Logics of independence and dependence GTS for FO GTS for IF Team - PowerPoint PPT Presentation

Logics of independence and dependence Jonni Virtema What is logic? Tarski Semantics Logics of independence and dependence GTS for FO GTS for IF Team semantics for FO Jonni Virtema Team semantics for IF University of Helsinki, Finland Teams as databases Phileth seminar @ Hokkaido University, 11th of May 2017 Dependence logic IF vs. D Relation to ESO Related works 1/ 46

Logics of Outline independence and dependence Jonni Virtema What is logic? Tarski Semantics GTS for FO Part 0: Introduction to first-order logic GTS for IF Part 1: Game theoretic semantics for first-order logic Team semantics for FO Part 2: Game theoretic semantics for independence-friendly logic Team semantics Part 3: Team semantics for independence-friendly logic for IF Teams as Part 4: Team semantics via database theory and dependence logic databases Dependence logic Part 5: Expressive power of dependence logic and IF-logic IF vs. D Relation to ESO Related works 2/ 46

Logics of independence and dependence Jonni Virtema What is logic? Tarski Semantics GTS for FO Introduction GTS for IF Team semantics for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 3/ 46

Logics of What is logic ? independence and dependence Jonni Virtema What is logic? Tarski Semantics GTS for FO GTS for IF Team semantics for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 4/ 46

Logics of What is logic ? independence and dependence Jonni Virtema What is logic? Tarski Semantics GTS for FO ◮ Mathematics, philosophy, linguistics, and theoretical computer science. GTS for IF Team semantics for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 4/ 46

Logics of What is logic ? independence and dependence Jonni Virtema What is logic? Tarski Semantics GTS for FO ◮ Mathematics, philosophy, linguistics, and theoretical computer science. GTS for IF Team semantics for FO ◮ Logic is an abstract machinery that can be used in describing the state and Team semantics behaviour of systems, and deduction related to these systems. for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 4/ 46

Logics of What is logic (model theoretic view)? independence and dependence Jonni Virtema What is logic? Tarski Semantics GTS for FO ◮ Mathematics, philosophy, linguistics, and theoretical computer science. GTS for IF Team semantics for FO ◮ Logic is an abstract machinery that can be used in describing the state and Team semantics behaviour of systems, and deduction related to these systems. for IF Teams as databases ◮ The core notions are models and formulae . Dependence logic IF vs. D Relation to ESO Related works 4/ 46

Logics of What are models and formulae? independence and dependence Jonni Virtema What is logic? ◮ A model is an abstraction of some state of affairs. Tarski Semantics GTS for FO GTS for IF ◮ A collection of points together with some arrows from points to points is a Team semantics model (e.g., modelling a rail network). for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 5/ 46

Logics of What are models and formulae? independence and dependence Jonni Virtema What is logic? ◮ A model is an abstraction of some state of affairs. Tarski Semantics GTS for FO GTS for IF ◮ A collection of points together with some arrows from points to points is a Team semantics model (e.g., modelling a rail network). for FO Team semantics for IF ◮ A formula is an abstraction of a claim related to some state of affairs. Teams as databases Dependence logic ◮ A sentence that describes a possible property of a model is a formula (e.g., IF vs. D a statement that there is a rail connection from Tokyo to Sapporo). Relation to ESO Related works 5/ 46

Logics of First-order models independence and dependence Jonni Virtema ◮ An n -ary relation R over a set A is a subset of A n . What is logic? ◮ For simplicity we now consider only 1-ary and 2-ary relations, i.e., subsets of Tarski Semantics GTS for FO A and { ( a , b ) | a , b ∈ A } , respectively. GTS for IF Team semantics for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 6/ 46

Logics of First-order models independence and dependence Jonni Virtema ◮ An n -ary relation R over a set A is a subset of A n . What is logic? ◮ For simplicity we now consider only 1-ary and 2-ary relations, i.e., subsets of Tarski Semantics GTS for FO A and { ( a , b ) | a , b ∈ A } , respectively. GTS for IF ◮ (We omit function symbols and constant symbols) Team semantics for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 6/ 46

