Tak (function)
Recursive function From Wikipedia, the free encyclopedia
In computer science, the Tak function is a recursive function, named after Ikuo Takeuchi . It is defined as follows:
def tak(x: int, y: int, z: int) -> int:
if y < x:
return tak(
tak(x - 1, y, z),
tak(y - 1, z, x),
tak(z - 1, x, y)
)
else:
return z
This function is often used as a benchmark for languages with optimization for recursion.[1][2][3][4]
tak() vs. tarai()
Summarize
Perspective
This section needs additional citations for verification. (September 2023) |
The original definition by Takeuchi was as follows:
def tarai(x: int, y: int, z: int) -> int:
if y < x:
return tarai(
tarai(x - 1, y, z),
tarai(y - 1, z, x),
tarai(z - 1, x, y)
)
else:
return y # not z!
tarai is short for たらい回し (tarai mawashi, "to pass around") in Japanese.
John McCarthy named this function tak() after Takeuchi.[5]
However, in certain later references, the y somehow got turned into the z. This is a small, but significant difference because the original version benefits significantly from lazy evaluation.
Though written in exactly the same manner as others, the Haskell code below runs much faster.
tarai :: Int -> Int -> Int -> Int
tarai x y z
| x <= y = y
| otherwise = tarai (tarai (x-1) y z)
(tarai (y-1) z x)
(tarai (z-1) x y)
One can easily accelerate this function via memoization yet lazy evaluation still wins.
The best known way to optimize tarai is to use a mutually recursive helper function as follows.
def laziest_tarai(x: int, y: int, zx: int, zy: int, zz: int) -> int:
if not y < x:
return y
else:
return laziest_tarai(
tarai(x-1, y, z),
tarai(y-1, z, x),
tarai(zx, zy, zz)-1, x, y)
def tarai(x: int, y: int, z: int) -> int:
if not y < x:
return y
else:
return laziest_tarai(
tarai(x-1, y, z),
tarai(y-1, z, x),
z-1, x, y)
Here is an efficient implementation of tarai() in C:
int tarai(int x, int y, int z)
{
while (x > y) {
int oldx = x, oldy = y;
x = tarai(x - 1, y, z);
y = tarai(y - 1, z, oldx);
if (x <= y) break;
z = tarai(z - 1, oldx, oldy);
}
return y;
}
Note the additional check for (x <= y
) before z (the third argument) is evaluated, avoiding unnecessary recursive evaluation.
References
External links
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