Computers for Mathematics: Theoretical Solution versus Constructive Solution

摘要

In this article the role of technology and its pivotal―in many cases, indispensable―role in solving real-world problems is underscored. It is demonstrated that the existence of a solution to a problem, which has been proven by any means, does not guarantee a constructive solution of the problem. In general, the problems are divided into two categories, namely, type Ⅰ and type Ⅱ. For any given typeⅠ problem, there is an algorithm for constructing the solution. Type I problems are divided into two classes Ⅰ_1 and Ⅰ_2. Ⅰ_1 includes those problems, for which the existing algorithm is a polynomial one, and Ⅰ_2 includes NP-complete problems, for which no polynomial time algorithms are known. For Type Ⅱ problems, there are not any known algorithms to construct the solution, that is, one may be able to find the solution(s) by trial and error (heuristic). In this article the crucial and positive effects of technology in constructing the solution to Type Ⅱ problems are surveyed. Also NP-complete problems and their role for constructing Public-Key Cryptosystems, and different existing Digital Schemes along with the role of technology and the advances in the field are discussed. Finally Mathematica (version 4) is being used to approximate or to construct the solutions to some of the Type Ⅱ and Type Ⅰ problems.