ALGEBRAIC SIMPLIFICATION A GUIDE FOR THE PERPLEXED by Joel Moses Project MAC, MIT Abstract and efficiency, a requirement which translates to a desire for a uniform representation of Algebraic simplification is examined first expressions utilizing a minimum number of from the point of view of a user needing to functions. Users tolerate, and in fact prefer, comprehend a large expression, and second from a certain amount of redundancy in an answer. the point of view of a designer who wants to For example, they usually desire to see expres- construct a useful and efficient system. sions containing the twelve trigonometric and First we describe various techniques akin to hyperbolic functions. Designers would prefer substitution. These techniques can be used to giving a user only the exponentials, sines and decrease the size of an expression and make it cosines, or just exponentials with both real more intelligible to a user. Then we deline- and complex arguments, or nothing but rational ate the spectrum of approaches to the design fuDctions. of automatic simplification capabilities in an There is one property of simplification algebraic manipulation system. Systems are about which both users and designers can agre~ divided into five types. Each type provides That is, that simplification changes only the different facilities for the manipulation and form or representation of an expression, but simplification of expressions. Finally we not its value. Changes of representation discuss some of the theoretical results relat~ occur in many problem solving domains. In ed to algebraic simplification. We describe fact, in the field of Artificial Intelligence several positive results about the existence one speaks of the Problem of Representation of powerful simplification algorithms and the which can be stated roughly as "how does one number-theoretic conjectures on which they transform the statement of the problem into a rely. Results about the non-existence of form which is more readily solved." Thus an algorithms for certain classes of expressions ideal, but not very helpful, way to describe are included. simplification is that it is the process which transforms expressions into a form with which Table of Contents the remaining steps of the problem can be taken most efficiently. 1.0 Introduction The Problem of Representation for alge- 2.0 Simplification for the Sake of braic expressions is especially acute because Comprehension - The Needs of Users there are so many equivalent ways to represent 2.1 Conventional Lexicographic Ordering an expression. Frequently one of these of Expressions equivalent forms is much more useful than an- 2.2 Substitution as an Aid to Comprehen- other, and just as frequently, it is a non- sion trivial problem to recognize the equivalence. 3.0 Simplification for the Sake of Effic- For example, it is rare that we do not want to ient Manipulation - What Designers Provide recognize that an expression is equivalent to 3.1 The Politics of Simplification 0. However, many of us have difficulty in 3.1.1 The Radicals recognizing the following identities. 3.1.2 The New Left 3.1.3 The Liberals log(e 2x + 2e x + i) - 2 log(e x + i) = 0 3.1.4 The Conservatives 3.1.5 The Catholics or 3.2 Intermediate Expression Swell 3.3 Canonical Simplifiers and Theoretical (21/3 + 41/3) 3 - 6(21/3 + 41/3 ) - 6 = 0 Results - The Radicals Revisited 3.3.1 Simplification Algorithms for or Expressions with Nested Exponentials 3.3.2 Expressions involving Exponen- log tan(~ + 7) - sinh -I tan x = 0 tials and Logarithms 3.3.3 Roots of Polynomials Consider how much more difficult the problems 3.3.4 Unsolvability Results become when we deal with expressions which are 4.0 Prospects for the Future several pages long. Yet expressions of such References size are quite common in algebraic manipula- Figures tion! An additional difficulty is that the usual manipulatory algorithms can easily 1.0 Introduction magnify a bad choice of representation. For example, the derivative of a product of n Simplification is the most pervasive pro- factors can be a sum of n terms each of n or cess in algebraic manipulation. It is also more factors. Thus a bad representation of the most controversial. Much of the contro- the product or one of its factors is propaga- versy is due to the difference between the ted and magnified n-fold. desires of a user and those of a system design- Another issue which arises in discussions er. The user wants expressions which he can of simplification is related to the local or comprehend, a requirement which usually means global nature of the problem. If expression that the expressions presented to the user A is deemed simpler than its equivalent should be small. The designer wants expres- expression B in one context, then is A to be sions which can be manipulated with great ease considered simpler than B in every context? Z82 A perfectly strict answer is no. For example, concentrate in section 2 on the users' need for comprehension of expressions, and then in x 7 section 3 on the designers' facilities for 12 manipulating expressions. x + 1 In discussing simplification for the sake of comprehension, we shall describe the is a more compact representation of the ration- technique of substituting labels for subexpres- al function it represents than sions in simplifying large expressions. We shall also describe principles of conventional i/4(4x3)x 4 lexicographic ordering of expressions. A (x4) 3 + 1 system can hinder the comprehension of a user by displaying expressions in unconventional The former is usually easier to manipulate and order. Unfortunately, no current system pays comprehend. However, when integrating, the sufficient heed to this point. latter expression indicates a p@ttern which In section 3 we describe five different suggests the substitution y = x ~ which yields approaches to the design of simplification facilities in algebraic manipulation systems. f dy , We describe the facilities which are usually y3+l offered, and indicate the advantages and disadvantages in them. The section ends with a much simpler integration problem than that a discussion of very powerful algorithms based which is posed by the first expression. on formalizations of the concepts of simplifi- Designers would prefer a system in which the cation. simplification steps are the same in every context. Users clearly would prefer a system 2.0 Simplification for the Sake of Compreq which could take contextual information into hension - The Needs of Users account in deriving a simplified expression. Designers can take comfort in the fact that One of the most common complaints of while the simplified form of an expression is users of algebraic manipulation systems is that not the same in every context, it is so in the expressions obtained as results of a calcu- many contexts. Nonetheless, the task of deriv- lation are incomprehensible and therefore ing a compromise between the wishes of users essentially useless. In order to understand and the requirements of an efficient system is the importance of such a complaint we have to likely to keep designers talking to themselves differentiate between two major classes of and to each other for quite some time. users. Some users are only interested in the A related aspect of simplification is the value of a calculation. For example, those extent to which the concept can be formalized. who use symbolic differentiation as a step in The point that we made above is that the simp- a numerical calculation do not care very much lest form of an expression depends on one's about the form of the symbolic derivative.! goals or, in other words, on the context. One For such users the problem of simplification would be hard put to formalize the goals of reduces to keeping the intermediate expressions all potential users. However, one can obtain in a calculation in such a form as to optimize theoretical results for simplification algo- the use of space and time in the calculation. rithms which have usef,11 properties. One such We do not wish to underestimate the difficulties property is that the algorithm simplifies to in the simplification problem for such users. zero any expression equivalent to 0. A stron- However, in this section we will be mainly ger property is that the simplifier reduces concerned with the needs of users who do care all equivalent expressions to a single (canon- about the form of expressions which result from ical) expression. a symbolic calculation. Historically, simplification was required The latter class of users include those in algebraic manipulation systems because the who need to make "physical sense" of an expres- manipulatory algorithms produced sloppy results, sion. Perhaps such a user is studying a pro- For example, the unsimplified result of cess and wants to learn about some property of differentiating the process by symbolically manipulating a model of it. For example, he might be inter- 2 ested in the manner in which the value of an X ax + xe expression varies as one of its variables increases in value. He could, of course, plot with respect to x is an expression such as the value of the expression for several values 2 2 of the variables, but this method may not be 0.x + a.l + l.e x + x-e x -2-x very useful if-there are many variables in the expression. Other users might need to examine Simplifying the derivative above would yield an expression in order to know what the next an expression like manipulatory step should be. A simple instance of such a situation occurs when the next step a + e x2 + 2x 2 e x2 in the calculation depends on whether the expression is linear or quadratic in a given With the ever growing use of algebraic manipu~ variable.