properties of Ackermann function

In this entry, we derive some basic properties of the Ackermann function $A(x,y)$, defined by double recursion, as follows:

 $A(0,y)=y+1,\quad A(x+1,0)=A(x,1),\quad A(x+1,y+1)=A(x,A(x+1,y)).$

These properties will be useful in proving that $A$ is not primitive recursive.

1. 1.

$A$ is total ($\operatorname{dom}(A)=\mathbb{N}^{2}$).

2. 2.

$A(1,y)=y+2$.

3. 3.

$A(2,y)=2y+3$.

4. 4.

$y.

5. 5.

$A(x,y).

6. 6.

$A(x,y+1)\leq A(x+1,y)$.

7. 7.

$A(x,y).

8. 8.

$A(r,A(s,y))

9. 9.

For any $r,s$, $A(r,y)+A(s,y) for some $t$ not depending on $y$.

Most of the proofs are done by induction.

Proof.
1. 1.

Induct on $x$. First, $A(0,y)=y+1$ is well-defined, so $(0,y)\in\operatorname{dom}(A)$ for all $y$. Next, suppose that for a given $x$, $(x,y)\in\operatorname{dom}(A)$ for all $y$. We want to show that $(x+1,y)\in\operatorname{dom}(A)$ for all $y$. To do this, induct on $y$. First, $(x+1,0)\in\operatorname{dom}(A)$, since $A(x+1,0)=A(x,1)$ is well-defined. Next, assume that $(x+1,y)\in\operatorname{dom}(A)$. Then $A(x,A(x+1,y))=A(x+1,y+1)$ is well-defined. so $(x+1,y+1)\in\operatorname{dom}(A)$ as well.

2. 2.

Induct on $y$. First, $A(1,0)=A(0,1)=2$. Next, assume $A(1,y)=y+2$. Then $A(1,y+1)=A(0,y+2)=y+3=(y+1)+2$.

3. 3.

Induct on $y$. First, $A(2,0)=A(1,1)=1+2=3$. Next, assume $A(2,y)=2y+3$. Then $A(2,y+1)=A(1,A(2,y))=A(2,y)+2=(2y+3)+2=2(y+1)+3$.

4. 4.

Induct on $x$. First, $y. Next, assume $y, where $x>0$. Then $y+1\leq A(x,y).

5. 5.

Induct on $x$. First, $A(0,y)=y+1. Next, assume that $A(x,y). Then $A(x+1,y).

6. 6.

Induct on $y$. First, $A(x,1)=A(x+1,0)$. Next, assume that $A(x,y+1)\leq A(x+1,y)$. Then $A(x,y+2)\leq A(x,A(x,y+1))\leq A(x,A(x+1,y))=A(x+1,y+1)$.

7. 7.

Induct on $x$. First, $A(0,y)=y+1. Next, assume that $A(x,y). There are two cases: $y=0$. Then $A(x+1,0)=A(x,1). Otherwise, $y=z+1$, so that $A(x+1,y)=A(x+1,z+1)=A(x,A(x+1,z))\leq A(x,A(x,z+1))=A(x,A(x,y)).

8. 8.

$A(r,A(s,y)).

9. 9.

Let $z=\max\{r,s\}$. Then $A(r,y)+A(s,y)\leq 2A(z,y)<2A(z,y)+3=A(2,A(z,y)). The proof is completed by setting $t=4+z$.

With respect to the recursive property of $A$, we see that $A$ is recursive, since, by Church’s Thesis, $A$ can be effectively computed (in fact, a program can be easily written to compute $A(x,y)$). We also have the following:

Proposition 1.

Define $A_{n}:\mathbb{N}\to\mathbb{N}$ by $A_{n}(m)=A(n,m)$. Then $A_{n}$ is primitive recursive for each $n$.

Proof.

$A_{0}(m)=m+1=s(m)$, so is primitive recursive. Now, assume $A_{n}$ is primitive recursive, then $A_{n+1}(0)=A(n+1,0)=A(n,1)=A_{n}(1)=k$, and $A_{n+1}(m+1)=A(n+1,m+1)=A(n,A(n+1,m))=A_{n}(A(n+1,m))=A_{n}(A_{n+1}(m))$, so that $A_{n+1}$ is defined by primitive recursion via the constant function $\operatorname{const}_{k}$, and $A_{n}$, which is primitive recursive by the induction hypothesis. Therefore $A_{n+1}$ is primitive recursive also. ∎

The most important fact about $A$ concerning recursiveness is that $A$ is not primitive recursive. Due to the length of its proof, it is demonstrated in this entry (http://planetmath.org/AckermannFunctionIsNotPrimitiveRecursive).

Title properties of Ackermann function PropertiesOfAckermannFunction 2013-03-22 19:07:25 2013-03-22 19:07:25 CWoo (3771) CWoo (3771) 9 CWoo (3771) Result msc 03D75