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The Expressive Power of Modal Dependence Logic Jonni Virtema The Expressive Power of Backround Modal Dependence Logic Modal logic Team semantics Modal dependence logic Jonni Virtema Modal definability Succinctness Japan Advanced Institute of Science and Technology, Japan University of Tampere, Finland Bibliography jonni.virtema@uta.fi Scandinavian Logic Symposium 2014 25th of August, 2014

The Expressive Motivation and history Power of Modal Dependence Logic Logical modelling of uncertainty, imperfect information and functional Jonni Virtema dependence in the framework of modal logic. Backround Modal logic The ideas are transfered from first-order dependence logic (and Team semantics independence-friendly logic) to modal logic. Modal dependence logic Historical development: Modal definability ◮ Branching quantifiers by Henkin 1959. Succinctness ◮ Independence-friendly logic by Hintikka and Sandu 1989. Bibliography ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ IF modal logic by Tulenheimo 2003. ◮ Dependence logic by V¨ a¨ an¨ anen 2007. ◮ Modal dependence logic by V¨ a¨ an¨ anen 2008.

The Expressive Motivation and history Power of Modal Dependence Logic Jonni Virtema Backround In IF modal logic, diamonds can be slashed by boxes that precede them: Modal logic � 1 ( ♦ 2 / � 1 ) ϕ. Team semantics Modal dependence The idea in modal dependence logic ( MDL ) is quite different than in IF modal logic logic: dependences are not between states, but truth values of propositions. Modal definability Succinctness MDL is not able to express temporal dependencies; to remedy this, Ebbing et Bibliography al. 2013 introduced extended modal dependence logic ( EMDL ). Propositional dependence logic is closely related to the Inquisitive logic of Groenendijk 2007.

The Expressive Syntax for modal logic Power of Modal Dependence Logic Jonni Virtema Backround Definition Modal logic Team semantics Let Φ be a set of atomic propositions. The set of formulae for standard modal Modal dependence logic ML (Φ) is generated by the following grammar logic Modal definability ϕ ::= p | ¬ p | ( ϕ ∨ ϕ ) | ( ϕ ∧ ϕ ) | ♦ ϕ | � ϕ, Succinctness Bibliography where p ∈ Φ. Note that formulas are assumed to be in negation normal form: negations may occur only in front of atomic formulas.

The Expressive Kripke structures Power of Modal Dependence Logic Jonni Virtema Backround Modal logic Definition Team semantics Let Φ be a set of atomic propositions. A Kripke model K over Φ is a tuple Modal dependence logic Modal definability K = ( W , R , V ) , Succinctness Bibliography where W is a nonempty set of worlds , R ⊆ W × W is a binary relation, and V is a valuation V : Φ → P ( W ).

The Expressive Semantics for modal logic Power of Modal Dependence Logic Definition Jonni Virtema Kripke semantics for ML is defined as follows. Backround Modal logic K , w | ⇔ w ∈ V ( p ) . Team semantics = p Modal dependence logic K , w | = ¬ p ⇔ w �∈ V ( p ) . Modal definability Succinctness K , w | = ϕ ∨ ψ ⇔ K , w | = ϕ or K , w | = ψ. Bibliography K , w | = ϕ ∧ ψ ⇔ K , w | = ϕ and K , w | = ψ. K , w ′ | = ϕ , for some w ′ s.t. xRw ′ . K , w | = ♦ ϕ ⇔ = ϕ , for all w ′ s.t. xRw ′ . K , w | ⇔ K , w | = � ϕ

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema Backround Modal logic Team semantics Modal dependence logic Modal definability Succinctness Bibliography

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Modal logic Kripke model then T ⊆ W is a team of K . Team semantics Modal dependence logic Modal definability Succinctness Bibliography

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Modal logic Kripke model then T ⊆ W is a team of K . Team semantics 2. The standard semantics for modal logic is given with respect to pointed Modal dependence models K , w . In team semantics the semantics is given for models and logic teams, i.e., with respect to pairs K , T , where T is a team of K . Modal definability Succinctness Bibliography