Logics of First-order models independence and dependence Jonni Virtema ◮ An n -ary relation R over a set A is a subset of A n . What is logic? ◮ For simplicity we now consider only 1-ary and 2-ary relations, i.e., subsets of Tarski Semantics GTS for FO A and { ( a , b ) | a , b ∈ A } , respectively. GTS for IF ◮ (We omit function symbols and constant symbols) Team semantics for FO Definition Team semantics for IF Let τ = { P , R } where P is a 1-ary relation symbol and R a 2-ary relation Teams as databases symbol. A τ -model A is a tuple ( A , P , R ), where Dependence logic ◮ A is a nonempty set called the domain of A , IF vs. D ◮ R ⊆ A × A is a binary relation over A , and Relation to ESO Related works ◮ P ⊆ A is a unary relation over A . 6/ 46

Logics of First-order models independence and dependence Jonni Virtema What is logic? Tarski Semantics Example GTS for FO GTS for IF ◮ Let A be the set of all cities in Japan. Team semantics for FO ◮ Let R = { ( Sapporo , Hakodate ) , ( Hakodate , Hirosaki ) , ( Hirosaki , Sapporo ) } . Team semantics for IF ◮ Let P = { Hakodate , Hirosaki } . Teams as databases Now ( A , R , P ) is a first-order model that describes my travel during the Golden Dependence logic Week. IF vs. D Relation to ESO Related works 7/ 46

Logics of First-order language independence and dependence Jonni Virtema Definition What is logic? The formulae for first-order logic FO over vocabulary { R , P } is generated by Tarski Semantics the following grammar: GTS for FO GTS for IF ϕ ::= P ( x ) | R ( x , y ) | x = y | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ∃ x ϕ | ∀ x ϕ, Team semantics for FO where x and y are variable symbols. Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 8/ 46

Logics of First-order language independence and dependence Jonni Virtema Definition What is logic? The formulae for first-order logic FO over vocabulary { R , P } is generated by Tarski Semantics the following grammar: GTS for FO GTS for IF ϕ ::= P ( x ) | R ( x , y ) | x = y | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ∃ x ϕ | ∀ x ϕ, Team semantics for FO where x and y are variable symbols. Team semantics for IF Teams as Example databases Dependence logic Using the example model of the last slide, the sentence: IF vs. D I visited two Japanese cities during the Golden Week can be written in first-oder Relation to ESO logic as follows: Related works ∃ x ∃ y ( ¬ x = y ∧ P ( x ) ∧ P ( y )) 8/ 46

Logics of The connection between models and formulae independence and dependence Jonni Virtema What is logic? Tarski Semantics A model describes some system and a formula describes some possible property GTS for FO of that system. GTS for IF Team semantics for FO Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 9/ 46

Logics of The connection between models and formulae independence and dependence Jonni Virtema What is logic? Tarski Semantics A model describes some system and a formula describes some possible property GTS for FO of that system. GTS for IF Team semantics The question: for FO Does the property described hold in the system or not? Team semantics for IF Teams as databases Dependence logic IF vs. D Relation to ESO Related works 9/ 46

Logics of The connection between models and formulae independence and dependence Jonni Virtema What is logic? Tarski Semantics A model describes some system and a formula describes some possible property GTS for FO of that system. GTS for IF Team semantics The question: for FO Does the property described hold in the system or not? Team semantics for IF Is the formula true in the model? Teams as databases Dependence logic IF vs. D Relation to ESO Related works 9/ 46

Logics of The connection between models and formulae independence and dependence Jonni Virtema What is logic? Tarski Semantics A model describes some system and a formula describes some possible property GTS for FO of that system. GTS for IF Team semantics The question: for FO Does the property described hold in the system or not? Team semantics for IF Is the formula true in the model? Teams as databases Dependence logic Formally: IF vs. D Does A , s | = ϕ hold? Relation to ESO Related works 9/ 46