(a) Show that the set of recursive bijections from ℕ onto ℕ is a subgroup of the group of permut

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

(a) Show that the set of recursive bijections from ℕ onto ℕ...

(a) Show that the set of recursive bijections from $\mathbb{N}$ onto $\mathbb{N}$ is a subgroup of the group of permutations of $\mathbb{N}$. The remainder of this exercise is devoted to showing that this assertion is false if we replace recursive by primitive recursive. (b) Let $\phi$ be a (total) function of one variable that is recursive but not primitive recursive; let $e$ be an index for a machine $\mathcal{M}$ that computes $\phi$. Consider the function $T$ which associates, with $x$, the time required by $\mathcal{M}$ to compute $\phi(x) ;$ more precisely, $\left.T(x)=\mu t\left[(e, t, x) \in B^{1}\right)\right]$. Show that if $f$ is a function that satisfies $f(x) \geq T(x)$ for all $x \in \mathbb{N}$, then $f$ is not primitive recursive; also show that the graph $G$ of $T$ is primitive recursive. (c) We set $$ g(x)=\sup \{T(y): y \leq x\}+2 x . $$ Show that $g$ is a strictly increasing recursive function and that it is not primitive recursive. Show that the graph $G_{1}$ and the range $I$ of $g$ are primitive recursive sets. (d) Show that there is a unique strictly increasing primitive recursive function $g^{\prime}$ whose range is the complement of $I$. (e) Define the function $h$ by $$ \begin{aligned} h(2 x) &=g(x) \\ h(2 x+1) &=g^{\prime}(x) \end{aligned} $$ where $g$ and $g^{\prime}$ are the functions defined in (c) and (d) above. Show that $h$ is a bijective recursive function that is not primitive recursive. Show that its inverse, $h^{-1}$, is primitive recursive.

(a) Show that the set of recursive bijections from $\mathbb{N}$ onto $\mathbb{N}$ is a subgroup of the group of permutations of $\mathbb{N}$.
The remainder of this exercise is devoted to showing that this assertion is false if we replace recursive by primitive recursive.
(b) Let $\phi$ be a (total) function of one variable that is recursive but not primitive recursive; let $e$ be an index for a machine $\mathcal{M}$ that computes $\phi$. Consider the function $T$ which associates, with $x$, the time required by $\mathcal{M}$ to compute $\phi(x) ;$ more precisely, $\left.T(x)=\mu t\left[(e, t, x) \in B^{1}\right)\right]$.
Show that if $f$ is a function that satisfies $f(x) \geq T(x)$ for all $x \in \mathbb{N}$, then $f$ is not primitive recursive; also show that the graph $G$ of $T$ is primitive recursive.
(c) We set
$$
g(x)=\sup \{T(y): y \leq x\}+2 x .
$$
Show that $g$ is a strictly increasing recursive function and that it is not primitive recursive. Show that the graph $G_{1}$ and the range $I$ of $g$ are primitive recursive sets.
(d) Show that there is a unique strictly increasing primitive recursive function $g^{\prime}$ whose range is the complement of $I$.
(e) Define the function $h$ by
$$
\begin{aligned}
h(2 x) &=g(x) \\
h(2 x+1) &=g^{\prime}(x)
\end{aligned}
$$
where $g$ and $g^{\prime}$ are the functions defined in (c) and (d) above. Show that $h$ is a bijective recursive function that is not primitive recursive. Show that its inverse, $h^{-1}$, is primitive recursive.

Key Concepts

Time Complexity in Computability

Time complexity in computability theory examines the amount of time a machine takes to compute a function. By associating functions with the computation time, one can relate the intuitive notion of efficiency with the computability properties of functions. This concept is used to show that certain time bounding functions exceed the confines of the primitive recursive class.

Graph Representations in Computability

In computability theory, the graph of a function is the set of input-output pairs and is a useful tool to study functions from a decision-problem perspective. A function’s graph being primitive recursive means that there is an effective decision procedure to verify whether a given pair is in the graph, which is key for analyzing the computability and complexity properties of functions.

Minimization (µ-operator)

The minimization operator, or µ-operator, extends the scope of functions that can be defined recursively by allowing searches for the least number that satisfies a given condition. This operator is crucial in defining general recursive functions but is not available in the framework of primitive recursion, thus leading to functions that are recursive but not primitive recursive.

Constructing Non-Primitive Recursive Functions

The construction of functions that are recursive but not primitive recursive often involves combining simpler functions in ways that exploit the limits of primitive recursion, such as using a supremum or minimization over computational steps. These constructions help illustrate the boundaries between the classes of computable functions and are essential for understanding hierarchies in computational complexity.

Primitive Recursive Functions

Primitive recursive functions are a strict subset of recursive functions that are defined using only basic initial functions and the operations of composition and primitive recursion. They exclude the unbounded minimization (µ-operator), which is why not every recursive function is primitive recursive. This distinction is key in understanding hierarchies within computable functions.

Recursive Functions

Recursive functions are those that can be computed by some algorithm or Turing machine. They are the foundation of computability theory and capture the intuitive notion of an effectively calculable function. This concept extends to various computational models and is central to understanding which problems are algorithmically solvable.

Subgroups in Permutations

The subgroup concept in permutations refers to a set of bijections (permutations) that not only includes the identity function but is also closed under composition and inversion. This property is important in algebra and ensures that operations within the set maintain structure, which is here applied to understand the closure properties of computable bijections.

Inverse Functions in Computability

Investigating the invertibility of functions within computability theory involves determining whether the inverse of a given function is computable. This concept is important because a function might be defined in a complex manner, yet its inverse may have a simpler computational structure. Understanding the relationship between a function and its inverse sheds light on the subtle interplay between different classes of computable functions.

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