### 21.06.15 - 26.06.15, Seminar 15261

# Logics for Dependence and Independence

### Diese Seminarbeschreibung wurde vor dem Seminar auf unseren Webseiten veröffentlicht und bei der Einladung zum Seminar verwendet.

## Motivation

Dependence and independence are interdisciplinary notions that are pervasive in many areas of science. They appear in domains such as mathematics, computer science, statistics, quantum physics, and game theory. The development of logical and semantical structures for these notions provides an opportunity for a systematic approach, which can expose surprising connections between different areas, and may lead to useful general results.

Dependence Logic and more generally logics with team semantics are new tools for modeling dependencies and interaction in dynamical scenarios. Reflecting this, it has higher expressive power and complexity than classical logics used for these purposes previously. Algorithmically, first-order dependence logic corresponds exactly to the complexity class NP and to the so-called existential fragment of second-order logic. Since the introduction of dependence logic in 2007, the framework has been generalized, e.g., to the contexts of modal, intuitionistic, and probabilistic logic. Moreover, interesting connections have been found to complexity theory, database theory, statistics, and dependence logic has been applied in areas such as linguistics, social choice theory, and physics. Although significant progress has been made in understanding the computational side of these formalisms, still many central questions remain unsolved so far.

**Applications of logic with team semantics**

Dependence logic has interesting connections to computer science, linguistics, game theory, social choice theory, philosophy, and physics. It is expected to be very fruitful to bring researchers in fundamental issues together with researchers in the application areas, since this will influence the development of further generalizations and variants of dependence logic. An interesting new connection was made in 2013 when it was realized that the logical framework of so-called Inquisitive Semantics, that analyzes information exchange through communication, is essentially equivalent to the team semantics of dependence logic. This connection boosted the development of propositional variants of dependence logic, which were well understood in the inquisitive semantics research community.

**Model theory for variants of logics with team semantics**

As mentioned above, first-order dependence logic is equal in expressibility to existential second-order logic. With, e.g., the addition of the classical negation or the intuitionistic implication the expressive power of the logic increases up to full second-order logic. However there are variants of dependence logic whose expressive power and many model theoretical properties are not yet well understood. One such example is the so-called first-order inquisitive logic. Other examples come from the area of modal team-based logics, e.g., modal independence logic.

**Complexity issues in logics with team semantics**

For many of the generalizations of dependence logic, basic algorithmic questions such as satisfiability and model checking have not been classified from a computational complexity point of view. Important in this context is the concept of succinctness. E.g., modal dependence logic is known to be equal in expressive power to usual modal logic, however, formulas of the first logic tend to be much shorter than those of the second, which implies a considerable raise in complexity of the satisfiability problem (from PSPACE to NEXPTIME). There are numerous further open complexity issues in team-based logics on Kripke structures such as modal dependence and independence logic.

Also, very little is known at the moment about tractable fragments of dependence logic (except the recent result of Galliani and Hella stating the correspondence between inclusion logic and PTIME). For the applications areas (to be mentioned below) it is very important that the main algorithmic issues (satisfiability, validity, inference, model checking) have efficient solutions. Hence, it is of immense importance to identify fragments of dependence logic and its variants that on the one side allow efficient algorithms and on the other side are still expressive enough for the applications. A seminar as the one proposed here is the ideal opportunity to attack questions like these, since it will bring together researchers working in fundamental areas as well as researchers from the application areas.

**Game theory, category theory, and team-based logics**

The semantics of first-order dependence logic and modal dependence logic can be formulated in game theoretical terms. In fact, historically the semantics of independence friendly logic was first formulated in terms of games only. In 1997, Hodges developed a compositional team semantics for independence friendly logic, the analogue of which is also used for dependence logic. The tight connection between dependence logic and game theory, however, has not been exploited so far from a computational point of view.

Abramsky and Väänänen in a paper from 2009 point out, how many semantical issues in dependence theory can conveniently be formalized and studied in the language of category theory. This connection deserves further study as well.