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 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.