ELEMENTARY
↑ Mazzanti, S., "Plain Bases for Classes of Primitive Recursive Functions", Mathematical Logic Quarterly, 48 (2002) 93-104
↑ Lauri Hella and José María Turull-Torres (2006), “Computing queries with higher-order logics”, Theoretical Computer Science (Essex, UK: Elsevier Science Publishers Ltd.) 355 (2): 197–214, doi:10.1016/j.tcs.2006.01.009, ISSN 0304-3975, http://portal.acm.org/citation.cfm?id=1142890.1142897
Rose, H.E., "Subrecursion: Functions and hierarchies", Oxford University Press, New York, USA, 1984. ISBN 0-19-853189-3
限定和: f ( x → , y ) = ∑ z < y g ( x → , z ) {\displaystyle f({\overrightarrow {x}},y)=\sum \limits _{z<y}g({\overrightarrow {x}},z)}
限定積: f ( x → , y ) = ∏ z < y g ( x → , z ) {\displaystyle f({\overrightarrow {x}},y)=\prod \limits _{z<y}g({\overrightarrow {x}},z)}
初等的関数の例としては次のものがある:
乗算関数: x ⋅ y = ∑ z < y x {\displaystyle x\cdot y=\sum \limits _{z<y}x}
加算関数: x + y = S ( x ) ⋅ S ( y ) − ̇ S ( x ⋅ y ) {\displaystyle x+y=S(x)\cdot S(y){\dot {-}}S(x\cdot y)}
冪乗関数: x y = ∏ z < y x {\displaystyle x^{y}=\prod \limits _{z<y}x}
素数列: p n = 2 , 3 , 5 , 7 , 11 , ... {\displaystyle p_{n}=2,3,5,7,11,\ldots }