Recent Progress on the Complexity of the Decision Problem for the Reals

This paper concerns recent progress on the computational complexity of decision methods and quantifier elimination methods for the first order theory of the reals. The paper begins with a quick introduction of terminology followed by a short survey of some complexity highlights. We then discuss ideas leading to the most (theoretically) efficient algorithms known. The discussion is necessarily simplistic as a rigorous development of the algorithms forces one to consider a myriad of details.