/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 The purpose of this exercise is ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 21

The purpose of this exercise is to prove the following fact: (*) There exists a (total) recursive function \(\psi(x, y)\) such that if we set \(\psi_{x}=\) \(\lambda y . \psi(x, y)\), then the set \(\left\\{\psi_{x}: x \in \mathbb{N}\right\\}\) is precisely the set of all primitive recursive functions of one variable. (a) Show that if \(f \in \mathcal{F}_{p}\), then the following two conditions are equivalent: (i) \(f\) is primitive recursive; (ii) there exists an index \(i\) and a primitive recursive function \(g \in \mathcal{F}_{p}\) such that the machine whose index is \(i\) computes \(f\) and the computation time \(T\left(i, x_{1}, x_{2}, \ldots, x_{p}\right)\) is less than or equal to \(g\left(x_{1}, x_{2}, \ldots, x_{p}\right)\). (b) We will make use of Ackerman's function \(\zeta\) and of the functions \(\zeta_{n}=\) \(\lambda x . \zeta(n, x)\). Show that if \(f\) is a primitive recursive function of one variable, then there exist two integers \(n\) and \(A\) such that, for all \(x\), we have $$ f(x) \leq \sup \left(A, \zeta_{n}(x)\right) . $$ (c) Let \(g\) be the function of four variables defined by $$ \begin{gathered} g(i, A, n, x)=\mu y \leq \sup (A, \zeta(n, x)) \\ {\left[\exists t \leq \sup (A, \zeta(n, x))(i, t, x, y) \in C^{1}\right] ;} \end{gathered} $$ [recall that \((i, t, x, y) \in C^{1}\) means that when the machine whose index is \(i\) is set in operation with \(x\) on its first band, it will halt at the instant \(t\) with output \(y\) ]. Show that, for all \(i, A\), and \(n\), the function \(\lambda x \cdot g(i, A, n, x)\) is primitive recursive and that, conversely, if \(f\) is any primitive recursive function of one variable, then there exist integers \(i, A\), and \(n\) such that \(f=\lambda x . g(i, A, n, x)\). (d) Use these results to prove \((*)\). (e) Show that there exists a recursive set that is not primitive recursive.

Short Answer

Expert verified
The proof demonstrates a strong relationship between recursive functions and primitive recursive functions. It implies there exists a total recursive function that can account for every primitive recursive function of one variable. Furthermore, it shows some recursive functions cannot be primitive.

Step by step solution

01

Proving the equivalence of the two conditions

The equivalence of the two conditions means that if 'f' is a primitive recursive function, there exists a Turing machine that computes 'f'. Furthermore, the computation time of the Turing machine will be less than or equal to 'g', which is also another primitive recursive function.
02

Proving the existence of 'n' and 'A'

Proof requires showing that there are integer values of 'n' and 'A' that satisfy the Ackermann function conditions. This entails proving that if 'f' is a primitive recursive function of one variable, then there are 'n' and 'A' such that for all 'x', \(f(x) \leq \sup (A, \zeta(n, x))\) .
03

Defining the function 'g' and proving its primitiveness

The function 'g' of four variables is defined as per given. The subsequent step requires proving that this function is primitive recursive.
04

Proving that variable modifications of function 'g' returns any primitive recursive function

Assumption of 'f' to be any primitive recursive function of one variable, and then proving that there exist integers 'i', 'A', and 'n' such that \(f= \lambda x . g(i, A, n, x)\). This step essentially demonstrates that by modifying the values of constants, function 'g' can return all different primitive recursive functions of a single variable.
05

Proving the main statement (*)

The previous steps, when put together, would essentially be proving that the function 'g' corresponds to the set of all primitive recursive functions of a variable. Hence, it provides an idea about the existence of a recursive function agreeing with the main statement (*).
06

Proving the existence of a recursive set for the unproven recursive primitive function

Final step involves the proof indicating that there exists a recursive set that is not primitive recursive. This could be demonstrated by producing a function that halts for all inputs, but the halting or computation time does not have an upper bound primitive recursive function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recursion Theory
Recursion theory, also known as computability theory, is a branch of mathematical logic that deals with questions related to computation in abstract terms. It considers the capabilities and limitations of algorithms and computational devices. This theory helps us distinguish between problems that can be solved algorithmically and those that cannot. One of the central concepts within recursion theory is the classification of functions based on their computational complexity.

