Algebraic simplification a guide for the perplexed

Abstract

Algebraic simplification is examined first from the point of view of a user who needs to comprehend a large expression, and second from the point of view of a designer who wants to construct a useful and efficient system. First we describe various techniques akin to substitution. These techniques can be used to decrease the size of an expression and make it more intelligible to a user. Then we delineate the spectrum of approaches to the design of automatic simplification capabilities in an algebraic manipulation system. Systems are divided into five types. Each type provides different facilities for the manipulation and simplification of expressions. Finally we discuss some of the theoretical results related to algebraic simplification. We describe several positive results about the existence of powerful simplification algorithms and the number-theoretic conjectures on which they rely. Results about the nonexistence of algorithms for certain classes of expressions are included.

Simulation Bounds for Equivalence Verification of Arithmetic Datapaths with Finite Word-Length Operands

“…The concept of vanishing polynomials is used in [13] to derive a symbolic approach to test equivalence of polynomial functions. Both approaches utilize some form of algebraic simplification, which suffers from the well-known intermediate-expression swell problem [25]. In addition, these techniques cannot provide an error trace whenever non-equivalence is detected.…”

Computer Algebra Algorithms

“…GCD and factorization algorithms on such multivariate sparse polynomials were studied both theoretically and empirically as early as a decade ago (Wang 1978) [177], and randomization became one of the major ingredients to sparsity preserving operations (Zippel 1979) [188]. Early on Moses (1971b) [125] pointed out, however, that sparsity is not measured in terms of nonzero coefficients alone, as his example…”

“…The notion of simplicity itself is subjective,a nd different approaches are compared by [125] and [23]. One such concept prescribes simplification to canonical forms (as in Caviness 1970) [30], which means that the simplified versions of different expressions with the same meaning must be identical.…”

Understanding expression simplification

“…A short conversation with end users or a survey of Maple worksheets (or Mathematica notebooks) quickly confirms this impression. But if one instead scours the scientific literature to find papers relating to simplification, a few are easily found: a few early general papers [4,6,19] [and the earlier work they reference], some on elementary functions like [3], as well as papers on nested radicals [17,26], but even dedicated searches found little more. Looking at the standard textbooks on Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.…”