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Richardson's theorem

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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.

Contents

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.

Statement of the theorem

Richardson's theorem can be stated as follows: 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: e a x 2 has an antiderivative in the elementary functions if and only if a = 0.)

    • 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. P. S. Wang 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 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.

    References

    Richardson's theorem Wikipedia
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