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In computability theory, primitive recursive functions are a class of functions that are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions (µ-recursive functions are also called partial recursive). Primitive recursive functions form an important building block on the way to a full formalization of computability. These functions are also important in proof theory. Most of the functions normally studied in number theory are primitive recursive. For example: addition, division, factorial, exponential and the ''n''th prime are all primitive recursive. So are many approximations to real-valued functions.〔Brainerd and Landweber, 1974〕 In fact, it is difficult to devise a total recursive function that is ''not'' primitive recursive, although some are known (see the section on Limitations below). The set of primitive recursive functions is known as PR in computational complexity theory. == Definition == The primitive recursive functions are among the number-theoretic functions, which are functions from the natural numbers (nonnegative integers) to the natural numbers. These functions take ''n'' arguments for some natural number ''n'' and are called ''n''-ary. The basic primitive recursive functions are given by these axioms: #Constant function: The 0-ary constant function 0 is primitive recursive. #Successor function: The 1-ary successor function ''S'', which returns the successor of its argument (see Peano postulates), is primitive recursive. That is, ''S''(''k'') = ''k'' + 1. #Projection function: For every ''n''≥1 and each ''i'' with 1≤''i''≤''n'', the ''n''-ary projection function ''P''''i''''n'', which returns its ''i''-th argument, is primitive recursive. More complex primitive recursive functions can be obtained by applying the operations given by these axioms: #Composition: Given ''f'', a ''k''-ary primitive recursive function, and ''k'' ''m''-ary primitive recursive functions ''g''1,...,''g''''k'', the composition of ''f'' with ''g''1,...,''g''''k'', i.e. the ''m''-ary function is primitive recursive. #Primitive recursion: Given ''f'', a ''k''-ary primitive recursive function, and ''g'', a (''k''+2)-ary primitive recursive function, the (''k''+1)-ary function ''h'' is defined as the primitive recursion of ''f'' and ''g'', i.e. the function ''h'' is primitive recursive when #: and #: The primitive recursive functions are the basic functions and those obtained from the basic functions by applying these operations a finite number of times. 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「primitive recursive function」の詳細全文を読む スポンサード リンク
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