Expressibility as a Complexity Measure: Results and Directions

Given a property, S, one can discuss the computational complexity of checking whether or not an input satisfies S. One can also ask, "What is the complexity of expressing the property S?" It is natural that these two questions are related. However, it is startling how closely tied they are when the second question refers to expressing the property in first-order logic. In this article we survey some work relating first-order expressibility to computational complexity, and present a number of open problems. Most complexity theory is done from the point of view of Turing machines, or boolean circuits. We believe that first-order logic provides a valuable different point of view, adding fresh insights to well studied problems.