Richardson's theorem concerns the class 
of expressions generated by
1. The rational numbers and the two real numbers 
and 
,
2. The variable 
,
3. The operations of addition, multiplication,
and function composition, and
4. The sine, exponential,
and absolute value functions.
It states that no algorithm can decide, given 
, whether 
for every real number 
. This identity problem is recursively
undecidable (Richardson 1968, p. 520).
See also
Constant Problem,
Hidden Zero,
Integer Relation,
Integration
Problem,
Recursion,
Recursively
Undecidable,
Rice's Theorem,
Schanuel's
Conjecture,
Undecidable,
Zero
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References
Caviness, B. F. "On Canonical Forms and Simplification." J. Assoc. Comp. Mach. 17, 385-396, 1970.Davenport, J. H.
"Equality in Computer Algebra and Beyond." J. Symb. Comput. 34,
259-270, 2002.Petkovšek, M.; Wilf, H. S.; and Zeilberger,
D. A=B.
Wellesley, MA: A K Peters, 1996. https://www2.math.upenn.edu/~wilf/AeqB.html.Richardson,
D. "Some Undecidable Problems Involving Elementary Functions of a Real Variable."
J. Symbolic Logic 33, 514-520, 1968. https://doi.org/10.2307/2271358.Richardson,
D. "How to Recognize Zero." J. Symb. Comput. 24, 627-645,
1997.Richardson, D. "The Uniformity Conjecture." In Computability
and Complexity in Analysis: 4th International Workshop, CCA 2000, Swansea, UK, September
17-19, 2000 (Ed. J. Blanck, V. Brattka, and P. Hertling).
Berlin, Germany: Springer-Verlag, pp. 253-272, 2001. https://doi.org/10.1007/3-540-45335-0_15.Trott,
M. The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2005. https://www.mathematicaguidebooks.org/.Referenced
on Wolfram|Alpha
Richardson's Theorem
Cite this as:
Weisstein, Eric W. "Richardson's Theorem."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RichardsonsTheorem.html
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