Fixed-parameter decidability: Extending parameterized complexity analysis

Abstract

We extend the reach of fixed-parameter analysis by introducing classes of parameterized sets defined based on decidability instead of complexity. Known results in computability theory can be expressed in the language of fixed-parameter analysis, making use of the landscape of these new classes. On the one hand this unifies results that would not otherwise show their kinship, while on the other it allows for further exchange of insights between complexity theory and computability theory. In the landscape of our fixed-parameter decidability classes, we recover part of the classification of real numbers according to their computability. From this, using the structural properties of the landscape, we get a new proof of the existence of P-selective bi-immune sets. Furthermore, we show that parameter values in parameterized sets in our uniformly fixed-parameter decidability classes interact with both instance complexity and Kolmogorov complexity. By deriving a parameter based upper bound on instance complexity, we demonstrate how parameters convey a sense of randomness. Motivated by the instance complexity conjecture, we show that the upper bound on the instance complexity is infinitely often also an upper bound on the Kolmogorov complexity.

Levelable Sets and the Algebraic Structure of Parameterizations

“…The following theorem gives a time bounded version of Lemma 6.1 in [Witteveen and Torenvliet, 2016].…”