Article

Pitambar Sai Goyal, Loyola – ICAM College of Engineering and Technology (LICET)

There is a provision in the rules of set theory and the set theoretic definition of 'algorithm' which allows for condensation of the existing representations of algorithms, drawing on certain classical results of Cantor and Godel. This condensation was also completely effective on formalizations of programs running on a Turing Machine. An equivalence is demonstrated between an algorithmic process and a dynamic physical process involving the motion of a particle in a certain impulse field. This does not condense the problem in the context of modern digital computers, but greatly condenses it for the human brain and certain analog systems, as will be demonstrated. Two theorems will be presented, regarding formalism of 'algorithm', one by which we will see that all computationally solvable problems, with finite sized inputs, are equivalent to the solving of an equation, f(x) =0 for an integer valued function f on the integers. The second theorem will demonstrate that a large subclass of these problems are reduced to the factorization of an integer. It is understood that the invention of alternate formal representations of algorithms may be critical to several problems of computation, including the P vs NP problem.