Solving Problems by Formula Manipulation in Logic and Linear Inequalities

Using formal logic, many iproblems from the general area of linear inequalities co~ be expressed in the elementary theory o f addition on the real numbers (EAR). We describe a method for eliminatit~ quantiflers in EAR which has been programmed arm demonstrate its usefulness in solving some problems related to linear programming. In the area o f mechanical mathematics this kind o f approach has been neglected in favor o f more generalized method r based on Herbrand expansion. However, in a restricted area, such as linear inequalities, the use o f these ~pecialized methods can increaye e.O~cieney by several orders of maonitud~ over an axiomatic Herbrand approach, aJut make practical problems accessible. As is common in a~ificial intelligence, the work reported here is of an interdisciplinary nature. It involves mathematical logic, linear inequalities, and symbolic mathematics on a computer. For the sake of a~gument, let us distinguish two kinds of workers in the area of linear inequalities. There is the theoretician, who is developing new methods and discovering new theorems. Then there is the user, who is faced with a practical problem which can be expressed in sorae way at least piecewise linearly. As a simple-minded distinction between the theoretician and the user we can say that the latter is interested in questions involving a fixed number of variables,, while the former is concerned with questions involving an arbitrary number of variables. Using terminology from iogie to be made more precise below ~this means that the ,•ser is generally working within the elementary J~eory of additi<m on the reals while the ~Lheoretician is generally working on a higher level. * This is a revised version of a paper pres+;nted at the Inter~mional Joint Confe]x'm-e on Artificial Intelligence, London, September 1971. Artificial Intelligenee 3 (1972), 165--i74 Copyright ~) 1972 by North-Holland Publishing Company