In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For exampl...

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the...

We show pro-definability of spaces definable types in various classical complete first order theories, including o-minimal Presburger arithmetic, $p$-adically closed fields, real and algebraically valued fields ordered differential fields. Furthermore, we prove other distinguished subspaces, some which have an interesting geometric interpretation. Our general strategy consists showing that are ...