The functions we often encounter are structured through definitive rules or processes that produce a result for given inputs. In the context of recursion theory, certain functions are called recursive if they can be computed by an algorithm. A subset of recursive functions are primitive recursive functions鈥攖hese are functions that can be defined using basic arithmetic and composition, but they are also limited in the sense they must always halt, or in other words, they cannot represent all calculable functions. The textbook exercise we're examining explores the proof of the existence of a total recursive function that encapsulates all primitive recursive functions of one variable.
Ackermann's Function
Ackermann's function is a well-known example of a computable function that is not primitive recursive, which makes it important in illustrating the limitations of primitive recursion. It is a two-argument function that grows extraordinarily rapidly with relatively small inputs, exceeding the growth rate of functions like exponentiation.

In the exercise, Ackermann's function, denoted by \(\zeta(n, x)\), is utilized to demonstrate that for any primitive recursive function of one variable, there exist constants \(n\) and \(A\) such that the function is bounded by \(\sup(A, \zeta(n, x))\) for all \(x\). This bound utilizes the rapid growth properties of Ackermann's function to provide an upper estimate for the outputs of any primitive recursive function, thereby helping establish its primitiveness.
Turing Machine Computation
A Turing machine is an abstract computational model that consists of an infinite tape and a head that moves along the tape, reading and writing symbols according to a set of rules. The concept is fundamental in computer science as it provides a simple model that can simulate the logic of any computer algorithm. Turing machine computation refers to the process of the Turing machine following its programmed instructions to manipulate symbols on the tape.

In the provided exercise, Turing machines are discussed in the context of primitive recursive functions. It is shown that for every primitive recursive function, there exists a Turing machine that can compute the function's output in a time that is less than or equal to another primitive recursive function. This relationship indicates a strong connection between primitive recursive functions and the computations that can be performed by Turing machines.
G枚del's Theorems
G枚del's theorems, particularly the Incompleteness Theorems, are central to mathematical logic and recursion theory. The first of these theorems states that every sufficiently powerful and consistent axiomatic system cannot prove all truths about the arithmetic of natural numbers. The second theorem shows that such a system cannot demonstrate its own consistency.

These theorems intersect with recursion theory as they imply the existence of statements within systems鈥攖hat are true but cannot be proven within the system. They also suggest the existence of problems that no algorithm can solve. In the context of our exercise, G枚del's ideas resonate with the concept of primitive recursive functions. Although primitive recursive functions are powerful and can compute a wide array of numerical functions, G枚del's theorems assure us that there are always functions beyond their reach鈥攕pecifically, functions that are recursive but not primitive recursive.

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Most popular questions from this chapter

Let \(A\) and \(B\) be two subsets of \(\mathbb{N}\). We say that \(A\) is reducible to \(B\) and write \(A \leq B\) if there exists a (total) recursive function \(f\) such that \(x \in A \quad\) if and only if \(f(x) \in B\). (a) Show that the relation \(\leq\) is reflexive and transitive. (b) Assume that \(A\) is reducible to \(B\). Show that if \(B\) is recursively enumerable, then \(A\) is recursively enumerable; show also that if \(B\) is recursive, then so is \(A\). Set $$ \begin{aligned} X &=\left\\{x: \phi^{1}(x, x) \text { is defined }\right\\} \\ Y &=\left\\{\alpha_{2}(x, y): \phi^{1}(x, y) \text { is defined }\right\\} \end{aligned} $$ (c) Show that a set \(A \subseteq \mathbb{N}\) is recursively enumerable if and only if \(A \leq Y\). (d) Let \(A\) and \(B\) be two subsets of \(\mathbb{N}\). Let $$ C=\\{2 n: n \in A\\} \cup\\{2 n+1: n \in B\\} . $$ Show that \(A\) and \(B\) are reducible to \(C\) and that if \(D\) is a subset of \(\mathbb{N}\) such that \(A\) and \(B\) are both reducible to \(D\), then \(C\) is reducible to \(D\). (e) We will say that \(A\) is self-dual if \(A \leq \mathbb{N}-A\). Show that for every \(B \subseteq \mathbb{N}\), there exists a \(C \subseteq \mathbb{N}\) that is self-dual and is such that \(B \leq C\). (f) Let \(\mathcal{T}\) be a set of partial recursive functions of one variable that is not empty and is not equal to the set of all partial recursive functions of one variable. Set $$ A=\left\\{x: \phi_{x}^{1} \in \mathcal{T}\right\\} . $$ (i) Show that if the partial.function whose domain is empty belongs to \(\mathcal{T}\), then \(X \leq \mathbb{N}-A\). (ii) Show that, in the opposite case, \(X \leq A\). (iii) Show that \(A\) is not self-dual. (g) Show that \(Y \leq X\).

