In this exercise, we will give an alternate proof of the...

In this exercise, we will give an alternate proof of the fact that there exists a primitive recursive function $\beta$ of one variable such that, for all $i$, $$ \phi_{i}^{1}=\phi_{\beta(i)}^{1} \quad \text { and } \quad \beta(i)>i . $$ (See Theorem 5.51.) This proof is based only on the fixed point theorems and no longer involves Turing machines. (a) Show that there exists a primitive recursive function $\delta$ such that, for all $n$, $\phi_{\delta(n)}^{1}$ is the constant function equal to $n$. (b) Define the function $\gamma(n, t, z)$ by $$ \gamma(n, t, z)= \begin{cases}\delta(n) & \text { if } z<t \\ t & \text { otherwise. }\end{cases} $$ By applying the third version of the fixed point theorem (see Theorem 5.55) to this function, show that there exists a primitive recursive function $h(n, t)$ such that $$ \phi_{h(n, t)}^{1}= \begin{cases}\phi_{\delta(n)}^{1} & \text { if } h(n, t) \leq t \\ \phi_{t}^{1} & \text { otherwise. }\end{cases} $$ (c) Show that, for all $t$, the set $A_{t}=\{n: h(n, t) \leq t\}$ has at most $t+1$ elements. Use this to conclude that the desired function $\beta$ exists.

In this exercise, we will give an alternate proof of the fact that there exists a primitive recursive function $\beta$ of one variable such that, for all $i$,
$$
\phi_{i}^{1}=\phi_{\beta(i)}^{1} \quad \text { and } \quad \beta(i)>i .
$$
(See Theorem 5.51.) This proof is based only on the fixed point theorems and no longer involves Turing machines.
(a) Show that there exists a primitive recursive function $\delta$ such that, for all $n$, $\phi_{\delta(n)}^{1}$ is the constant function equal to $n$.
(b) Define the function $\gamma(n, t, z)$ by
$$
\gamma(n, t, z)= \begin{cases}\delta(n) & \text { if } z<t \\ t & \text { otherwise. }\end{cases}
$$
By applying the third version of the fixed point theorem (see Theorem 5.55) to this function, show that there exists a primitive recursive function $h(n, t)$
such that
$$
\phi_{h(n, t)}^{1}= \begin{cases}\phi_{\delta(n)}^{1} & \text { if } h(n, t) \leq t \\ \phi_{t}^{1} & \text { otherwise. }\end{cases}
$$
(c) Show that, for all $t$, the set $A_{t}=\{n: h(n, t) \leq t\}$ has at most $t+1$ elements. Use this to conclude that the desired function $\beta$ exists.

Key Concepts

Self-Reference in Computability

Self-reference in computability arises when a function is allowed to refer to its own index in its definition. This technique, justified by fixed point theorems, is crucial in proofs that require a function to have properties or behaviors that are defined in relation to its own description. It is a powerful tool for constructing functions with delicate or ‘diagonal’ features, which often play a key role in proving fundamental results about the limits and structure of computability.

Indexing of Computable Functions

The indexing or enumeration of computable functions is a method used to assign unique natural number codes (indices) to the functions computed by formal devices like Turing machines or primitive recursive methods. This concept allows researchers to discuss properties and relationships among computable functions in an abstract, uniform way, facilitating proofs that involve transforming or comparing these functions based on their indices.

Primitive Recursive Functions

Primitive recursive functions form a class of total computable functions that are defined using basic functions (such as the zero, successor, and projection functions) and are closed under operations of composition and primitive recursion. They are fundamental in computability theory because they represent an effective way to construct functions by step?by?step procedures without resorting to potentially unbounded search, thus ensuring totality and computational simplicity.

Fixed Point Theorems in Computability

Fixed point theorems, particularly Kleene’s Recursion Theorem, assert that for any computable transformation there exists an index such that the function it defines is equivalent to the function obtained by applying the transformation to its own index. This theorem is pivotal in recursive function theory because it enables self-referential constructions and provides a robust method for proving equivalences between different encodings of functions.

Transcript

00:01 We're asked to show that the function b of n from the busy beaver problem cannot be computed by any turing machine.

00:14 So to do this, we'll use a proof by contradiction.

00:18 So assume that there is a turing machine t that does compute b of n in binary in particular.

00:59 Now, let t prime be the turing machine that starts with a blank tape, writes n in binary, then computes b of n, again in binary, and then converts b of n from binary to unary.

02:32 Now since t exists, t prime must exist as well, because there do exist t machines that write n in binary, starting with a blank tape.

03:00 And there do exist touring machines that convert from binary to unary.

03:03 So for example, for a turning machine that would write n in binary, starting with a blank tape, we would use one state per one that needs to be written down.

03:42 And, for example, for a machine that converts binary to unary, we could have a machine where the machine erases the zeros, and we replace 1 in the nth position.

04:21 So this would indicate 2 to the n by n plus 1 ones in the same position, or instead of n plus 1, just n1s.

04:54 For example, if we have the binary code 1 ,0, we're going to remove that 0, so we just have one left, and we'll replace this 1, which is in the second position, by 2 ones.

05:14 And so in unary, this will give us the integer 1.

05:22 In binary, 1 0 is 2.

05:41 This should be n plus 1 .1 is my mistake...

Please give Ace some feedback