Semantics 2 Strict and Lax Operators Game Theoretic Semantics Expressivity 3 Exclusion Logic Inclusion/Exclusion Logic Definability in I/E Logic 4 Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Independence Logic Independence Atoms (Grädel, Väänänen) t 3 if and only if, for all s , s ′ ∈ X such that = X = � t 1 � M | t 2 ⊥ � t 1 � s ′ � there exists a s ′′ ∈ X such that � t 1 � s � = � � t 1 � s ′′ � � t 2 � s ′′ � = � t 1 � s � � t 2 � s � , � t 1 � s ′′ � � t 3 � s ′′ � = � t 1 � s ′ � � t 3 � s ′ � . Independence Logic I I = First Order Logic + Independence Atoms Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Properties of Independence Logic Properties of Independence Logic (Grädel, Väänänen) Contains Dependence Logic; As expressive as Dependence Logic over sentences; More expressive on open formulas (no downwards closure). Open Problem What classes of teams are definable by open formulas in Independence Logic I ? This talk will answer this. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Properties of Independence Logic Properties of Independence Logic (Grädel, Väänänen) Contains Dependence Logic; As expressive as Dependence Logic over sentences; More expressive on open formulas (no downwards closure). Open Problem What classes of teams are definable by open formulas in Independence Logic I ? This talk will answer this. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Outline Non-Functional Dependencies 1 Independence Atoms Inclusion and Exclusion Atoms Semantics 2 Strict and Lax Operators Game Theoretic Semantics Expressivity 3 Exclusion Logic Inclusion/Exclusion Logic Definability in I/E Logic 4 Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Inclusion Dependencies Definition R relation, � x ,� y tuples of attributes, | � x | = | � y | . y if and only if for all r ∈ R there exists an r ′ ∈ R = � x ⊆ � Then R | such that x ) = r ′ ( � r ( � y ) . Fairly well studied; Sound and complete axiomatization. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Example of Inclusion Dependency Professor University Person Date of Birth Hilbert Königsberg Hilbert 23/01/1862 Hilbert Göttingen Gauss 30/04/1777 Gauss Göttingen Torvalds 28/12/1969 R | = Professor ⊆ Person; R �| = Person ⊆ Professor. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Example of Inclusion Dependency Professor University Person Date of Birth Hilbert Königsberg Hilbert 23/01/1862 Hilbert Göttingen Gauss 30/04/1777 Gauss Göttingen Torvalds 28/12/1969 R | = Professor ⊆ Person; R �| = Person ⊆ Professor. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Exclusion Dependencies Definition R relation, � x ,� y tuples of attributes, | � x | = | � y | . y if and only if, for all r , r ′ ∈ R , = � x | � Then R | x ) � = r ′ ( � r ( � y ) . Often, not used explicity; Very commonly used implicitly, for typing of attributes; Sound and complete axiomatization together with inclusion dependencies. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Example of Exclusion Dependency Professor University Person Date of Birth Hilbert Königsberg Hilbert 23/01/1862 Hilbert Göttingen Gauss 30/04/1777 Gauss Göttingen Torvalds 28/12/1969 R | = University | Date of Birth; R �| = Professor | Person. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Example of Exclusion Dependency Professor University Person Date of Birth Hilbert Königsberg Hilbert 23/01/1862 Hilbert Göttingen Gauss 30/04/1777 Gauss Göttingen Torvalds 28/12/1969 R | = University | Date of Birth; R �| = Professor | Person. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Inclusion and Exclusion Logic Inclusion Atoms = X � t 1 ⊆ � t 2 if and only if { ( � t 1 � s � ,� = � t 1 ⊆ � M | t 2 � s � ) : s ∈ X } | t 2 ; Exclusion Atoms = X ¬ ( � t 1 | � t 2 ) if and only if { ( � t 1 � s � ,� = � t 1 | � M | t 2 � s � ) : s ∈ X } | t 2 . Inclusion/Exclusion Logic I/E Logic = FOTeam ( ⊆ , | ) . Inclusion Logic = only inclusion atoms, Exclusion Logic = only exclusion atoms. Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Independence Atoms Expressivity Inclusion and Exclusion Atoms Definability in I/E Logic Direct Definitions for Tuple Existence Literals Semantics Inclusion Atoms t 2 if and only if for all s ∈ X there exists a s ′ ∈ X = X � t 1 ⊆ � M | such that � t 1 � s � = � t 2 � s ′ � ; Exclusion Atoms t 2 if and only if, for all s , s ′ ∈ X , = X � t 1 | � M | � t 1 � s � � = � t 2 � s ′ � . Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Strict and Lax Operators Expressivity Game Theoretic Semantics Definability in I/E Logic Outline Non-Functional Dependencies 1 Independence Atoms Inclusion and Exclusion Atoms Semantics 2 Strict and Lax Operators Game Theoretic Semantics Expressivity 3 Exclusion Logic Inclusion/Exclusion Logic Definability in I/E Logic 4 Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Strict and Lax Operators Expressivity Game Theoretic Semantics Definability in I/E Logic Two Semantics for Disjuction A lax semantics = X ψ 1 ∨ L ψ 2 ⇔ ∃ Y , Z s.t. X = Y ∪ Z , M | M | = Y ψ 1 and M | = Z ψ 2 ; A strict semantics = X ψ 1 ∨ S ψ 2 ⇔∃ Y , Z s.t. X = Y ∪ Z , X ∩ Y = ∅ , M | M | = Y ψ 1 and M | = Z ψ 2 ; D is usually given with ∨ L (or even: X ⊆ Y ∪ Z !). Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Strict and Lax Operators Expressivity Game Theoretic Semantics Definability in I/E Logic Two Semantics for Disjuction A lax semantics = X ψ 1 ∨ L ψ 2 ⇔ ∃ Y , Z s.t. X = Y ∪ Z , M | M | = Y ψ 1 and M | = Z ψ 2 ; A strict semantics = X ψ 1 ∨ S ψ 2 ⇔∃ Y , Z s.t. X = Y ∪ Z , X ∩ Y = ∅ , M | M | = Y ψ 1 and M | = Z ψ 2 ; D is usually given with ∨ L (or even: X ⊆ Y ∪ Z !). Pietro Galliani Inclusion and exclusion atoms in team semantics

Non-Functional Dependencies Semantics Strict and Lax Operators Expressivity Game Theoretic Semantics Definability in I/E Logic In Dependence Logic, Lax = Strict No difference for D (or for T − ) = X ψ 1 ∨ S ψ 2 iff M | = X ψ 1 ∨ L ψ 2 . If ψ 1 , ψ 2 ∈ D , M | Proof. = X ψ 1 ∨ S ψ 2 , M | = X ψ 1 ∨ L ψ 2 ; If M | = X ψ 1 ∨ L ψ 2 then X = X 1 ∪ X 2 , M | If M | = X 1 ψ 1 , M | = X 2 ψ 2 . Take Y = X 2 \ X 1 : by downwards closure , M | = Y ψ 2 , = X ψ 1 ∨ S ψ 2 . X 1 ∪ Y = X , so M | Pietro Galliani Inclusion and exclusion atoms in team semantics