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Chapter 5: Problem 15

Let \(\alpha\) be a recursive function that is injective. We set $$ \begin{aligned} &A=\operatorname{Ran}(\alpha) \\ &B=\\{x: \text { there exists } y>x \text { such that } \alpha(y)<\alpha(x)\\} \end{aligned} $$ (a) Show that \(B\) is recursively enumerable and that its complement is infinite. (b) Assume that there exists an infinite recursively enumerable subset \(C \subset \mathbb{N}\) that is disjoint from \(B\). Show that \(A\) is recursive. (c) Show that there exists a recursively enumerable set which has (1) a nonempty intersection with every infinite recursively enumerable set, and (2) an infinite complement.

Short Answer

Expert verified
The set \( B \) is recursively enumerable and its complement is infinite. If there exists an infinite recursively enumerable subset \( C \subset \mathbb{N} \) that is disjoint from \( B \), then \( A \) is recursive. There exists a recursively enumerable set which has a nonempty intersection with every infinite recursively enumerable set, and an infinite complement.

Step by step solution

01

Understand and Define Sets

Let's understand the sets A and B which have been defined for us. Set \( A \) is the range of function \( \alpha \), and set \( B \) contains elements \( x \) such that there exists a number \( y > x \) and \( \alpha(y) < \alpha(x) \). This means \( B \) consists of some \( x \) for which the recursive function \( \alpha \) evaluated at a higher number \( y \) gives a smaller result.
02

Prove B's Properties

To prove that \( B \) is recursively enumerable, consider the existence of a machine that enumerates pairs \( (x, y) \) with \( y > x \). For each pair, the machine computes \( \alpha(y) \) and \( \alpha(x) \) and accepts if \( \alpha(y) < \alpha(x) \). This means we can enumerate elements in \( B \). To prove that its complement is infinite, we assume the contrary i.e., \( B \)'s complement is finite. But \( \alpha \) is a recursive function that is injective, and there will always be a larger \( x \) so that \( \alpha(y) < \alpha(x) \), which is a contradiction.
03

Assume C Exists and Prove A's Properties

We assume that an infinite recursively enumerable subset \( C \subset \mathbb{N} \) that is disjoint from \( B \) exists. To prove that \( A \) is recursive, we define a function \( \beta \) that acts like \( \alpha \) on \( B \) and the undefined for \( \bar{B} \). Therefore, \( \beta \) is a total recursive function. We check for an input \( x \) if \( x \in B \), if yes, we output \( \beta(x) \), otherwise we enumerate \( C \) until we find a \( y \) that has not been the argument to \( \beta \), and we output \( \beta(y) \).
04

Prove the Existence of a Particular Set

To prove the existence of a recursively enumerable set which has a non-empty intersection with every infinite recursively enumerable set, and an infinite complement, we observe that the set of all even natural numbers satisfies these conditions. It is recursively enumerable, it intersects with every infinite recursively enumerable set because every infinite set contains at least one even number, and its complement, the set of odd natural numbers, is infinite.

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Key Concepts

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

Injective functions
An injective function, sometimes called one-to-one, maps every element of a set uniquely to another element. If two different elements in the domain are mapped to the same element in the codomain, the function is not injective. This uniqueness property is crucial.

With a focus on injective functions in the context of recursively enumerable sets, we establish that the function retains a unique mapping throughout the enumeration process.

  • Injective functions ensure that there are no duplicates in the range, simplifying the task of enumerating or listing such sets because every input corresponds to a unique output.
  • This property helps avoid contradictions when applying recursive methods. It ensures clear results from computational processes, such as identifying elements belonging to specific sets.
Recursive functions
Recursive functions are defined on natural numbers and follow a rule where they can call themselves within their own definition. This feature makes them paramount in computational theory.

Recursive functions are capable of solving problems by breaking them down into simpler subproblems (problems of the same type).

  • They can be computed by an algorithm, meaning there exists a step-by-step method to derive output for inputs from a recursive function.
  • When applied to recursively enumerable sets, recursive functions can help enumerate these sets effectively because every step in computation is clearly defined.
By using recursive functions, we can determine properties about the sets they help define. This includes checking membership in sets or generating lists of elements that belong to a set in a systematic way.
Infinite sets
Infinite sets are those sets that do not have a finite number of elements. In the context of recursion theory and recursively enumerable sets, infinite sets are particularly interesting because they continue to provide elements indefinitely.

Understanding infinite sets helps in grasping how recursive enumeration of sets works, particularly when discussing subsets of the natural numbers.

  • In infinite sets, if we remove a finite number of elements, the set still remains infinite.
  • The properties of infinite sets are used to prove certain aspects of recursively enumerable sets, such as having an infinite complement as discussed in recursively enumerable set theory problems.
This concept is crucial when discussing problems related to recursively enumerable and recursive sets, as it lays the groundwork for understanding what can or cannot be computed or listed by an algorithm.
Recursion theory
Recursion theory, or computability theory, is the study of computable functions and their properties. It plays a significant role in understanding what can be calculated mechanically鈥攅ssentially the theoretical foundation of computer science.

Recursion theory not only deals with function computation but also with the computational complexity and classification of different types of problems. This includes sets such as recursively enumerable ones, and their characteristics.

  • It focuses on the strategies and methodologies for calculating given functions or determining memberships in sets, which aids in exploring the limitations and extents of algorithmic processes.
  • By examining recursively enumerable sets through recursion theory, we can determine which parts of a set can be computed or enumerated solely by mechanical means without additional external intervention.
Understanding this topic allows you to grasp how different functions and sets interact in terms of computability and helps in addressing problems that revolve around these compute aspects.

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Most popular questions from this chapter

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 \(x, \phi_{x}^{1}=\theta_{\varepsilon(x)}\).

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

For each of the following functions, construct a Turing machine that computes it: (a) \(\lambda x \cdot x^{2}\), (b) \(\lambda x y \cdot x y\), (c) \(\lambda x \cdot x-1\) (d) \(\lambda x y \cdot x-y\)

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

(a) Show that if a partial function \(f \in \mathcal{F}_{1}^{*}\) is \(T\)-computable, then it is computable by a Turing machine that has exactly three bands. (b) Consider the set \(\mathcal{M}_{n}\) of Turing machines that have three bands and \(n\) states. Set these machines in operation with an initial configuration in which all bands are clean. If machine \(\mathcal{M}\) halts, we let \(\sigma(\mathcal{M})\) be the number of strokes written on its second band at the instant it halts; otherwise, we set \(\sigma(\mathcal{M})=0\). Show that the set $$ \left\\{\sigma(\mathcal{M}): \mathcal{M} \in \mathcal{M}_{n}\right\\} $$ is bounded. We will denote the upper bound of this set by \(\Sigma(n)\). (c) Let \(f\) be a partial function of one variable that is computable by a machine \(\mathcal{M}\) in \(\mathcal{M}_{n} .\) For every integer \(p\), construct a machine \(\mathcal{N}_{p}\) with three bands which, when started in an initial configuration in which all bands are clean, begins by writing \(p\) strokes on its first band, then returns its head to the beginning of the tape, and continues to behave exactly as \(\mathcal{M}\) would. How many states does \(\mathcal{N}_{p}\) have? (d) Show that the function \(\Sigma\) is not \(T\)-computable.
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