Approximation and dependence via multiteam semantics

Abstract

We define a variant of team semantics called multiteam semantics based on multisets and study the properties of various logics in this framework. In particular, we define natural probabilistic versions of inclusion and independence atoms and certain approximation operators motivated by approximate dependence atoms of Väänänen.

Characterizing Strongly First Order Dependencies: The Non-Jumping Relativizable Case

“…What is then the point of Team Semantics? In brief, Team Semantics allows us to extend First Order Logic in new ways, like for instance by adding new types of atoms describing dependencies between different assignments: 3 We need at least two elements in our model in order to encode disjunctions in terms of existential quantifications in Proposition 21 and Theorem 30. The case in which only one element exists is in any case trivial, and may be dealt with separately if required.…”

“…It is also of more direct practical interest, because of the connections between Team Semantics and Database Theory (see for instance [16,24]). Probabilistic variants of Team Semantics have recently gathered attention (see for instance [3,4,15]).…”

Continuous Team Semantics

“…The introduction of Independence Logic (FO(⊥)) in [9] and inclusion (x ⊆ y) and exclusion atoms (x | y) (and the corresponding logics) [8] demonstrated the versatility of the framework and have led to several studies on the applications of team semantics in areas such as database theory, model theory, and quantum information theory (see, e.g. [10,13,6,16,12,1]).…”

“…In the metric context, it is very natural to consider approximate versions of the usual dependence and independence atoms (see, e.g., [15,23,6] for related previous work on so-called metric functional dependencies and approximate dependencies in (non-metric) team semantics). As an added bonus, the approximate versions are compatible with our notions of well-behaved formulae, and thus greatly increase what we can express directly without leaving the realm of tameness.…”

Polyteam Semantics

“…Multiteam semantics is motivated by the fact that multisets are widely assumed in database theory and occur in applications. Multiteam semantics can be also used to model and study database, probabilistic, and approximate dependencies in a unified framework (see [3,23]).…”