Characterizing Relative Frame Definability in Team Semantics via The Universal Characterizing Relative Frame Definability in Modality Jonni Virtema Team Semantics via The Universal Modality Definability Modal logic Frame definability Jonni Virtema What do we study? GbTh theorem University of Helsinki, Finland jonni.virtema@gmail.com Team semantics Extensions of ML Joint work with Katsuhiko Sano, JAIST, Japan Frame definability in team semantics Conclusion WoLLIC 2016 17th of August 2016 1/ 28
Characterizing Relative 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 Extensions of ML Frame definability in team semantics Conclusion 2/ 28
Characterizing Definability Relative 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? GbTh theorem ◮ Set of formulae: Team semantics � {∃ x 1 . . . x n ¬ x i = x j | n ∈ N } Extensions of ML i � = j ≤ n Frame definability in team semantics defines the class of infinite graphs. Conclusion 3/ 28
Characterizing Relative Definability Relative Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Which properties of graphs can be described with a given logic L if restricted to Definability some class of graphs C . Modal logic Frame definability Example monadic second-order logic on finite ordered graphs G = ( V , E ): What do we study? ◮ Single formula: GbTh theorem � � � ∃ X ∃ xy min ( x ) ∧ X ( x ) ∧ max ( y ) ∧ X ( y ) Team semantics Extensions of ML �� � ∧∀ xy succ ( x , y ) → ( X ( x ) ↔ ¬ X ( y )) Frame definability in team semantics defines the class { ( V , E ) | | V | is odd } relative to finite ordered graphs. Conclusion 4/ 28
Characterizing Modal logic Relative 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 ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � ϕ. Modal logic Frame definability What do we study? Semantics via pointed Kripke structures ( W , R , V ) , w . Nonempty set W , binary GbTh theorem relation R ⊆ W 2 , valuation V : Φ → P ( W ), point w ∈ W . Team semantics Extensions of ML 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 . ◮ 5/ 28
Characterizing Validity in models and frames Relative Frame Definability in Team Semantics via The Universal Modality ◮ Pointed model ( K , w ): ( W , R , V ) , w . Jonni Virtema ◮ Model ( K ): ( W , R , V ). Definability ◮ Frame ( F ): ( W , R ). Modal logic Frame definability We write: What do we study? GbTh theorem ◮ ( W , R , V ) | ( W , R , V ) , w | = ϕ holds for every w ∈ W . = ϕ iff Team semantics ◮ ( W , R ) | = ϕ iff ( W , R , V ) | = ϕ holds for every valuation V . Extensions of ML 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 } . ◮ Fr (Γ) := { ( W , R ) | ∀ ϕ ∈ Γ : ( W , R ) | = ϕ } . 6/ 28
Characterizing Frame definability Relative Frame Definability in Team Semantics via The Universal Modality Which classes of frames are definable by a (set of) modal formulae. Jonni Virtema Which classes are definable by a (set of) modal formulae within the class F fintra Definability of finite transitive frames. Modal logic Frame definability Examples: What do we study? Formula Property of R GbTh theorem � p → p Reflexive ∀ w ( wRw ) Team semantics p → �♦ p Symmetric ∀ wv ( wRv → vRw ) Extensions of ML � p → �� p ∀ wvu (( wRv ∧ vRu ) → wRu ) Frame definability Transitive in team semantics ♦ p → �♦ p ∀ wvu (( wRv ∧ wRu ) → vRu ) Euclidean Conclusion � p → ♦ p ∀ w ∃ v ( wRv ) Serial � ( � p → p ) → � p Irreflexive w.r.t F fintra ∀ wv ¬ ( wRv ) 7/ 28
Characterizing Goldblatt-Thomason Theorem (1975) Relative Frame Definability in Team Semantics via The Universal Set Φ of atomic propositions. The formulae of ML (Φ) are generated by: Modality Jonni Virtema ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � ϕ. Definability Modal logic Frame definability Theorem What do we study? GbTh theorem An elementary frame class is ML -definable iff Team semantics ◮ it is closed under taking Extensions of ML ◮ bounded morphic images Frame definability ◮ generated subframes in team semantics ◮ disjoint unions Conclusion ◮ and its complement is closed under taking ◮ ultrafilter extensions. 8/ 28
