1. Introduction

Abstract

During the last half a dozen years computational complexity has been strongly influenced and advanced by the investigation of efficient reductions between problems and problem classes. In particular, the study of the deterministic and nondeterministic polynomial time computations, P and NP, revealed the existence of “natural” complete problems in the class NP to which all other problems in this class can be efficiently reduced. Furthermore, it was shown that many problems of practical importance are contained in the class NP and that many of these problems are complete, thus showing, quite surprisingly, that a fast algorithm for any one of these (complete) problems can be effectively translated into fast algorithms for all other problems in this class [11], [31].Subsequent research on feasible computations has progressed very rapidly and has shown that there exist rich structural relations, revealed by efficient reductions, between different classes of computations, and it has increased our understanding feasible computations quite dramatically. As a matter of fact, this work has revealed a deep unity of this research area and has identified several super problems whose solution seems to be essential for a complete understanding of the structure of feasible computations. Thus these results have not only added to our understanding of the quantitative aspect of computing but have also unified a central field of research in computer science and are very likely to strongly influence its further development by creating a consensus about what problems are important and should be thoroughly investigated [1], [11], [24], [25], [31], [41].It may only be a slight exaggeration to claim that in the 1930s we started to understand what is and is not effectively computable and that in the 1970s we started to understand what is and is not practically or feasibly computable. There is no doubt that the results about what can and cannot be effectively computed or formalized in mathematics have had a profound influence on mathematics, and, even more broadly, they have influenced our view of our scientific methods. We believe that the results about what can and cannot be practically computed will also have a major influence on computer science, mathematics and, even though more slowly, will affect other research areas and influence how we think about scientific theories.The purpose of this monograph is to give an overview of some of the more recent developments in the study of the structure of feasible computations and in the investigation of provable properties of complexity of computations and their relations to feasible computations. This monograph treats only a select set of topics trying to show some of the techniques of this research, giving some insight into the results and emphasizing the unity of this research area.