# Chapter 3: The Church-Turing Thesis

## Chapter 3.3, The Definition of an Algorithm

### A Simple Polynomial Problem

• Problem Statement: Does a polynomial with only one variable such as 2x2 -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.