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...defined to be true in a team T of a Kripke model K if and only if for every w ∈ T there is v ∈ T such that, for all 1≤ i≤ n, K,w |= ϕi ⇐⇒ K,v |= ψi. Team semantics was originally introduced by Hodges =-=[10]-=-, who used sets of assignments of firstorder variables to define a compositional semantics for the independence-friendly logic of Hintikka and Sandu [9]. Later Väänänen observed that a dependence b...

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...ted Φ-model (K,w) and every k ∈ N there is a formula that characterises (K,w) completely up to k-equivalence. These Hintikka-formulas, which are also called characteristic formulas, are defined as in =-=[6]-=-. Definition 2.9. Assume that Φ is a finite set of proposition symbols. Let k ∈N and let (K,w) be a pointed Φ-model. The k-th Hintikka-formula χkK,w of (K,w) is defined recursively as follows: • χ0K,w...

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...h as physics, statistics, and even social choice theory. A team can also be seen as a relational database. In database theory a great variety of constraints over relations have been studied. Galliani =-=[5]-=- imported inclusion dependencies of database theory into the team semantics setting by defining inclusion atoms of the form x1 . . .xn ⊆ y1 . . .yn. This atom is satisfied in a team T if for every ass...

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.... 132 The expressive power of modal logic with inclusion atoms Instead of defining the satisfaction relation with respect to single nodes of the Kripke model, as in regular Kripke semantics (see e.g. =-=[1]-=-), here it is defined with respect to teams. Definition 2.3. The team semantics for ML is due to [17] and defined as follows: K,T |= p ⇐⇒ T ⊆V (p) K,T |= ¬p ⇐⇒ T ∩V (p) = /0 K,T |= ϕ ∧ψ ⇐⇒ K,T |= ϕ an...

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...fragments of MDL. Ebbing and Lohmann [4] showed that the model checking problem for MDL is NP-complete. Also, in recent years research on complexity questions has been active; see e.g. [11], [15] and =-=[18]-=-. The complexity of the satisfiability problem for modal inclusion logic is studied in [7]. The expressive power of MDL has been studied more systematically only recently. In [3], Ebbing et al. observ...

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...s. Sevenster showed in [14] that the satisfiability problem for MDL is complete for NEXPTIME. In [12], Lohmann and Vollmer proved complexity results for syntactic fragments of MDL. Ebbing and Lohmann =-=[4]-=- showed that the model checking problem for MDL is NP-complete. Also, in recent years research on complexity questions has been active; see e.g. [11], [15] and [18]. The complexity of the satisfiabili...

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... . . . , pn,q in dependence atoms. Furthermore, it was shown in [3] that, in terms of expressive power, EMDL is a sublogic of ML(6), the extension of modal logic with intuitionistic disjunction 6. In =-=[8]-=-, Hella et al. proved that the converse is also true, and hence EMDL and ML(6) have equal expressive power. Moreover, they gave a complete characterisation for the expressive power of these two logics...

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...c, MDL. Initially, the research on modal dependence logic was mainly concerned with complexity questions. Sevenster showed in [14] that the satisfiability problem for MDL is complete for NEXPTIME. In =-=[12]-=-, Lohmann and Vollmer proved complexity results for syntactic fragments of MDL. Ebbing and Lohmann [4] showed that the model checking problem for MDL is NP-complete. Also, in recent years research on ...

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...ow that a formula holds, while the other player tries to refute the formula. The following idea of using a function F as a winning strategy in the semantic game is due to Jonni Virtema and is used in =-=[2]-=-. Raine Rönnholm [13] introduced the idea in the context of inclusion and exclusion logic. Definition 5.1. Let K = (W,R,V ) be a Kripke model and T a team in W . Let ϕ ∈ML(▽). Let F be a function map...

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...NP-complete. Also, in recent years research on complexity questions has been active; see e.g. [11], [15] and [18]. The complexity of the satisfiability problem for modal inclusion logic is studied in =-=[7]-=-. The expressive power of MDL has been studied more systematically only recently. In [3], Ebbing et al. observed that purely propositional dependence atoms =(p1, . . . , pn,q) are too weak to express ...

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... for syntactic fragments of MDL. Ebbing and Lohmann [4] showed that the model checking problem for MDL is NP-complete. Also, in recent years research on complexity questions has been active; see e.g. =-=[11]-=-, [15] and [18]. The complexity of the satisfiability problem for modal inclusion logic is studied in [7]. The expressive power of MDL has been studied more systematically only recently. In [3], Ebbin...

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...tion Another way to express nonemptiness is to define a disjunction which requires that each of the disjuncts is satisfied in a nonempty subset. This connective was first suggested by Raine Rönnholm =-=[13]-=-. Definition 4.10. K,T |= ϕ ⊲⊳∨ψ if and only if T = /0 or there exist T1,T2 6= /0 such that T = T1 ∪ T2 and K,T1 |= ϕ and K,T2 |= ψ . Closure under k-bisimulation, closure under unions and the empty t...