Richardson's Theorem

Let  be 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 composition, and

4. The sine, exponential, and absolute value functions.

Then if , the predicate "" is recursively undecidable.

REFERENCES

Caviness, B. F. "On Canonical Forms and Simplification." J. Assoc. Comp. Mach. 17, 385-396, 1970.

Davenport, J. H. J. Symb. Comput. 34, 259, 2002.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. 

http://www.cis.upenn.edu/~wilf/AeqB.html.Richardson, D. "Some Unsolvable Problems Involving Elementary Functions of a Real Variable." J. Symbolic Logic 33, 514-520, 1968.Richardson, D. J. Symb. Comput. 24, 627, 1997.

Richardson, D. In Computability and Complexity in Analysis (Ed. J. Blanck, V. Brattka, and P. Hertling). Berlin: Springer-Verlag, 2000.

Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2005. http://www.mathematicaguidebooks.org/.