A view of structural complexity theory

At several recent conferences, the question “What is Structural Complexity Theory?” has been the source of some lively discussions. At this time there does not exist one commonly accepted answer but the intersection of almost all answers is nonempty. The purpose of this paper is to describe one answer to this question. We will not describe in detail recent technical results, although some will be mentioned as examples, but rather will provide comments about themes and paradigms which may be useful in organizing much of the material. We assume that the reader is familiar with (or has access to) the book Structural Complexity I, by Balcazar, Diaz, and Gabarro [BDG88]. What is desired in the formulation of a theory of computational complexity is a method for dealing with the quantitative aspects of computing. Such a method would depend upon a general theory that would provide a means for defining and studying the “inherent difficulty” of computing functions (or, more generally, solving problems). Such a theory would explain the relationships among assorted computational models and among the various complexity measures that can be defined in the context of the models and their different modes of operation, and explain why some functions are inherently difficult to compute. While any such theory must necessarily be mathematical in nature, it cannot be mathematics as such; rather, it must reflect aspects of real computing and contribute to the formal development of computer science. From the study of specific problems, it has become a widely accepted notion that a problem is not “feasible” unless it can be solved using at most polynomial space and a problem is not “tractable” unless it can be solved using at most polynomial time. Much of the effort in complexity theory has been placed on determining just what functions are