Definability in modal logics with team semantics Jonni Virtema Logics Definability in modal logics with team semantics Expressivity Frame definability What do we study? Jonni Virtema GbTh theorem Frame definability in team semantics Leibniz Universit¨ at Hannover, Germany Conclusion jonni.virtema@gmail.com References Logic Colloquium 2015 August 4th 2015 1/ 25
Definability in modal logics with team semantics Jonni Virtema Logics Expressivity Frame definability What do we study? Prologue GbTh theorem Frame definability in team semantics Conclusion References 2/ 25
Definability in Modal logic modal logics with team semantics Jonni Virtema Logics Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Expressivity Frame definability ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ♦ ϕ | � ϕ What do we study? GbTh theorem Frame definability Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary in team semantics relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Conclusion References E.g., K , w | = p iff w ∈ V ( p ), ◮ K , w | = ♦ ϕ iff K , v | = ϕ for some v s.t. wRv . ◮ 3/ 25
Definability in Modal logic modal logics with team semantics Jonni Virtema Logics Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Expressivity Frame definability ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ♦ ϕ | � ϕ What do we study? GbTh theorem Frame definability Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary in team semantics relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Conclusion References E.g., K , w | = p iff w ∈ V ( p ), ◮ K , w | = ♦ ϕ iff K , v | = ϕ for some v s.t. wRv . ◮ 3/ 25
Definability in Modal logic modal logics with team semantics Jonni Virtema Logics Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Expressivity Frame definability ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ♦ ϕ | � ϕ What do we study? GbTh theorem Frame definability Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary in team semantics relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Conclusion References E.g., K , w | = p iff w ∈ V ( p ), ◮ K , w | = ♦ ϕ iff K , v | = ϕ for some v s.t. wRv . ◮ 3/ 25
Definability in Modal logics with team semantics modal logics with team semantics Jonni Virtema Logics Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Expressivity Frame definability ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ♦ ϕ | � ϕ. What do we study? GbTh theorem Frame definability Semantics via team-pointed Kripke structures ( W , R , V ) , T . Nonempty set W , in team semantics binary relation R ⊆ W 2 , valuation V : Φ → P ( W ), team T ⊆ W . Conclusion References E.g., K , T | = p iff T ⊆ V ( p ), ◮ iff K , T ′ | = ϕ for some T ′ such that K , T | = ♦ ϕ ◮ ∀ w ∈ T ∃ v ∈ T ′ : wRv and ∀ v ∈ T ′ ∃ w ∈ T : wRv . 4/ 25
Definability in Modal logics with team semantics modal logics with team semantics Jonni Virtema Logics Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Expressivity Frame definability ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ♦ ϕ | � ϕ. What do we study? GbTh theorem Frame definability Semantics via team-pointed Kripke structures ( W , R , V ) , T . Nonempty set W , in team semantics binary relation R ⊆ W 2 , valuation V : Φ → P ( W ), team T ⊆ W . Conclusion References E.g., K , T | = p iff T ⊆ V ( p ), ◮ iff K , T ′ | = ϕ for some T ′ such that K , T | = ♦ ϕ ◮ ∀ w ∈ T ∃ v ∈ T ′ : wRv and ∀ v ∈ T ′ ∃ w ∈ T : wRv . 4/ 25
Definability in Logics of interest modal logics with team semantics Jonni Virtema Logics Extensions of modal logic with: Expressivity ◮ Propositional dependence atoms: MDL Frame definability What do we study? dep ( p 1 , . . . , p n , q ) GbTh theorem ◮ Modal dependence atoms: EMDL Frame definability dep ( ϕ 1 , . . . , ϕ n , ψ ) in team semantics Conclusion ◮ Inclusion atoms: ML ( ⊆ ) References ◮ Intuitionistic disjunction: ML ( � ) K , T | = ϕ � ψ iff K , T | = ϕ or K , T | = ψ ◮ Universal modality: ML ( � u ) 5/ 25
