Discontinuous Phenomena and Turing Definability

Definability, automorphisms, and dynamic properties of computably enumerable sets

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of FelixKlein’sErlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and theMartin Invariance Conjecture on their Tur...

Fixed-Parameter Tractability, Definability, and Model-Checking

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...

A World Where Relative Definability Coincides With Relative Recursiveness (i.e., Turing Reducibility)

Turing Definability in the Ershov Hierarchy

We obtain the first nontrivial d.c.e. Turing approximation to the class of computably enumerable (c.e.) degrees. This depends on the following extension of the splitting theorem for the d.c.e. degrees: For any d.c.e. degree a, any c.e. degree b, if b < a, then there are d.c.e. degrees x0,x1 such that b < x0,x1 < a and a = x0 ∨ x1. The construction is unusual in that it is incompatible with uppe...

Global Properties of the Turing Degrees and the Turing Jump

We present a summary of the lectures delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of the Turing jump within D.