Team Semantics and Recursive Enumerability

TL;DR

The paper introduces an extension of dependence logic, D*, capable of defining all recursively enumerable classes of finite models, thereby offering a novel logical framework for computation beyond Turing machines.

Contribution

It presents D*, an extension of dependence logic with an operator for domain extension, capturing all recursively enumerable classes, and explores generalized quantifiers in team semantics.

Findings

D* characterizes all recursively enumerable classes.

The domain extension operator enables new computational expressiveness.

Generalized quantifiers can be eliminated via dependence atoms.

Abstract

It is well known that dependence logic captures the complexity class NP, and it has recently been shown that inclusion logic captures P on ordered models. These results demonstrate that team semantics offers interesting new possibilities for descriptive complexity theory. In order to properly understand the connection between team semantics and descriptive complexity, we introduce an extension D* of dependence logic that can define exactly all recursively enumerable classes of finite models. Thus D* provides an approach to computation alternative to Turing machines. The essential novel feature in D* is an operator that can extend the domain of the considered model by a finite number of fresh elements. Due to the close relationship between generalized quantifiers and oracles, we also investigate generalized quantifiers in team semantics. We show that monotone quantifiers of type (1) can be canonically eliminated from quantifier extensions of first-order logic by introducing corresponding generalized dependence atoms.