Sudan function
In the theory of computation, the Sudan function is an example of a function that is recursive, but not primitive recursive. This is also true of the better-known Ackermann function. The Sudan function was the first function having this property to be published.
It was discovered (and published[1]) in 1927 by Gabriel Sudan, a Romanian mathematician who was a student of David Hilbert.
Definition
Value tables
| y\x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 3 | 4 | 5 | 6 | 7 |
| 3 | 3 | 4 | 5 | 6 | 7 | 8 |
| 4 | 4 | 5 | 6 | 7 | 8 | 9 |
| 5 | 5 | 6 | 7 | 8 | 9 | 10 |
| 6 | 6 | 7 | 8 | 9 | 10 | 11 |
| y\x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 1 | 3 | 5 | 7 | 9 | 11 | 13 |
| 2 | 4 | 8 | 12 | 16 | 20 | 24 | 28 |
| 3 | 11 | 19 | 27 | 35 | 43 | 51 | 59 |
| 4 | 26 | 42 | 58 | 74 | 90 | 106 | 122 |
| 5 | 57 | 89 | 121 | 153 | 185 | 217 | 249 |
| 6 | 120 | 184 | 248 | 312 | 376 | 440 | 504 |
In general, F1(x, y) is equal to F1(0, y) + 2y x.
| y\x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 8 | 27 | 74 | 185 | 440 |
| 2 | 19 | F1(8, 10) = 10228 | F1(27, 29) ≈ 1.55 ×1010 | F1(74, 76) ≈ 5.74 ×1024 | F1(185, 187) ≈ 3.67 ×1058 | F1(440, 442) ≈ 5.02 ×10135 |
References
- Cristian Calude, Solomon Marcus, Ionel Tevy, The first example of a recursive function which is not primitive recursive, Historia Mathematica 6 (1979), no. 4, 380–384 doi:10.1016/0315-0860(79)90024-7
- ↑ Bull. Math. Soc. Roumaine Sci. 30 (1927), 11 - 30; Jbuch 53, 171
External links
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