Definability in Expressive power of modal logics modal logics with team semantics Jonni Virtema Theorem (Gabbay, van Benthem) Logics A class C of pointed Kripke models is definable in ML if and only if C is closed Expressivity under k-bisimulation for some k ∈ N . Frame definability What do we study? Theorem (Hella, Stumpf 2015) GbTh theorem Frame definability A nonempty class C of team-pointed Kripke models is definable in ML ( ⊆ ) if in team semantics Conclusion and only if C is union closed and there exists k ∈ N such that C is closed under References team k-bisimulation. Theorem (Hella, Luosto, Sano, V. 2014) A nonempty class C of team-pointed Kripke models is definable in ML ( � ) if and only if C is downward closed and there exists k ∈ N such that C is closed under team k-bisimulation. 6/ 25
Definability in Expressive power of modal logics modal logics with team semantics Jonni Virtema Theorem (Gabbay, van Benthem) Logics A class C of pointed Kripke models is definable in ML if and only if C is closed Expressivity under k-bisimulation for some k ∈ N . Frame definability What do we study? Theorem (Hella, Stumpf 2015) GbTh theorem Frame definability A nonempty class C of team-pointed Kripke models is definable in ML ( ⊆ ) if in team semantics Conclusion and only if C is union closed and there exists k ∈ N such that C is closed under References team k-bisimulation. Theorem (Hella, Luosto, Sano, V. 2014) A nonempty class C of team-pointed Kripke models is definable in ML ( � ) if and only if C is downward closed and there exists k ∈ N such that C is closed under team k-bisimulation. 6/ 25
Definability in Expressive power of modal logics modal logics with team semantics Jonni Virtema Theorem (van Benthem’s theorem) Logics A class C of pointed Kripke models is definable in ML if and only if C is Expressivity definable in FO and closed under bisimulation. Frame definability What do we study? Via a recent result of Kontinen, M¨ uller, Schnoor, and Vollmer on ML ( ∼ ): GbTh theorem Corollary Frame definability in team semantics Conclusion A nonempty class C of team-pointed Kripke models is definable in ML ( ⊆ ) if and References only if C is union closed, definable in FO , and closed under team bisimulation. Corollary A nonempty class C of team-pointed Kripke models is definable in ML ( � ) if and only if C is downward closed, definable in FO , and closed under team bisimulation. 7/ 25
Definability in Expressive power of modal logics modal logics with team semantics Jonni Virtema Theorem (van Benthem’s theorem) Logics A class C of pointed Kripke models is definable in ML if and only if C is Expressivity definable in FO and closed under bisimulation. Frame definability What do we study? Via a recent result of Kontinen, M¨ uller, Schnoor, and Vollmer on ML ( ∼ ): GbTh theorem Corollary Frame definability in team semantics Conclusion A nonempty class C of team-pointed Kripke models is definable in ML ( ⊆ ) if and References only if C is union closed, definable in FO , and closed under team bisimulation. Corollary A nonempty class C of team-pointed Kripke models is definable in ML ( � ) if and only if C is downward closed, definable in FO , and closed under team bisimulation. 7/ 25
Definability in Expressive power modal logics with team semantics Jonni Virtema Logics Extended modal dependence logic EMDL : Expressivity Frame definability K , T | = dep ( ϕ 1 , . . . , ϕ n , ψ ) iff ∀ w 1 , w 2 ∈ T : What do we study? � GbTh theorem � � � � { w 1 } ∈ V ( ϕ i ) ⇔ { w 2 } ∈ V ( ϕ i ) ⇒ { w 1 } ∈ V ( ψ ) ⇔ { w 2 } ∈ V ( ψ ) . Frame definability i ≤ n in team semantics Conclusion References Theorem (Hella, Luosto, Sano, V. 2014) A class of team-pointed Kripke models is definable in EMDL if and only if it is definable in ML ( � ) . 8/ 25
Definability in Validity in models and frames modal logics with team semantics Jonni Virtema ◮ Pointed model ( K , w ): Logics ( W , R , V ) , w Expressivity ◮ Model ( K ): ( W , R , V ) Frame definability ◮ Frame ( F ): ( W , R ) What do we study? GbTh theorem We write: Frame definability in team semantics ◮ ( W , R , V ) | = ϕ iff ( W , R , V ) , w | = ϕ holds for every w ∈ W Conclusion ◮ ( W , R ) | = ϕ iff ( W , R , V ) | = ϕ holds for every valuation V References Every (set of) ML -formula defines the class of frames in which it is valid. ◮ Fr ( ϕ ) := { ( W , R ) | ( W , R ) | = ϕ } . ◮ Fr (Γ) := { ( W , R ) | ∀ ϕ ∈ Γ : ( W , R ) | = ϕ } . 9/ 25
Recommend
More recommend
