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 (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 expressions in the variable x which contains x and, as constant expressions, all rational numbers, and is such that if A(x) and B(x) are in E, then A(x) + B(x), A(x) - B(x), A(x)B(x), and A(B(x)) are also in E. Then:
- if x, log 2, π, ex, sin x ∈ E, then the problem of determining, for an expression A(x) in E, whether A(x) < 0 for some x is unsolvable;
- if also |x| ∈ E then the problem of determining whether A(x) = 0 for all x is also unsolvable;
- if furthermore there is a function B(x) ∈ E without an antiderivative in E then the integration problem is unsolvable. (Example:
has an elementary antiderivative if and only if a=0 in the elementary functions.)
Extensions
After Hilbert's Tenth Problem was solved in 1970, B. F. Caviness observed that the use of ex and log 2 could be removed.[3] P. S. Wang[4] later noted that under the same assumptions under which the question of whether there was x with A(x) < 0 was insoluble, the question of whether there was x with A(x) = 0 was also insoluble.
Miklós Laczkovich[5] removed also the need for π and reduced the use of composition. In particular, given an expression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn), both the question of whether A(x) > 0 for some x and whether A(x) = 0 for some x are unsolvable.
By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely.
See also
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 33, #4 (1968), pp. 514-520, JSTOR 2271358.
- ↑ On Canonical Forms and Simplification, B. F. Caviness, JACM, 17, #2 (April 1970), pp. 385-396.
- ↑ 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.
Further reading
- Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron (1996). A = B. A. K. Peters. p. 212. ISBN 1-56881-063-6.
- Richardson, Daniel (1968). "Some undecidable problems involving elementary functions of a real variable". Journal of Symbolic Logic 33 (4) (Association for Symbolic Logic). pp. 514–520. doi:10.2307/2271358. JSTOR 10.2307/2271358.