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Characterizing Frame Definability in Team Semantics via The Universal Modality Characterizing Frame Definability in Team Semantics Jonni Virtema via The Universal Modality Definability Modal logic Frame definability Jonni Virtema What do

Characterizing Frame Definability in Team Semantics via The Universal Modality Characterizing Frame Definability in Team Semantics Jonni Virtema via The Universal Modality Definability Modal logic Frame definability Jonni Virtema What do we study? GbTh theorem Leibniz Universit¨ at Hannover, Germany Team semantics jonni.virtema@gmail.com Modal dependence logic Joint work with Katsuhiko Sano, JAIST, Japan Frame definability in team semantics Conclusion WoLLIC 2015 References 20th of July 2015 1/ 36
Characterizing Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Definability Modal logic Frame definability PART I What do we study? GbTh theorem Team semantics Modal dependence logic Frame definability in team semantics Conclusion References 2/ 36
Characterizing Definability Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which properties of graphs can be described with a given logic L . Definability Modal logic Example first-order logic on graphs G = ( V , E ): Frame definability ◮ Single formula: ∃ x ∃ y ¬ x = y defines the class { ( V , E ) | | V | ≥ 2 } . What do we study? ◮ Set of formulae: GbTh theorem � {∃ x 1 . . . x n ¬ x i = x j | n ∈ N } Team semantics Modal dependence i � = j ≤ n logic defines the class of infinite graphs. Frame definability in team semantics A class of structures is called elementary, if there exists a set of FO -formulae Conclusion that defines the class. References 3/ 36
Characterizing Definability Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which properties of graphs can be described with a given logic L . Definability Modal logic Example first-order logic on graphs G = ( V , E ): Frame definability ◮ Single formula: ∃ x ∃ y ¬ x = y defines the class { ( V , E ) | | V | ≥ 2 } . What do we study? ◮ Set of formulae: GbTh theorem � {∃ x 1 . . . x n ¬ x i = x j | n ∈ N } Team semantics Modal dependence i � = j ≤ n logic defines the class of infinite graphs. Frame definability in team semantics A class of structures is called elementary, if there exists a set of FO -formulae Conclusion that defines the class. References 3/ 36
Characterizing Definability Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which properties of graphs can be described with a given logic L . Definability Modal logic Example first-order logic on graphs G = ( V , E ): Frame definability ◮ Single formula: ∃ x ∃ y ¬ x = y defines the class { ( V , E ) | | V | ≥ 2 } . What do we study? ◮ Set of formulae: GbTh theorem � {∃ x 1 . . . x n ¬ x i = x j | n ∈ N } Team semantics Modal dependence i � = j ≤ n logic defines the class of infinite graphs. Frame definability in team semantics A class of structures is called elementary, if there exists a set of FO -formulae Conclusion that defines the class. References 3/ 36
Characterizing Definability Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which properties of graphs can be described with a given logic L . Definability Modal logic Example first-order logic on graphs G = ( V , E ): Frame definability ◮ Single formula: ∃ x ∃ y ¬ x = y defines the class { ( V , E ) | | V | ≥ 2 } . What do we study? ◮ Set of formulae: GbTh theorem � {∃ x 1 . . . x n ¬ x i = x j | n ∈ N } Team semantics Modal dependence i � = j ≤ n logic defines the class of infinite graphs. Frame definability in team semantics A class of structures is called elementary, if there exists a set of FO -formulae Conclusion that defines the class. References 3/ 36
Characterizing Modal logic Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Definability Modal logic ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � ϕ. Frame definability What do we study? GbTh theorem Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary Team semantics relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Modal dependence logic E.g., Frame definability in team semantics K , w | = p iff w ∈ V ( p ), ◮ Conclusion K , w | = ♦ ϕ iff K , v | = ϕ for some v s.t. wRv . ◮ References 4/ 36
