ALGEBRAIC SIMPLIFICATION a guide for perplexed

Introduction S i m p l i f i c a t i o n for the Sake of C o m p r e h e n s i o n - The Needs of Users 2.1 Conventional Lexicographic Ordering of E x p r e s s i o n s 2.2 S u b s t i t u t i o n as an Aid to C o m p r e h e n sion 3.0 S i m p l i f i c a t i o n for the Sake of Efficient M a n i p u l a t i o n - W h a t D e s i g n e r s P r o v i d e 3.1 The P o l i t i c s of S i m p l i f i c a t i o n 3.1.1 The R a d i c a l s 3.1.2 The New Left 3.1.3 The L i b e r a l s 3.1.4 The C o n s e r v a t i v e s 3.1.5 The C a t h o l i c s 3.2 I n t e r m e d i a t e E x p r e s s i o n Swell 3.3 C a n o n i c a l S i m p l i f i e r s and T h e o r e t i c a l Results - The R a d i c a l s R e v i s i t e d 3.3.1 S i m p l i f i c a t i o n A l g o r i t h m s for Expressions with Nested Exponentials 3.3.2 Expressions involving Exponentials and L o g a r i t h m s 3.3.3 Roots of P o l y n o m i a l s 3.3.4 U n s o l v a b i l i t y Results 4.0 P r o s p e c t s for the F u t u r e References Figures 1.0 Introduction
or
+ 41/3 ) - 6 = 0
and e f f i c i e n c y , a r e q u i r e m e n t w h i c h t r a n s l a t e s to a d e s i r e for a u n i f o r m r e p r e s e n t a t i o n of e x p r e s s i o n s u t i l i z i n g a m i n i m u m n u m b e r of functions. Users tolerate, and in fact prefer, a c e r t a i n amount of r e d u n d a n c y in an answer. For example, they u s u a l l y d e s i r e to see e x p r e s sions c o n t a i n i n g the twelve t r i g o n o m e t r i c and h y p e r b o l i c functions. Designers would prefer g i v i n g a user only the e x p o n e n t i a l s , sines and cosines, or just e x p o n e n t i a l s w i t h both real and c o m p l e x arguments, or n o t h i n g b u t r a t i o n a l fuDctions. T h e r e is one p r o p e r t y of s i m p l i f i c a t i o n about w h i c h both users and d e s i g n e r s can a g r e ~ That is, that s i m p l i f i c a t i o n changes only the form or r e p r e s e n t a t i o n of an e x p r e s si o n , but not its value. Changes of r e p r e s e n t a t i o n occur in m a n y p r o b l e m solving domains. In fact, in the field of A r t i f i c i a l I n t e l l i g e n c e one speaks of the P r o b l e m of R e p r e s e n t a t i o n w h i c h can be stated r o u g h l y as "how does one t r a n s f o r m the s t a t e m e n t of the p r o b l e m into a form w h i c h is m o r e r e a d i l y solved." Thus an ideal, but not very helpful, w a y to d e s c r i b e s i m p l i f i c a t i o n is that it is the p r o c e s s w h i c h t r a n s f o r m s e x p r e s s i o n s into a form w i t h w h i c h the r e m a i n i n g steps of the p r o b l e m can be taken m o s t e f f i c i e n t l y . The P r o b l e m of R e p r e s e n t a t i o n for algeb r a i c e x p r e s s i o n s is e s p e c i a l l y acute b e c a u s e there are so many e q u i v a l e n t ways to r e p r e s e n t an e x p r e s s i o n . F r e q u e n t l y one of these e q u i v a l e n t forms is m u c h m o r e u s e f u l than another, and just as frequently, it is a nontrivial p r o b l e m to r e c o g n i z e the e q u i v a l e n c e . For example, it is rare that we do not w a n t to r e c o g n i z e that an e x p r e s s i o n is e q u i v a l e n t to 0. However, m a n y of us have d i f f i c u l t y in r e c o g n i z i n g the f o l l o w i n g identities. log(e 2x + 2e x + i) - 2 log(e x + i) = 0 or (21/3 + 41/3) 3 - 6(21/3
A GUIDE
ALGEBRAIC SIMPLIFICATION F O R THE P E R P L E X E D by Joel M o s e s P r o j e c t MAC, MIT
Abstract A l g e b r a i c s i m p lຫໍສະໝຸດ Baidui f i c a t i o n is e x a m i n e d first from the p o i n t of v i e w of a user n e e d i n g to c o m p r e h e n d a large expression, and second from the p o i n t of v i e w of a d e s i g n e r w h o w a n t s to c o n s t r u c t a u s e f u l and e f f i c i e n t system. F i r s t we d e s c r i b e v a r i o u s techniques akin to substitution. T h e s e techniques can be used to d e c r e a s e the size of an e x p r e s s i o n and m a k e it m o r e i n t e l l i g i b l e to a user. Then we d e l i n e ate the s p e c t r u m of approaches to the d e s i g n of a u t o m a t i c s i m p l i f i c a t i o n c a p a b i l i t i e s in an a l g e b r a i c m a n i p u l a t i o n system. Systems are d i v i d e d into five types. Each type p r o v i d e s d i f f e r e n t f a c i l i t i e s for the m a n i p u l a t i o n and s i m p l i f i c a t i o n of expressions. F i n a l l y we d i s c u s s some of the t h e o r e t i c a l results relat~ ed to a l g e b r a i c s i m p l i f i c a t i o n . We describe several p o s i t i v e results about the e x i s t e n c e of p o w e r f u l s i m p l i f i c a t i o n a l g o r i t h m s and the n u m b e r - t h e o r e t i c c o n j e c t u r e s on w h i c h they rely. Results a b o u t the n o n - e x i s t e n c e of a l g o r i t h m s for c e r t a i n classes of e x p r e s s i o n s are included. Ta b l e 1.0 2.0 of C o n t e n t s