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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.〔(The identity problem for elementary functions and constants by Richardson and Fitch ) (pdf file)〕 ==Statement of the theorem== Richardson's theorem can be stated as follows:〔("Some Undecidable Problems Involving Elementary Functions of a Real Variable" ), Daniel Richardson, ''J. Symbolic Logic'' 33, #4 (1968), pp. 514-520, .〕 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 antiderivative in the elementary functions if and only if .) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Richardson's theorem」の詳細全文を読む スポンサード リンク
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