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Chapter 5: Problem 20

Let \(A \subseteq \mathbb{N}\) be a recursively enumerable set that is not recursive; let \(f\) be a partial recursive function whose domain is \(A\) and let \(i\) be an index for a Turing machine that computes \(f\). Show that the function \(\lambda x \cdot T^{1}(i, x)\) cannot be extended to a total recursive function [here, \(T^{1}\) is the function defined in the proof of the enumeration theorem whose value is the time required to compute \(f(x)\) ].

Short Answer

Expert verified
The function \(\lambda x \cdot T^{1}(i, x)\) cannot be extended to a total recursive function, because this would imply that the set \(A\) is recursive, which contradicts the fact that \(A\) is not recursive.

Step by step solution

01

Understanding the function \(\lambda x \cdot T^{1}(i, x)\)

The function \(\lambda x \cdot T^{1}(i, x)\) is a function that calculates the amount of time \(T\) needed by the Turing machine \(i\) to calculate the value of \(f\) at point \(x\). This function uses only a finite amount of time if \(x\) is in the domain of \(f\), because \(f\) is defined there and therefore computable. However, if \(x\) is outside the domain of \(f\), the Turing machine doesn't stop, thus this function is not defined there.
02

Examining the case where \(\lambda x \cdot T^{1}(i, x)\) can be extended

Suppose, for contradiction, that this function could be extended to a total recursive function. That would mean there exists a recursive function \(h: \mathbb{N} \rightarrow \mathbb{N}\), such that for every \(x\) in the domain of \(f\), \(h(x) = T^{1}(i, x)\) and for every \(x\) outside the domain of \(f\), \(h(x)\) has some definition.
03

Using \(h\) to construct a recursive characteristic function

We could then define another function \(g(x) = T(i, x, h(x))\), where \(T\) is the characteristic function for 'does the Turing machine \(i\) halt when started with \(x\) in \(h(x)\) steps'. Since \(h\) and \(T\) are recursive, \(g\) will be recursive too.
04

Realizing the contradiction

Since \(i\) is an index for a Turing machine that computes \(f\), \(g(x)\) would be \(1\) exactly when \(x\) is in the domain of \(f\), which is the set \(A\). But that means \(g(x)\) is a recursive characteristic function for \(A\). However, \(A\) is given to be non-recursive, leading to a contradiction.

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Key Concepts

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

Turing Machines
Turing Machines are fundamental tools in the field of theoretical computer science and are often used to model the logic of computation. They provide a framework for understanding what can be computed and how efficient the computation can be. A Turing Machine consists of an infinite tape divided into cells, a head that can read and write symbols on the tape, and a set of rules that define the machine's behavior based on the current state and the symbol it is reading.

Turing Machines can be used to define the complexity of problems, as well as to classify sets of numbers based on how a machine recognizes or decides membership. The recursive enumerable sets recognized by Turing Machines are particularly important, as they include all the sets for which membership can be verified by a machine, even if the machine does not halt for non-members. This implies that some problems, while recognizable, cannot be solved by a Turing Machine in a finite amount of time for all inputs.

Understanding these sets and the behavior of Turing Machines helps students comprehend the limitations of computation, as well as the connection between the machines and different classes of functions, such as partial recursive functions.
Partial Recursive Functions
Partial Recursive Functions are types of mathematical functions that are essential to computational theories and the study of algorithmic processes. These functions are 'partial' because they are not required to provide an output for every possible input; there may be some inputs for which the function does not terminate or provide a result. This is closely related to Turing Machines, as every partial recursive function can be computed by some Turing Machine.

In the exercise mentioned, a partial recursive function plays a pivotal role in the argument demonstrating the limitations that arise when working with recursively enumerable sets. The function in question is defined only on the set, which is recursively enumerable but not recursive. This means there is no guaranteed method to determine uniformly and in finite time whether an arbitrary number belongs to the set or not.

Students grappling with the concept of partial recursive functions should remember that these functions, much like Turing Machines, embody the concept of computability but also underscore the existence of problems that may not yield answers for all cases due to their undecidability or semi-decidability.
G枚del's Theorems
G枚del's Theorems, particularly his Incompleteness Theorems, are landmark findings in mathematical logic and the philosophy of mathematics. The first theorem states that any consistent formal system, which is capable of arithmetic, contains statements that are true but cannot be proven within the system. The second theorem goes further, stating that no such system can prove its own consistency.

The exercise at hand indirectly touches on the essence of G枚del's work, as it deals with the limits of computation and formal systems. The notion that the function \(\lambda x \cdot T^{1}(i, x)\) cannot be extended to a total recursive function parallels the understanding that some truths (or computations, in this case) are unattainable within a given system - an echo of G枚del's first theorem.

When exploring G枚del's Theorems, students are often confronted with profound implications that shake the foundations of mathematics and our understanding of what is knowable. It's important to recognize the theorems' relevance not just in mathematical theory but also in practical applications like the theory of computation, where they influence our grasp of what machines can and cannot do.

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Most popular questions from this chapter

Construct a Turing machine that halts if and only if the integer represented on its first band at the initial instant is even.

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

(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.

Let \(f \in \mathcal{F}_{1} .\) Show that \(f\) is recursive if and only if its graph $$ G=\left\\{(x, y) \in \mathbb{N}^{2}: y=f(x)\right\\} $$ is recursive.

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 \(Y \leq X\).
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