Ackermann Function

The Ackermann function is a classic example of a recursive function, notable especially because it is not a primitive recursive function. It grows very quickly in value, as does the size of its call tree. The Ackermann function is usually defined as follows:

\[\begin{split}A(m, n) = \begin{cases} n+1 & \mbox{if } m = 0 \\ A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\ A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0. \end{cases}\end{split}\]

nCrypt has devised a way to calculate the value of the Ackermann function using native scripts. But it is definitely non-trivial. Below we present a much simpler version.

contract Ackermann {
    int a;  // a = 2
    int b;  // b = 1

    function ackermann(int m, int n): int {
        bytes stk = num2bin(m, 1);

        // run this function off chain to get the loop count and set it here
        // e.g., (2, 1) requires 14 loops, (3, 5) 42438
        loop (14) {
            if (len(stk) > 0) {
                bytes top = stk[0:1];
                m = unpack(top);

                // pop
                stk = stk[1:len(stk)];

                if (m == 0) {
                    n = n + m + 1;
                } else if (n == 0) {
                    n = n + 1;
                    m = m - 1;
                    // push
                    stk = num2bin(m, 1) + stk;
                } else {
                    stk = num2bin(m - 1, 1) + stk;
                    stk = num2bin(m, 1) + stk;
                    n = n - 1;
                }
            }
        }

        return n;
    }

    // y = 5
    public function unlock(int y) {
        require(y == this.ackermann(this.a, this.b));
    }
}