Computer assistance for "discovering" formulas in system engineering and operator theory

The objective of this paper is two-fold. First we present a methodology for using a combination of computer assistance and human intervention to disco ver highly algebraic theorems in operator, matrix, and linear systems engin eering theory. Since the methodology allows limited human intervention, it is slightly less rigid than an algorithm. We call it a strategy. The second objective is to illustrate the methodology by deriving four theorems. The presentation of the methodology is carried out in three steps. The first st ep is introducing an abstraction of the methodology which we call an ideali zed strategy. This abstraction facilitates a high level discussion of the i deas involved, idealized strategies cannot be implemented on a computer. Th e second and third steps introduce approximations of there abstractions whi ch we call prestrategy and strategy, respectively. A strategy is more gener al than a prestrategy and, in fact, every prestrategy is a strategy. The ab ove mentioned approximations are implemented on a computer. We stress that, since there is a computer implementation, the reader can use these techniq ues to attack their own algebra problems. Thus the paper might bt of both p ractical and theoretical interest to analysts, engineers, and algebraists. Now we give the idea of a prestrategy. A prestrategy relies almost entirely on two commands which we call NCProcess1 and NCProcess2. These two command s are sufficiently powerful so that, in many cases, when one applies them r epeatedly to a complicated collection of equations, they transform the coll ection of equations into an equivalent but substantially simpler collection of equations. A loose description of a prestrategy applied to a list of eq uations is: (1) Declare which variables are known and which are unknown. At the beginni ng of a prestrategy, the order in which the equations are listed is not imp ortant, since NCProcess1 and NCProcess2 will reorder them so that the simpl est ones appear first. (2) Apply NCProcess1 to the equations; the output is a set of equations, us ually some in fewer unknowns than before, carefully partitioned based upon which unknowns they contain. (3) The user must select "important equations," especially any which solve unknown, say x. (When an equation is declared to be important or a variable is switched from being an unknown to being a known, then the way in which NCProcess1 and NCProcess2 reorder the equations is modified.) (4) Switch x to being known rather than unknown. Go to (2) above or stop. When this procedure stops, it hopefully gives the "canonical" necessary con ditions for the original equations to have a solution. As a final step we r un NCProcess2 which aggressively eliminates redundant equations and partiti ons the output equations in a way which facilitates proving that the necess ary conditions are also sufficient. Many classical theorems in analysis can be viewed in terms of solving a collection of equations. We have found tha t this procedure actually discovers the classic theorem in a modest collect ion of classic cases involving factorization of engineering systems and mat rix completion problems. One might regard the question of which classical t heorems in analysis can be proven with a strategy as an analog of classical Euclidean geometry where a major question was what can bi constructed with a compass and ruler. Here the goal is to determine which theorems in syste ms and operator theory could be discovered by repeatedly applying NCProcess 1 and NCProcess2 (or their successors) and the (human) selection of equatio ns which are important. The major practical challenge addressed here is fin ding operations which, when implemented in software. present the user with crucial algebraic information about his problem while not overwhelming him with too much redundant information. This paper consists of two parts. A de scription of strategies, a high-level description of the algorithms, a desc ription of the applications to operator, matrix, and linear system engineer ing theory, and a description of how one would use a strategy to "discover" four different theorems are presented in the first part of the paper. Thus , one who seeks a conventional viewpoint for this rather unconventional pap er might think of this as providing a unified proof of four different theor ems. Thr theorems were selected fur their diverse proofs and because: they are widely known (so that many readers should be familiar with at least one of them). The NCProcess commands use noncommutative Grobner Basis algorith ms which have emerged in the last decade, together with algorithms for remo ving redundant equations and a method for assisting a mathematician in writ ing a (noncommutative) polynomial as a composition of polynomials. The read er needs to know nothing about Grobner Basis to understand the first part o f this paper. Descriptions involving the theory of Grobner Basis appear in the second part of the paper. (C) 1999 Academic Press.