Characterizing Modal logic Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Definability Modal logic ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � ϕ. Frame definability What do we study? GbTh theorem Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary Team semantics relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Modal dependence logic E.g., Frame definability in team semantics K , w | = p iff w ∈ V ( p ), ◮ Conclusion K , w | = ♦ ϕ iff K , v | = ϕ for some v s.t. wRv . ◮ References 4/ 36
Characterizing Modal logic Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Definability Modal logic ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � ϕ. Frame definability What do we study? GbTh theorem Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary Team semantics relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Modal dependence logic E.g., Frame definability in team semantics K , w | = p iff w ∈ V ( p ), ◮ Conclusion K , w | = ♦ ϕ iff K , v | = ϕ for some v s.t. wRv . ◮ References 4/ 36
Characterizing Validity in models and frames Frame Definability in Team Semantics via The Universal Modality Jonni Virtema ◮ Pointed model ( K , w ): ( W , R , V ) , w . Definability ◮ Model ( K ): ( W , R , V ). Modal logic ◮ Frame ( F ): ( W , R ). Frame definability What do we study? We write: GbTh theorem ◮ ( W , R , V ) | ( W , R , V ) , w | = ϕ holds for every w ∈ W . = ϕ iff Team semantics Modal dependence ◮ ( W , R ) | = ϕ iff ( W , R , V ) | = ϕ holds for every valuation V . logic Frame definability Every (set of) ML -formula defines the class of frames in which it is valid. in team semantics Conclusion ◮ Fr ( ϕ ) := { ( W , R ) | ( W , R ) | = ϕ } . References ◮ Fr (Γ) := { ( W , R ) | ∀ ϕ ∈ Γ : ( W , R ) | = ϕ } . 5/ 36
Characterizing Validity in models and frames Frame Definability in Team Semantics via The Universal Modality Jonni Virtema ◮ Pointed model ( K , w ): ( W , R , V ) , w . Definability ◮ Model ( K ): ( W , R , V ). Modal logic ◮ Frame ( F ): ( W , R ). Frame definability What do we study? We write: GbTh theorem ◮ ( W , R , V ) | ( W , R , V ) , w | = ϕ holds for every w ∈ W . = ϕ iff Team semantics Modal dependence ◮ ( W , R ) | = ϕ iff ( W , R , V ) | = ϕ holds for every valuation V . logic Frame definability Every (set of) ML -formula defines the class of frames in which it is valid. in team semantics Conclusion ◮ Fr ( ϕ ) := { ( W , R ) | ( W , R ) | = ϕ } . References ◮ Fr (Γ) := { ( W , R ) | ∀ ϕ ∈ Γ : ( W , R ) | = ϕ } . 5/ 36
Characterizing Validity in models and frames Frame Definability in Team Semantics via The Universal Modality Jonni Virtema ◮ Pointed model ( K , w ): ( W , R , V ) , w . Definability ◮ Model ( K ): ( W , R , V ). Modal logic ◮ Frame ( F ): ( W , R ). Frame definability What do we study? We write: GbTh theorem ◮ ( W , R , V ) | ( W , R , V ) , w | = ϕ holds for every w ∈ W . = ϕ iff Team semantics Modal dependence ◮ ( W , R ) | = ϕ iff ( W , R , V ) | = ϕ holds for every valuation V . logic Frame definability Every (set of) ML -formula defines the class of frames in which it is valid. in team semantics Conclusion ◮ Fr ( ϕ ) := { ( W , R ) | ( W , R ) | = ϕ } . References ◮ Fr (Γ) := { ( W , R ) | ∀ ϕ ∈ Γ : ( W , R ) | = ϕ } . 5/ 36
Characterizing Frame definability Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which classes of Kripke frames are definable by a (set of) modal formulae. Definability Which elementary classes are definable by a (set of) modal formulae. Modal logic Frame definability Examples: What do we study? Formula Property of R GbTh theorem Team semantics � p → p Reflexive ∀ w ( wRw ) Modal dependence p → �♦ p Symmetric ∀ w , v ( wRv → vRw ) logic � p → �� p Transitive ∀ w , v , u (( wRv ∧ vRu ) → wRu ) Frame definability in team semantics ♦ p → �♦ p Euclidean ∀ w , v , u (( wRv ∧ wRu ) → vRu ) Conclusion � p → ♦ p Serial ∀ w ∃ v ( wRv ) References 6/ 36
Characterizing Frame definability Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which classes of Kripke frames are definable by a (set of) modal formulae. Definability Which elementary classes are definable by a (set of) modal formulae. Modal logic Frame definability Examples: What do we study? Formula Property of R GbTh theorem Team semantics � p → p Reflexive ∀ w ( wRw ) Modal dependence p → �♦ p Symmetric ∀ w , v ( wRv → vRw ) logic � p → �� p Transitive ∀ w , v , u (( wRv ∧ vRu ) → wRu ) Frame definability in team semantics ♦ p → �♦ p Euclidean ∀ w , v , u (( wRv ∧ wRu ) → vRu ) Conclusion � p → ♦ p Serial ∀ w ∃ v ( wRv ) References 6/ 36

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