Inclusion and exclusion dependencies in team semantics — On some logics of imperfect information

We study model and frame definability of various modal logics. Let ML( u + ) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We show that a class of Kripke models is definable in ML( u + ) if and only if the class is elementary and closed under disjoint unions and surjective bisimulations. We also characterise the definability of ML( u + ) in the spirit of the well-known Goldblatt-Thomason theorem. We show that an elementary class F of Kripke frames is definable in ML( u + ) if and only if F is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition we study frame definability relative to finite transitive frames and give an analogous characterisation of ML( u + )-definability relative to finite transitive frames. Finally, we initiate the study of model and frame definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. We establish strict linear hierarchies with respect to model definability and frame definability, respectively. We show that, with respect to model and frame definability, the before mentioned team-based logics, except modal dependence logic, either coincide with ML( u + ) or plain modal logic ML. Thus as a corollary we obtain model theoretic characterisation of model and frame definability for the team-based logics.This article subsumes and extends the conference articles [30] and [31].

show abstract

“…The following proposition is proven in the same way as the analogous propositions for first-order dependence logic [33] and inclusion logic [9].…”

Section: Extensions Of Modal Logic With Atomic Dependency Notionsmentioning

confidence: 69%

Characterizing Frame Definability in Team Semantics via the Universal Modality

Sano

Virtema

2015

Logic, Language, Information, and Computation

View full text Add to dashboard Cite

show abstract

“…However, the first order logical consequences of dependence logic sentences can be axiomatized and such axioms are given in [14]. There are many other new atomic formulas that suggest themselves in the team semantics context, for example the independence atom [12] x?y with the meaning that x and y are "independent" (the values of x reveal nothing about the values of y), and the inclusion atom [7] x ✓ y with the meaning that values of x occur also as values of y,…”

Section: Multiverse and Team Semanticsmentioning

confidence: 99%

“…This is reminiscent of the formulax ?ỹ in [12] (see also [7]). The idea is that M 00 picks the truth value of ' from M and the truth value of from M 0 .…”

Section: ✓ ✓mentioning

confidence: 99%

See 1 more Smart Citation

Multiverse set theory and absolutely undecidable propositions

Väänánen¹

2014

Interpreting Gödel

View full text Add to dashboard Cite

“…First note that inclusion and dependence atoms can be expressed in FO(⊥ c ) [9,13]. Also it is easy to see that one can construct existential secondorder logic sentences that capture probabilistic inclusion and conditional independence atoms over teams of fixed domain.…”

Section: Probabilistic Notions In Team Semanticsmentioning

confidence: 99%