New methods for proving the impossibility to solve problems through reduction of problem spaces

Abstract

Problem solvers are computational systems which make use of different search algorithms for solving problems. Sometimes, while employing such search algorithms, problem solvers may prove to be inefficient and take too great an effort so as to showing that the problem has no solution. For such cases, in this paper we explain a technique which provides a quick proof that finding a solution is actually impossible. This technique results in reducing the number and simplifying the topology of the states which shape a problem space. Hence, we show and prove efficient new techniques intended to find such reductions which may result to be very useful for many problems.

On the existence of solutions in systems of linear Diophantine equations

“…Systems of linear Diophantine equations, although harder to solve, are especially important with regard to fields such as integer linear programming or to solve some puzzles [10]. Some research has been so far made concerning algorithms designed to find the minimal solutions for a system of the form Ax = 0, where x > 0 [1][2][3][4][5]7,11,14,17].…”