Consider the following sets: A={x: ϕx^1(0) is defined } B={x: ϕx^1 is a total functio

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

Consider the following sets: A={x: ϕx^1(0) is defined }...

Consider the following sets: $$ \begin{aligned} &A=\left\{x: \phi_{x}^{1}(0) \text { is defined }\right\} \\ &B=\left\{x: \phi_{x}^{1} \text { is a total function }\right\} \end{aligned} $$ (a) Show that the complement of $A$ is not recursively enumerable. (b) Show that there exists a primitive recursive function $f \in \mathcal{F}_{1}$ such that for all $i, i \in A$ if and only if $\alpha(i) \in B$. Show that the complement of $B$ is not recursively enumerable. (c) Let $F$ be the following partial function: $$ F(x, y)= \begin{cases}1 & \text { if for all } z<y, \neg B^{1}(e, z, x) \\ \text { undefined } & \text { otherwise }\end{cases} $$ where $B^{1}$ is the predicate defined in the proof of the enumeration theorem and $e$ is the index of a partial function whose domain is $A$. Show that the partial function $\lambda y \cdot F(x, y)$ is total if and only if $x \notin A$. Conclude from this that $B$ is not recursively enumerable. (d) By generalizing the results from (b) and (c), prove the following: Proposition Let $f$ be a partial recursive function of one variable whose domain is infinite; then neither the set $\left\{x: \phi_{x}^{1}=f\right\}$ nor its complement is recursively enumerable.

Consider the following sets:
$$
\begin{aligned}
&A=\left\{x: \phi_{x}^{1}(0) \text { is defined }\right\} \\
&B=\left\{x: \phi_{x}^{1} \text { is a total function }\right\}
\end{aligned}
$$
(a) Show that the complement of $A$ is not recursively enumerable.
(b) Show that there exists a primitive recursive function $f \in \mathcal{F}_{1}$ such that for all $i, i \in A$ if and only if $\alpha(i) \in B$. Show that the complement of $B$ is not recursively enumerable.
(c) Let $F$ be the following partial function:
$$
F(x, y)= \begin{cases}1 & \text { if for all } z<y, \neg B^{1}(e, z, x) \\ \text { undefined } & \text { otherwise }\end{cases}
$$
where $B^{1}$ is the predicate defined in the proof of the enumeration theorem and $e$ is the index of a partial function whose domain is $A$.

Show that the partial function $\lambda y \cdot F(x, y)$ is total if and only if $x \notin A$. Conclude from this that $B$ is not recursively enumerable.
(d) By generalizing the results from (b) and (c), prove the following:
Proposition Let $f$ be a partial recursive function of one variable whose domain is infinite; then neither the set $\left\{x: \phi_{x}^{1}=f\right\}$ nor its complement is recursively enumerable.

Key Concepts

Primitive Recursive Functions

Primitive recursive functions form a subclass of total recursive functions that are constructed using basic functions and closed under operations such as composition and primitive recursion. Their guaranteed termination makes them important in understanding computational complexity and contrasts with the more general class of recursive functions, where non-termination is possible.

Enumeration Theorem and Indexing

The enumeration theorem in computability theory underpins the assignment of indices or codes to partial recursive functions, establishing that there exists a universal partial recursive function that can simulate any other. This effective indexing is central to many proofs, including those involving reductions between sets, demonstrating properties such as non-recursive enumerability of particular complements.

Complement of Recursively Enumerable Sets

The complement of a recursively enumerable set is not necessarily recursively enumerable. This asymmetry is a significant result in computability theory, highlighting that while a Turing machine may list all elements of a set, the inability to list non-elements often reflects inherent undecidability in the underlying decision problems.

Recursively Enumerable (r.e.) Sets

Recursively enumerable sets, or r.e. sets, are those sets for which there exists a Turing machine that can list (enumerate) all the members of the set, possibly without halting for inputs not in the set. This concept is central in computability theory because it captures the idea of effective enumerability, but also because many important sets (such as the halting set) are r.e. without being decidable.

Total Functions

Total functions, in the context of recursive functions, are those functions which are defined for every possible input in their domain. The distinction between total and partial functions is critical when addressing problems in computability, as certain properties, such as the recursive enumerability of a set of indices, can depend on whether the underlying functions are total.

Partial Recursive Functions

Partial recursive functions are those computable functions that may not be defined for every input; that is, there can be inputs for which the computation never halts. This concept is fundamental in computability theory as it distinguishes between functions that are guaranteed to produce an output (total functions) and those that might not, forming the basis for many undecidability results.

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