Formulation of linear problems and solution by a universal machine

Abstract

Using the predicate language for ordered fields a class of problems referred to aslinear problems is defined. This class contains, for example, all systems of linear equations and inequalities, all linear programming problems, all integer programming problems with bounded variables, all linear complementarity problems, the testing of whether sets that are defined by linear inequalities are semilattices, all satisfiability problems in sentenial logic, the rank-computation of matrices, the computation of row-reduced echelon forms of matrices, and all quadratic programming problems with bounded variables. A single, one, algorithm, to which we refer as theUniversal Linear Machine, is described. It solves any instance of any linear problem. The Universal Linear Machine runs in two phases. Given a linear problem, in the first phase a Compiler running on a Turing Machine generates alinear algorithm for the problem. Then, given an instance of the linear problem, in the second phase the linear algorithm solves the particular instance of the linear problem. The linear algorithm is finite, deterministic, loopless and executes only the five ordered field operations -- additions, multiplications, subtractions, divisions and comparisons. Conversely, we show that for each linear algorithm there is a linear problem which the linear algorithm solves uniquely. Finally, it is shown that with a linear algorithm for a linear problem, one can solve certain parametric instances of the linear problem.