When we constructed the functions \(\phi^{p}\), we used a certain number of codings and, for this purpose, we had to make some completely arbitrary choices. In this exercise, our concern is to know what sort of functions would have been obtained instead of the \(\phi^{p}\) if our choices had been different. The only assumption we will make is that these choices are reasonable and sufficient for proving the enumeration theorem and the fixed point theorems. Let \(\Psi=\left\\{\psi^{p}: p \geq 1\right\\}\) be a family of partial recursive functions such that, for all \(p, \psi^{p} \in \mathcal{F}_{p+1}^{*}\). We set $$ \psi_{x}^{p}=\lambda y_{1} y_{2} \ldots y_{p} . \psi^{p}\left(x, y_{1}, y_{2}, \ldots, y_{p}\right) . $$ Consider the following conditions on the family \(\Psi\) : -(enu) For every \(p>0\), the set \(\left\\{\psi_{i}^{p}: i \in \mathbb{N}\right\\}\) is equal to the set of all partial recursive functions of \(p\) variables. \- (smn) For every pair of integers \(m\) and \(n\), there exists a total recursive function \(\sigma_{n}^{m}\) of \(n+1\) variables such that for all \(i, x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2}, \ldots, y_{m}\), we have $$ \begin{aligned} &\psi^{n+m}\left(i, x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2} \ldots, y_{m}\right) \\ &\quad=\psi^{m}\left(\sigma_{n}^{m}\left(i, x_{1}, x_{2}, \ldots, x_{n}\right), y_{1}, y_{2}, \ldots, y_{m}\right) \end{aligned} $$ (a) Let \(\theta\) be a partial recursive function of two variables. For every integer \(x\), we \(\operatorname{set} \theta_{x}=\lambda y . \theta(x, y)\). Show that the following two conditions are equivalent: (i) there exists a family \(\Psi=\left\\{\psi^{p}: p \geq 1\right\\}\) that satisfies conditions (enu) and \((\mathrm{smn})\) and is such that \(\psi^{1}=\theta\); (ii) there exists a recursive function \(\beta\) such that, for all \(x, \phi_{x}^{1}=\theta_{\beta(x)}\). (b) Assume once more that the family \(\Psi\) satisfies the conditions (enu) and \((\mathrm{smn})\). Show that the fixed point theorems are valid for the family \(\Psi\). (c) Assume that the function \(\theta\) satisfies conditions (i) or (ii) from (a). Show that there exist two injective recursive functions \(\alpha\) and \(\beta\) such that, for all \(x\), $$ \phi_{x}^{1}=\theta_{\beta(x)} \quad \text { and } \quad \theta_{x}=\phi_{\alpha(x)}^{1} $$ (d) (Difficult!) Under these same hypotheses, show that there exists a recursive function \(\varepsilon\) that is total and bijective and is such that, for all \(x, \phi_{x}^{1}=\theta_{\varepsilon(x)}\).

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 \(\gamma(\sigma)=x\).

Show that every infinite recursively enumerable set includes an infinite recursive set.

Let \(g, \alpha\), and \(h\) be partial recursive functions with \(g\) and \(\alpha\) in \(\mathcal{F}_{1}^{*}\) and \(h \in \mathcal{F}_{3}^{*}\). Show that there exists one and only one function \(f \in \mathcal{F}_{2}^{*}\) such that, for all \(x\) and \(y\), $$ \begin{aligned} f(0, y) &=g(y), \\ f(x+1, y) &=h(f(x, \alpha(y)), y, x), \end{aligned} $$ and \(f\) is partial recursive.
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