# Chapter 3: The Church-Turing Thesis

## For Your Enjoyment

## Chapter 3.3, The Definition of an Algorithm

### A Simple Polynomial Problem

- Problem Statement: Does a polynomial with only one variable such
as 2x
^{2} -7x + 6 = 0 have an integral root?
- This problem is decidable because upper and lower bounds on a solution
can be calculated.
- What algorithm solves this problem?

### Hilbert's 10th Problem

- Problem Statement: Does a polynomial such as 3xy + 4x - 5y - 2 = 0 have
integral roots?
- This problem is Turing recognizable, but not Turing decidable.
- What is the algorithm?

### Church-Turing Thesis

The intuitive notion of an algorithm is the same as a Turing Machine
algorithm.

### Turing Machine Descriptions

- A
**formal description** includes all details.
- An
**implementation description** uses English to describe
how the read/write head moves.
- A
**high-level description** uses English to describe the
algorithm (but ignores the details of the read/write head).
- Example 3.23 is an example of a high-level description.

### Active Learning Problems

- Problem 3.5, parts a, b, and c, on page 188.
- Problem 3.7 on page 188.
- Problem 3.8, part a, on page 188. Provide a high-level description.