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Modal logic Kripke model then T ⊆ W is a team of K . Team semantics 2. The standard semantics for modal logic is given with respect to pointed Modal dependence models K , w . In team semantics the semantics is given for models and logic teams, i.e., with respect to pairs K , T , where T is a team of K . Modal definability Succinctness 3. Some possible interpretations for K , w and K , T : Bibliography

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Modal logic Kripke model then T ⊆ W is a team of K . Team semantics 2. The standard semantics for modal logic is given with respect to pointed Modal dependence models K , w . In team semantics the semantics is given for models and logic teams, i.e., with respect to pairs K , T , where T is a team of K . Modal definability Succinctness 3. Some possible interpretations for K , w and K , T : Bibliography (a) K , w | = ϕ : The actual world is w and ϕ is true in w .

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Modal logic Kripke model then T ⊆ W is a team of K . Team semantics 2. The standard semantics for modal logic is given with respect to pointed Modal dependence models K , w . In team semantics the semantics is given for models and logic teams, i.e., with respect to pairs K , T , where T is a team of K . Modal definability Succinctness 3. Some possible interpretations for K , w and K , T : Bibliography (a) K , w | = ϕ : The actual world is w and ϕ is true in w . (b) K , T | = ϕ : The actual world is in T , but we do not know which one it is. The formula ϕ is true in the actual world.

The Expressive Team semantics? Power of Modal Dependence Logic Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Modal logic Kripke model then T ⊆ W is a team of K . Team semantics 2. The standard semantics for modal logic is given with respect to pointed Modal dependence models K , w . In team semantics the semantics is given for models and logic teams, i.e., with respect to pairs K , T , where T is a team of K . Modal definability Succinctness 3. Some possible interpretations for K , w and K , T : Bibliography (a) K , w | = ϕ : The actual world is w and ϕ is true in w . (b) K , T | = ϕ : The actual world is in T , but we do not know which one it is. The formula ϕ is true in the actual world. (c) K , T | = ϕ : We consider sets of points as primitive. The formula ϕ describes properties of collections of points.

The Expressive Team semantics for modal logic Power of Modal Dependence Logic Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Modal logic K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence logic K , w | = p ⇔ w ∈ V ( p ) . Modal definability K , w | = ¬ p ⇔ w / ∈ V ( p ) . Succinctness K , w | = ϕ ∧ ψ ⇔ K , w | = ϕ and K , w | = ψ. Bibliography K , w | = ϕ ∨ ψ ⇔ K , w | = ϕ or K , w | = ψ. K , w ′ | = ϕ for every w ′ s.t. wRw ′ . K , w | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔

The Expressive Team semantics for modal logic Power of Modal Dependence Logic Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Modal logic K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence logic K , T | = p ⇔ T ⊆ V ( p ) . Modal definability K , T | = ¬ p ⇔ T ∩ V ( p ) = ∅ . Succinctness K , T | = ϕ ∧ ψ ⇔ K , T | = ϕ and K , T | = ψ. Bibliography K , w | = ϕ ∨ ψ ⇔ K , w | = ϕ or K , w | = ψ. K , w ′ | = ϕ for every w ′ s.t. wRw ′ . K , w | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔

The Expressive Team semantics for modal logic Power of Modal Dependence Logic Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Modal logic K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence logic K , T | = p ⇔ T ⊆ V ( p ) . Modal definability K , T | = ¬ p ⇔ T ∩ V ( p ) = ∅ . Succinctness K , T | = ϕ ∧ ψ ⇔ K , T | = ϕ and K , T | = ψ. Bibliography K , T | = ϕ ∨ ψ ⇔ K , T 1 | = ϕ and K , T 2 | = ψ for some T 1 ∪ T 2 = T . K , w ′ | = ϕ for every w ′ s.t. wRw ′ . K , w | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