On the descriptive complexity of linear algebra

Abstract

The central open question in the field of descriptive complexity theory is whether or not there is a logic that expresses exactly the polynomial-time computable properties of finite structures. It is known, from the work of Cai, Fürer and Immerman that fixed-point logic with counting ( ) does not suffice for this purpose. Recent work has shown that natural problems involving systems of linear equations are not definable in this logic. This focuses attention on problems of linear algebra as a possible source of new extensions of the logic. Here, I explore the boundary of definability in with respect to problems from linear algebra and look at suggestions on how the logic might be extended. © 2008 Springer-Verlag Berlin Heidelberg.