On Extroverted Complexity Theory

Complexity theory has come a long way in the thirty years since its inception. It has identiied some of the most fundamental and deep problems related to computation, it has developed a powerful methodology for attacking them, and it is broadly considered as one of the most challenging mathematical frontiers. Considered as a pure mathematical discipline in the pursuit of mathematical insight and depth, complexity is successful and well-established. In this note, however, I would like to concentrate on complexity as an applied mathematical discipline whose function is to gain insights into the problems of the natural, social, and applied and engineering sciences. This aspect has been somewhat peripheral to what we usually mean by \complexity theory," but I believe it is important for its future. Understanding the position of complexity theory within the realm of scien-tiic inquiry is an important project which is, of course, well beyond the scope of this note; here I shall only refer anecdotally to six distinct ways in which complexity theory has reached out and touch other elds: Complexity as NP-completeness. One of the major achievements of complexity theory is the living connection it has forged between application problems and modes of resource-bounded computation. This is typically done through the key notion of completeness, of which NP-completeness is of course the most popular kind. Completeness comes so natural to a complexity theorist, that it is easy to forget what an important and innuential concept it has been. In fact, outside theoretical computer science, \complexity theory" is often understood |unjustly, to be sure| as synonymous to \NP-completeness." 2 Complexity as mathematical poverty. One of the fundamental theses that seems to be almost universally accepted and practiced in computer science is that algorithms |and eecient algorithms in particular| are the natural outtow of mathematical structure discovered in applications. 3 If we accept this implication , then we must also espouse the contrapositive one, namely, that complexity is the manifestation of mathematical nastiness. Complexity has been often and brilliantly used within computer science and mathematics in this allegorical way; a computational problem is formulated and proved hard for the sole purpose of pointing out the mathematical diiculties involved in an area or approach (see 6] for an early example from database theory). Adapted from the introduction of 8] 2 Among treatments of complexity theory by scientists outside our eld this is one is not the most ignorantly …