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| In mathematics, '''Richardson's theorem''' establishes a limit on the extent to which an [[algorithm]] can [[decision problem|decide]] whether certain mathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is [[Undecidable problem|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]].
| | The author's name is Christy Brookins. As a woman what she really likes is style and she's been doing it for quite a while. He works as a bookkeeper. For many years he's been residing in Alaska and he doesn't plan on altering it.<br><br>my web blog; online reader ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 www.machlitim.org.il]) |
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| Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number [[Pi|π]], the number [[natural logarithm|log 2]], the variable ''x'', the operations of addition, subtraction, multiplication, [[function composition|composition]], and the [[sine|sin]], [[exponential function|exp]], and [[absolute value|abs]] functions.
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| For some classes of expressions (generated by other primitives than in Richardson's theorem) there exist algorithms that can determine whether expression is zero.<ref>[http://portal.acm.org/citation.cfm?id=190429 The identity problem for elementary functions and constants by Richardson and Fitch] (pdf file)</ref>
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| ==Statement of the Theorem==
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| 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>
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| 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,
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| * if log(2), ''π'', ''e<sup>x</sup>'', sin ''x'' ∈ E, then ''A(x)'' ≥ 0 for all ''x'' is unsolvable;
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| * if also ''|x|'' ∈ ''E'' then ''A(x)'' = 0 is unsolvable.
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| If furthermore there is a function ''B(x)'' ∈ ''E'' without an antiderivative in ''E'' then the integration problem is unsolvable. Example: <math>e^{ax^2}</math> has an elementary antiderivative if and only if a=0 in the elementary functions.
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| ==See also==
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| *[[Constant problem]]
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| ==References==
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| <references />
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| ==Further reading==
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| *{{cite book |last1=Petkovšek |first1=Marko |authorlink1=Marko Petkovšek |last2=Wilf |first2=Herbert S. |authorlink2=Herbert S. Wilf |last3=Zeilberger |first3=Doron |authorlink3=Doron Zeilberger |title=A = B |publisher=[[A. K. Peters]] |url=http://www.cis.upenn.edu/~wilf/AeqB.html |year=1996 |isbn=1-56881-063-6 |pages=212}}
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| *{{Cite news |last=Richardson |first=Daniel |year=1968 |title=Some unsolvable problems involving elementary functions of a real variable |periodical=[[Journal of Symbolic Logic]] |volume=33 |issue=4 |pages=514–520 |url=http://www.jstor.org/pss/2271358 |doi=10.2307/2271358 |jstor=10.2307/2271358 |publisher=Association for Symbolic Logic}}
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| ==External links==
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| *{{MathWorld|urlname=RichardsonsTheorem|title=Richardson's theorem}}
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| [[Category:Theorems in the foundations of mathematics]]
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| [[Category:Computability theory]]
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| {{mathlogic-stub}}
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The author's name is Christy Brookins. As a woman what she really likes is style and she's been doing it for quite a while. He works as a bookkeeper. For many years he's been residing in Alaska and he doesn't plan on altering it.
my web blog; online reader (www.machlitim.org.il)