Constraint Algebra-A Supervisory Programming Technique and a Cognitive Process

Abstract

Mathematical models of complex physical or bionic systems involve many simultaneous nonlinear equations. These groups of relationships are difficult to manipulate and even simulalation on a computer is unwieldy because most computational paths are multidirectional and are either over-or under-constrained. The foundation and purposes for an algebra of contraints are outlined in this paper. A typical application of constraint algebra would be as a supervisory routine for a digital program that operates on the topological properties of the set of the equations and determines the allowable computational paths. At the conclusion of these logical operations, which are performed with the aid of a constraint matrix, normal programming can be employed for the quantitative operations on the allowable paths. Thus, one more rational function in the man/computer relationship-that of the generation of perfectly constrained relationships-can now be taken over by the computer. The inclusion of a theorem from thermodynamics allows quite a different application: new variables may be deduced from the constraints which, together with their corresponding equations, simplify the model. This ability to synthesize new concepts (variables) and relationships (equations) which tend to simplify models can be considered as an analog for the cognitive process of abstraction.