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) in 1927 by Gabriel Sudan, a Romanian mathematician who was a student of David Hilbert.

Definition

$F_{0}(x,y)=x+y,$

$F_{n+1}(x,0)=x,\ n\geq 0$

$F_{n+1}(x,y+1)=F_{n}(F_{n+1}(x,y),F_{n+1}(x,y)+y+1),\ n\geq 0.$

Value tables

Values of F₀(x, y)

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

Values of F₁(x, y)

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) + 2^y x.

Values of F₂(x, y)

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

External links

• OEIS: A260003, A260004