Starting with Cook’s pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time computability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability sometimes also yield interesting new practical algorithms. A typical example is the application of the ellipsoid algorithm to combinatorial optimization problems (see, for example, Lovász ). On the other hand, it has a strong influence on many different branches of mathematics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a constructive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of permutation groups.
KeywordsReal Function Complexity Theory Random Access Machine Finite String Recursive Analysis
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