Using Matching in Algebraic Equation Solving

This paper describes the use of powerful a lgebra ic matching techniques f o r apply ing rewr i t e ru les In equat ion s o l v i n g . A matcher Is presented tha t knows about the commutat iv l ty and a s s o c i a t i v i t y o f a d d i t i o n and m u l t i p l i c a t i o n , w i l l provide de fau l t s f o r missing summands and f a c t o r s , and i f necessary w i l l solve a l g e b r a i c a l l y f o r the value o f pa t te rn v a r i a b l e s . able to der ive the so lu t ions to several of these hard problems. The PRESS so lu t ions c o n s t i t u t e a r a t i o n a l reconst ruc t ion of part of the h i s t o r y of a lgebra : we are not saying that t h i s is how these so lu t ions were f i r s t d iscovered, but that our program does show how they could have been discovered by a r a t i o n a l process of ana lys is of the problem, in terms o f the too l s ava i l ab le fo r i t s s o l u t i o n . 1 . I n t r o d u c t i o n This paper describes the use of powerful matching techniques in a lgebra ic equat ion s o l v i n g . This work bu i l ds on the PRESS algebra system, a computer program f o r so lv ing equations and I n e q u a l i t i e s and f o r s i m p l i f y i n g expressions [ 4 ] . A powerful matcher, upon which the present work is based, I s proposed i n [ 3 ] ; t h i s repor t a lso describes many of the ideas in PRESS. PRESS and the matcher extensions are w r i t t e n in PROLOG [ 7 ] . The goal of the resesrch described here is to t e s t the search con t ro l technique of metaleve l inference and the powerful a lgebra ic manipulat ion methods of PRESS on some hard problems. We consider the problems of so lv ing the general quadrat ic and cubic equations and a general t r igonometr ic equat ion . The so lu t ions of these problems given in standard algebra textbooks a l l have a "magic" element. An expression is drawn out of a hat w i th a f l o u r i s h of "Consider the f o l l ow ing term • • • " • As a side e f f e c t of app ly ing the PRESS equation so lv ing methods, we show how the magic element can be understood. In f a c t , the program Is PRESS uses m u l t i p l e sets of rewr i te r u l e s , employing metaleve l reasoning and descr ip t ions to guide that a p p l i c a t i o n and hence con t ro l search. Some of the Important rewr i te ru le sets a re : I s o l a t i o n PRESS t r i e s apply ing I s o l a t i o n ru les when there is a s ing le occurrence of the unknown in the equat ion. I s o l a t i o n ru les are appl ied to s t r i p away surrounding funct ions and operators from the unknown, f i n a l l y r e s u l t i n g in an equation w i t h the unknown on one side by i t s e l f , and some expression ( f r e e of the unknown) on the o the r . A t y p i c a l i s o l a t i o n ru le is a rcs in x b -> x = s i n b. c o l l e c t i o n Co l l ec t i on ru les serve to reduce the number of occurrences of the unknown, so tha t i s o l a t i o n can be app l i ed . A t y p i c a l c o l l e c t i o n ru le i s uw + vw -> (u+v)w which c o l l e c t s r e l a t i v e to w. Computing resources f o r t h i s research were provided by Science Research Counci l grant number GR/A 37954. A. Borning was supported at the Un ive rs i t y of Edinburgh by a NATO Poatdoctoral Fel lowship from the Nat iona l Science Foundation. Current address: Computer Science Department, FR-35 Un ive rs i t y of Washington S e a t t l e , Washington 98195 USA a t t r a c t i o n A t t r a c t i o n ru les move occurrences of the unknown c loser together in the expression t r e e , so that perhaps a c o l l e c t i o n r u l e can be app l ied . A sample ru le is log bu + log b v -> log b uv , which a t t r a c t s u and v. To apply a rewr i t e ru le to an expression, PRESS uses a matcher tha t knows about the commutat iv l ty and a s s o c i a t i v i t y of a d d i t i o n and m u l t i p l i c a t i o n . For example, to apply the c o l l e c t i o n ru le uw + vw -> (u+v)w