Richardson's theorem

In mathematics, Richardson's theorem establishes a limit on the extent to which an algorithm can decide whether certain mathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is undecidable whether a particular expression E satisfies the equation E = 0, and similarly undecidable whether the functions defined by expressions E and F are everywhere equal. It was proved in 1968 by computer scientist Daniel Richardson of the University of Bath.

Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number log 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.

For some classes of expressions (generated by other primitives than in Richardson's theorem) there exist algorithms that can determine whether expression is zero.

Calculation

The Richardson’s theorem can be calculated as follows Let be E a set of real functions such that, if A(x), B(x) ∈ E, then A(x) ± B(x), A(x)B(x), A(B(x)) ∈ E. The rational numbers are contained as constant functions. Then for expressions A(x) in E,

• if log 2, π, e^x, sin x ∈ E, then A(x) ≥ 0 for all x is unsolvable;
• if also $\sqrt{x^2}$ ∈ E then A(x) = 0 is unsolvable.

If furthermore there is a function B(x) ∈ E without primitive in E then the integration problem is unsolvable. Example: exp(x²).

See also

• Constant problem

Further reading

• Petkovšek, Marko; Herbert S. Wilf, Doron Zeilberger (1996). A = B. A. K. Peters. pp. 212. ISBN 1568810636. http://www.cis.upenn.edu/~wilf/AeqB.html. 
• Richardson, Daniel (1968). "Some unsolvable problems involving elementary functions of a real variable". Journal of Symbolic Logic (Association for Symbolic Logic) 33 (4): pp. 514–520. doi:10.2307/2271358. JSTOR 10.2307/2271358. http://links.jstor.orgsici?sici=0022-4812%28196812%2933%3A4%3C514%3ASUPIEF%3E2.0.CO%3B2-H .

External links

• Weisstein, Eric W., "Richardson's theorem" from MathWorld.