Understanding expression simplification

Abstract

We give the first formal definition of the concept of simplification for general expressions in the context of Computer Algebra Systems. The main mathematical tool is an adaptation of the theory of Minimum Description Length, which is closely related to various theories of complexity, such as Kolmogorov Complexity and Algorithmic Information Theory. In particular, we show how this theory can justify the use of various "magic constants" for deciding between some equivalent representations of an expression, as found in implementations of simplification routines.

Automated simplification of large symbolic expressions

“…The paper (Carette, 2004) provides a formal description of simplification. The author discusses, using ideas including Kologorov complexity and minimum description length, a method for defining whether a version of an expression is more simple than another.…”

“…Searching for methods of simplification reveals many older papers as mentioned in (Carette, 2004). The papers (Buchberger and Loos, 1982;Casas et al, 1990;Caviness, 1970;Fateman, 1972;Fitch, 1973;Moses, 1971) explore formalism and technique for simplification.…”

“…1 All the algorithms (with the exception of substitution) only process the rational coefficients α i , so an expression m i=1 α i x i is simpler than n i=1 β i x i if m < n. This is in contrast to the general case discussed in (Carette, 2004), where the question of which version of a general expression is considered and formalized.…”

Closed Forms: What They Are and Why We Care

“…This is an idea of J. Carette [28] to the effect that solutions at algebraic end-points and algebraic initial points for holonomic ODEs-i.e., differential-equation systems with integer-polynomial coefficients-could be considered closed. One might say diffeoclosed.…”

An Intelligent Mathematics Assessment System

“…For b) in the example, the standard answer is 5 2 , while the teacher may treat it as totally correct when students enter 5 2 , 2.5, 1 1 2 as the answer. But if a student answer is 10 4 , 1 4 + 9 4 , 3 − 0.5, 5x x+x , or 00002.5000, the teacher will argue that the answer is not completely correct and will give a score of 80% or 70%.…”