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→Normal form theorem: use consistent notation |
m Removed unusual language ("operator" instead of "relation"), even though there is no risk of misunderstanding in this context, waters down the terminology unnecessarily |
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While some textbooks use the μ-operator as defined here,<ref name="Enderton.1972">Enderton, H. B., A Mathematical Introduction to Logic, Academic Press, 1972</ref><ref name="Boolos.Burgess.Jeffrey.2007">Boolos, G. S., Burgess, J. P., Jeffrey, R. C., Computability and Logic, Cambridge University Press, 2007</ref> others like<ref name="Jones.1997">Jones, N. D., Computability and Complexity: From a Programming Perspective, The MIT Press, Cambridge, Massachusetts, London, England, 1997</ref><ref>Kfoury, A. J., R. N. Moll, and M. A. Arbib, A Programming Approach to Computability, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1982</ref> demand that the μ-operator is applied to ''total'' functions {{mvar|f}} only. Although this restricts the μ-operator as compared to the definition given here, the class of μ-recursive functions remains the same, which follows from Kleene's Normal Form Theorem (see [[#Normal form theorem|below]]).<ref name="Enderton.1972"/><ref name="Boolos.Burgess.Jeffrey.2007"/> The only difference is, that it becomes undecidable whether a specific function definition defines a μ-recursive function, as it is undecidable whether a computable (i.e. μ-recursive) function is total.<ref name="Jones.1997"/>
The ''[[strong equality]]''
:<math>f(x_1,\ldots,x_k) \simeq g(x_1,\ldots,x_l)</math>
holds if and only if for any choice of arguments either both functions are defined and their values are equal or both functions are undefined.
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