Design machines that compute the following relations. You...

Design machines that compute the following relations. You may use the macros and machines constructed in Sections 12.2 through 12.4 and the machines constructed in Exercise 2. a) $g t(n, m)= \begin{cases}1 & \text { if } n>m \\ 0 & \text { otherwise }\end{cases}$ b) $\operatorname{persq}(n)= \begin{cases}1 & \text { if } n \text { is a perfect square } \\ 0 & \text { otherwise }\end{cases}$ c) divides $(n, m)= \begin{cases}1 & \text { if } n>0, m>0, \text { and } m \text { divides } n \\ 0 & \text { otherwise }\end{cases}$

Design machines that compute the following relations. You may use the macros and machines constructed in Sections 12.2 through 12.4 and the machines constructed in Exercise 2.
a) $g t(n, m)= \begin{cases}1 & \text { if } n>m \\ 0 & \text { otherwise }\end{cases}$
b) $\operatorname{persq}(n)= \begin{cases}1 & \text { if } n \text { is a perfect square } \\ 0 & \text { otherwise }\end{cases}$
c) divides $(n, m)= \begin{cases}1 & \text { if } n>0, m>0, \text { and } m \text { divides } n \\ 0 & \text { otherwise }\end{cases}$

Key Concepts

Turing Machines

A Turing machine is an abstract mathematical model that defines computation. It manipulates symbols on an infinite tape according to a finite set of rules. This model forms the basis for studying algorithms, computability, and the limits of what can be computed mechanically.

Macro Operations

Macro operations refer to the practice of building complex Turing machine behaviors by combining smaller, well-defined subroutines or operations. This modular approach simplifies the construction and analysis of Turing machines by allowing designers to reuse and compose preconstructed components.

Computable Decision Problems

Computable decision problems are questions with yes/no answers for which there exists a mechanical procedure or algorithm that provides a correct answer for every input. In the context of Turing machines, these problems illustrate the concept of decidability, where a machine is constructed to output 1 or 0 based on whether the input satisfies a certain property.

Comparison Operations

Comparison operations are computational techniques used to determine the relative size or order of numbers, such as checking if one number is larger than another. These operations are fundamental in many algorithms and are essential when designing machines to decide numerical relations.

Arithmetic Property Testing

Arithmetic property testing involves algorithms that determine whether numbers satisfy specific mathematical properties, such as being a perfect square or one number dividing another without a remainder. Such tests are key examples of decision procedures in arithmetic and play a central role in computational number theory.

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