Set 𝒮^*=⋃p>0 ℕ^p and define a map αfrom 𝒮^* into ℕ as follows: if σis a sequence of integers of

step 1 So for this problem, we're going to recursively define two sequences by initializing C not to eq

Set 𝒮^*=⋃p>0 ℕ^p and define a map αfrom 𝒮^* into ℕ as...

Set $\mathcal{S}^{*}=\bigcup_{p>0} \mathbb{N}^{p}$ and define a map $\alpha$ from $\mathcal{S}^{*}$ into $\mathbb{N}$ as follows: if $\sigma$ is a sequence of integers of length $p$, then $\alpha(\sigma)=\alpha_{2}\left(p, \alpha_{p}(\sigma)\right)$. (a) Show that the function $\alpha$ is injective and that its range is a primitive recursive set. (b) Show that there exists a primitive recursive function $g$ such that if $\sigma=$ $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ and if $b=\sup \left(n, a_{1}, a_{2}, \ldots, a_{n}\right)$, then $\alpha(s) \leq g(b) .$ (c) Show that the function $\phi$ defined by $$ \phi(p, i, x)= \begin{cases}\beta_{p}^{i}(x) & \text { if } 1 \leq i \leq p \\ 0 & \text { otherwise }\end{cases} $$ is primitive recursive. (d) We now define another coding: let $\gamma$ be the function which, with every $\left(a_{0}, a_{1}, a_{2}, \ldots, a_{p}\right) \in \mathcal{S}^{*}$, associates the integer $$ \gamma\left(\left(a_{0}, a_{1}, \ldots, a_{p}\right)\right)=\pi(0)^{a_{0}+1} \cdot \pi(1)^{a_{1}+1} \cdot \ldots \cdot \pi(p)^{a_{p}+1} $$ it is understood that the value of $\gamma$ on the empty sequence is 1 . Show that $\gamma$ is an injective map and that its range is a primitive recursive set. (e) Show that the two codings can be obtained from one another in a primitive recursive way; more precisely, show that there exist two primitive recursive functions $f$ and $h$ of one variable such that (i) for all $x$ in the range of $\alpha, f(x)=\gamma(\sigma)$, where $\sigma$ is the non-empty sequence satisfying $\alpha(\sigma)=x$; (ii) for all $x$ in the range of $\gamma, h(x)=\alpha(\sigma)$, where $\sigma$ is the non-empty sequence satisfying $\gamma(\sigma)=x$.

Set $\mathcal{S}^{*}=\bigcup_{p>0} \mathbb{N}^{p}$ and define a map $\alpha$ from $\mathcal{S}^{*}$ into $\mathbb{N}$ as follows: if $\sigma$ is a sequence of integers of length $p$, then $\alpha(\sigma)=\alpha_{2}\left(p, \alpha_{p}(\sigma)\right)$.
(a) Show that the function $\alpha$ is injective and that its range is a primitive recursive set.
(b) Show that there exists a primitive recursive function $g$ such that if $\sigma=$ $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ and if $b=\sup \left(n, a_{1}, a_{2}, \ldots, a_{n}\right)$, then $\alpha(s) \leq g(b) .$
(c) Show that the function $\phi$ defined by
$$
\phi(p, i, x)= \begin{cases}\beta_{p}^{i}(x) & \text { if } 1 \leq i \leq p \\ 0 & \text { otherwise }\end{cases}
$$
is primitive recursive.
(d) We now define another coding: let $\gamma$ be the function which, with every $\left(a_{0}, a_{1}, a_{2}, \ldots, a_{p}\right) \in \mathcal{S}^{*}$, associates the integer
$$
\gamma\left(\left(a_{0}, a_{1}, \ldots, a_{p}\right)\right)=\pi(0)^{a_{0}+1} \cdot \pi(1)^{a_{1}+1} \cdot \ldots \cdot \pi(p)^{a_{p}+1}
$$
it is understood that the value of $\gamma$ on the empty sequence is 1 . Show that $\gamma$ is an injective map and that its range is a primitive recursive set.
(e) Show that the two codings can be obtained from one another in a primitive recursive way; more precisely, show that there exist two primitive recursive functions $f$ and $h$ of one variable such that
(i) for all $x$ in the range of $\alpha, f(x)=\gamma(\sigma)$, where $\sigma$ is the non-empty sequence satisfying $\alpha(\sigma)=x$;
(ii) for all $x$ in the range of $\gamma, h(x)=\alpha(\sigma)$, where $\sigma$ is the non-empty sequence satisfying $\gamma(\sigma)=x$.

Key Concepts

Encoding of Sequences

This concept involves representing finite sequences of natural numbers by single natural numbers using methods such as Gödel numbering or prime factorization. Encoding allows one to handle finite sequences using the machinery of number theory and recursion theory. The encoding functions are typically designed to be injective so that every sequence has a unique number associated with it, which is crucial for effective decoding and further manipulations.

Transformation Between Codings

This concept refers to the ability to convert between different encodings (or codings) of sequences using primitive recursive functions. It demonstrates that alternative representations of the same data can be interrelated in a computationally effective manner. Such transformations are essential for comparing different formal systems and for ensuring that properties proven in one coding scheme carry over to others.

Bounds and Supremum in Coding

When coding sequences, it is often necessary to show that the numerical codes are bounded by a function of the maximum value present in the sequence (or related parameters). Establishing such bounds guarantees that the codes remain within a predictable range, ensuring that related computations (like decoding) are manageable and primitive recursive in nature.

Primitive Recursive Functions

These are functions built from basic initial functions (such as the zero function, successor function, and projection functions) using composition and primitive recursion. They are total and computable, forming a fundamental class in computability theory. Their importance is that many naturally defined functions, including those used to encode sequences or manipulate numbers, can be shown to be primitive recursive, ensuring that these operations are algorithmically feasible.

Injective Mappings

An injective (or one-to-one) mapping is a function that assigns distinct outputs to distinct inputs. In the context of coding sequences, injectivity ensures that each coded number corresponds to a unique sequence. This uniqueness is essential for reliable decoding and for showing that the range of the coding function is well-defined and amenable to further algorithmic treatment.

Primitive Recursive Sets

A set is called primitive recursive if its characteristic function is primitive recursive. This means that membership in the set can be decided algorithmically within the bounds of primitive recursion. Showing that the range of a coding function is a primitive recursive set implies that the encoded values can be effectively recognized and manipulated, which is vital in formal theories of computation.

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