Characterizing Goldblatt-Thomason Theorem (Goranko, Passy 1992) Relative Frame Definability in Team Semantics via The Universal u ) are generated by: The formulae of ML ( � Modality Jonni Virtema u ϕ. ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � ϕ | � Definability Modal logic u ϕ Frame definability K , w | ↔ ∀ v ∈ W : K , v | = � = ϕ. What do we study? GbTh theorem Team semantics Theorem Extensions of ML u ) -definable iff An elementary frame class is ML ( � Frame definability in team semantics ◮ it is closed under taking Conclusion ◮ bounded morphic images ◮ and its complement is closed under taking ◮ ultrafilter extensions. 9/ 28
Characterizing Goldblatt-Thomason Theorem (Sano, V. 2015) Relative Frame Definability in Team Semantics via The Universal u + ) are generated by: The formulae of ML ( � Modality Jonni Virtema u ϕ. ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | � ϕ | ♦ ϕ | � Definability Modal logic Frame definability Theorem What do we study? u + ) -definable iff GbTh theorem An elementary frame class is ML ( � Team semantics ◮ it is closed under taking Extensions of ML ◮ bounded morphic images Frame definability ◮ generated subframes in team semantics ◮ and it reflects Conclusion ◮ ultrafilter extensions, ◮ finitely generated subframes. 10/ 28
Characterizing Goldblatt-Thomason Theorem in the Finite Relative Frame Definability in Team Semantics via The Universal Modality Theorem (van Benthem 1988) Jonni Virtema A class of finite transitive frames is ML -definable within the class F fintra of all Definability Modal logic finite transitive frames if and only if it is closed under taking Frame definability ◮ bounded morphic images, What do we study? ◮ generated subframes, GbTh theorem ◮ disjoint unions. Team semantics Extensions of ML Frame definability Theorem (Gargov, Goranko 1993) in team semantics Conclusion u ) -definable within the class F fin of all finite A class of finite frames is ML ( � frames if and only if it is closed under taking bounded morphic images. 11/ 28
Characterizing What do we study? Relative Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Definability ◮ Frame definability in ML ( � u + ) within finite transitive frames. Modal logic ◮ Frame (model) definability of particular team based modal logics: Frame definability ◮ Modal dependence logics MDL and EMDL . What do we study? ◮ Modal inclusion logics MINC and EMINC . GbTh theorem ◮ Modal team logic MT L . Team semantics u + ) coincides with that of ◮ Note: Frame (model) definability of ML ( � Extensions of ML EMDL (Sano, V. 2015). Frame definability in team semantics Conclusion 12/ 28
Characterizing What do we show? Relative Frame Definability in Team Semantics via The Universal Modality ◮ Variant of the Goldblatt-Thomason theorem for ML ( � u + ) within F fintra . Jonni Virtema Definability ◮ We show the following trichotomy with respect to model definability: Modal logic Frame definability u + ) , MT L} {ML , MINC , EMINC} < MDL < {EMDL , ML ( � What do we study? GbTh theorem Team semantics ◮ We show the following dichotomy with respect to frame definability: Extensions of ML Frame definability u + ) , MT L} . {ML , MINC , EMINC} < {MDL , EMDL , ML ( � in team semantics Conclusion The expressive powers of all of the logics above differ. 13/ 28
u + ) within finite transitive frames Frame definability in ML ( � Characterizing Relative Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Definability Theorem Modal logic u + ) -definable within the class F fintra of A class of finite transitive frames is ML ( � Frame definability all finite transitive frames if and only if it is closed under taking What do we study? ◮ bounded morphic images, GbTh theorem Team semantics ◮ generated subframes. Extensions of ML Frame definability u ¬ ϕ F , w . The proof uses Jankov-Fine formulas ϕ F of the type � ( F ) � in team semantics w ∈ dom Conclusion 14/ 28
Characterizing Relative Frame Definability in Team Semantics via The Universal Modality Jonni Virtema Definability Modal logic Frame definability PART II What do we study? GbTh theorem Team semantics Extensions of ML Frame definability in team semantics Conclusion 15/ 28
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