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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 (in fact E = F if and only if E - F = 0). 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 an expression is zero.^[1]

Statement of the theorem

Richardson's theorem can be stated as follows:^[2] Let E be 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 x, log(2), π, e^x, sin x ∈ E, then A(x) ≥ 0 for all x is unsolvable;
if also |x| ∈ E then A(x) = 0 for all x is unsolvable.

If furthermore there is a function B(x) ∈ E without an antiderivative in E then the integration problem is unsolvable. Example: $e^{ax^{2}}$ has an elementary antiderivative if and only if a=0 in the elementary functions.

Extensions

Wang^[3] removed the assumption that log(2), e^x, and |x| ∈ E, so that only π and sin x were needed. Laczkovich^[4] also removed the need for π and reduced the use of composition. In particular, in the ring generated by the integers, x, sin xⁿ, and sin(x sin xⁿ), both A(x) ≥ 0 for all x and A(x) = 0 for all x are unsolvable.

By contrast, the Tarski–Seidenberg theorem says that the reals alone are decidable, so it is not possible to remove the sine function entirely.

References

^ The identity problem for elementary functions and constants by Richardson and Fitch (pdf file)
^ "Some Undecidable Problems Involving Elementary Functions of a Real Variable", Daniel Richardson, J. Symbolic Logic Volume 33, Issue 4 (1968), pp. 514-520, JSTOR 2271358.
^ P. S. Wang, The undecidability of the existence of zeros of real elementary functions, Journal of the Association for Computing Machinery 21:4 (1974), pp. 586–589.
^ Miklós Laczkovich, The removal of π from some undecidable problems involving elementary functions, Proc. Amer. Math. Soc. 131:7 (2003), pp. 2235–2240.

External links

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

This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.

[1] The identity problem for elementary functions and constants by Richardson and Fitch (pdf file)

[2] "Some Undecidable Problems Involving Elementary Functions of a Real Variable", Daniel Richardson, J. Symbolic Logic Volume 33, Issue 4 (1968), pp. 514-520, JSTOR 2271358.

[3] P. S. Wang, The undecidability of the existence of zeros of real elementary functions, Journal of the Association for Computing Machinery 21:4 (1974), pp. 586–589.

[4] Miklós Laczkovich, The removal of π from some undecidable problems involving elementary functions, Proc. Amer. Math. Soc. 131:7 (2003), pp. 2235–2240.

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 ==Statement of the theorem==
-Richardson's theorem can be stated as follows.<ref>{{cite web |title=Tensor Computer Algebra |date=December 22, 2008 |url=http://metric.iem.csic.es/Martin-Garcia/slides/salamanca.pdf |format=PDF}}{{dead link|date=December 2013}}</ref>
+Richardson's theorem can be stated as follows:<ref>[http://projecteuclid.org/euclid.jsl/1183736504 "Some Undecidable Problems Involving Elementary Functions of a Real Variable"], Daniel Richardson, J. Symbolic Logic Volume 33, Issue 4 (1968), pp. 514-520, {{JSTOR|2271358}}.</ref>
 Let ''E'' be 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 x, log(2), π, ''e<sup>x</sup>'', sin ''x'' ∈ E, then ''A(x)'' ≥ 0 for all ''x'' is unsolvable;