--- Waldek Hebisch <[EMAIL PROTECTED]> wrote: > I have a web page where I put references to some basic literature > about symbolic computations: > > http://www.math.uni.wroc.pl/~p-wyk4/malgo/literat.html > > For your question fundamental result is Richardson 1968, where he > proved that when expresion are build from arithmetic operations, > polynonials and trigonometic functions then already the problem > of finding out if a given expression in zero is undecidable. > > Note that sound algorithm should _prove_ that expression is zero. > For some expressions we can show that there is no proof that > the expression is zero, but also no proof that the expression is > nonzero. Naive idea would be to classify expression into > provably zero, provably nonzero and and "undecidable". The catch > is that proving that an expression is "undecidable" is even > harder problem that proving that it is zero. So in constructive > direction we are trying to find out large classes of decidable > expressions, but without any hope of completeness (more effort > is likely to give bigger class). > > In constructive direction there is again result of Richardson > (How to recognize zero) where he proposes a method which > hopefully can decide equality of elementary numbers (numbers > produced using algebraic operations, exponential and trigonometric > functions). For elementary functions (functions build from > polynomials, > exponential and trigonometric functions using composition and > algebraic > operations -- absolute value is excluded) there is Risch (1979) > structure > theorem which reduces the problem to constansts. > > Axiom problem is that its notion of equality is inconsitent: Axiom > sometimes treats roots (in particular square root) as multivalued, > but in other places assumes that expressions form a field which > requires building abstract algebraic extension (or choosing a single > value). This dual notion of equality is responsible for bug 290: > Risch algorithm requires a field, but since Axion do not apply > simplification to roots we in fact get ring with zero divisors. > > -- > Waldek Hebisch > [EMAIL PROTECTED] >
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