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Chapter 5: Problem 12

Show that if \(f\) is a function of one variable that is recursive and increasing, then its range is a recursive set. Conversely, show that every infinite recursive set is the range of a strictly increasing recursive function.

Short Answer

Expert verified
Both parts of the statement are true. The range of a recursive, increasing function will always form a recursive set and any infinite recursive set is the range of a strictly increasing recursive function.

Step by step solution

01

Proof Part 1

Assume \(f\) is a recursive, increasing function. The range of such a function is a recursive set if it can be computed by a terminating algorithm. Since \(f\) is recursive, there is an algorithm that can compute its values for any given input. Moreover, due to its increasing property, this algorithm halts in a finite number of steps. Therefore, all the values of \(f\) form a recursive set.
02

Proof Part 2

Assume \(A\) is an infinite recursive set. To show that \(A\) is the range of a strictly increasing recursive function, define the function \(f\) as follows: for any positive integer \(n\), \(f(n)\) is the \(n\)-th smallest element of \(A\). Because \(A\) is recursive, there exists an algorithm that lists all elements of \(A\), and thus can compute the \(n\)-th smallest element in a finite number of steps. With this definition, \(f\) is proved to be strictly increasing and recursive, and its range is the set \(A\).

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Key Concepts

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

Recursive Set
A recursive set, also known as a decidable or computable set, is the cornerstone of mathematical logic and theoretical computer science. It refers to a particular set of numbers that can be fully determined by a computer program. This program should be able to answer whether a given number is part of the set or not after a finite amount of time.
In our context, when a function is recursive and increasing, its range鈥攖he set of all outputs it can produce鈥攊s a recursive set. This means that we can develop a procedure to verify if a particular number is an output of our function. Since the function's values can be computed and it trends upward, we can sequentially check outputs for every input until we match or surpass the number in question. If there is a match, it's confirmed that the number is in the set.
This concept is not just an abstract idea; it has concrete applications in defining permissible operations within a programming environment and understanding what outcomes we can expect from those operations.
Increasing Function
An increasing function, in simple terms, always goes up as you move along the x-axis. It's like climbing a hill where each step takes you higher than the last. Formally, a function \( f \) is said to be increasing if for any two numbers \( a \) and \( b \) where \( a < b \) then \( f(a) \leq f(b) \). If \( f(a) < f(b) \) for all values of \( a \) and \( b \) where \( a < b \) it's called strictly increasing.
This trait is especially useful when proving characteristics about recursive sets because it gives a predictable pattern to the outputs of a function. When a recursive function is increasing, you can determine an order to its outputs which, in turn, helps establish the recursive nature of its range as described in the exercise.
G枚del's Theorems
G枚del's theorems are landmark insights from the logician Kurt G枚del. His incompleteness theorems, for instance, tell us about the limitations of mathematical systems. The first theorem says that in any consistent mathematical system, there are truths that can't be proved within that system. The second informs us that a system can't demonstrate its own consistency.
While the exercise on recursive sets and functions might not directly deal with G枚del's theorems, understanding these concepts can provide us with a profound appreciation for the limitations and capabilities of formal systems. Knowing that there are bounds to what can be computed or decided influences our understanding of recursion and computability, challenging us to think critically about the functions and algorithms we study.
Recursion Theory

Understanding Recursion Theory

Recursion theory, also known as computability theory, delves into which mathematical problems are solvable using algorithms鈥攕tep-by-step computational procedures. This field posits that if a problem can be solved by an algorithm, then it's said to be 'computable' or 'decidable'.
In our exercise, the conversion between a recursive set and a recursive function is an example of recursion theory in action. The theory backs the understanding of why the range of a recursive, increasing function is itself a recursive set and vice versa. It deals with the core principles of how mathematical operations are processed and come to a conclusion, essentially defining the boundaries of algorithmic computing.
Recursion theory is the scaffold that holds up key concepts in computer science and logic, including the design of programming languages, the efficiency of algorithms, and the theory behind artificial intelligence. It provides a framework to analyze and understand recursive functions and sets within the realm of computation.

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Most popular questions from this chapter

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 \(f(x)\) ].

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.

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.

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

(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.
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