Class 35: On Computable Numbers

cs1120: Introduction to Computing
Explorations in Language, Logic, and Machine
University of Virginia, Spring 2016

http://xplorecs.org/class35

Class 35:
How many Turing Machines?
On Computable Numbers
Countable Sets
The smallest natural number that cannot be described in eleven words.

Transcript

1. Class 35: Computability Introduction to Computing: Explorations in Language, Logic, and Machines cs1120 Spring 2016 David Evans University of Virginia Halting Problems Hockey Team
2. Church-Turing Thesis A Turing Machine can simulate any “mechanical computer”. Alonzo Church, 1903-1995 Alan Turing, 1912-1954 1
3. 7 0 0 A 1 0 4 w 0 # 0 $ 1 4 w 0 # Recap: Turing Machine Symbols: a finite set of things you can write in a square States: a finite set of the “states of mind” of the machine Rules: a list of quintuples: (read symbol, state) è (write symbol, move, new state) Read symbol in current square Based on rule for current state and symbol read: Write a symbol into current square Move left or right one position Change state of mind How many Turing machines are there?
5. Defining Computable Numbers A number is computable if its decimal can be written down by a machine. All green text: taken directly from Turing’s Paper I give some arguments showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, …, the numbers , e, etc.
6. Is computable?
7. Is computable? Computable convergence: (Defined carefully by Turing, but not by us.)
8. How many computable numbers are there?
9. How many whole numbers are there?
10. Are there more integers?
11. How many bitstrings in are there?
12. How many real numbers?
13. How many real numbers? “Je le vois, mais je ne le crois pas.” Georg Cantor
14. Countable and Uncountable Infinities Countable: Natural Numbers Any set that has 1-to-1 mapping to whole numbers Not Countable: Real Numbers Set of strings in {0, 1}*
15. 7 0 0 A 1 0 4 w 0 # 0 $ 1 4 w 0 # How many Turing Machines? Symbols: a finite set of things you can write in a square States: a finite set of the “states of mind” of the machine Rules: a list of quintuples: (read symbol, state) è (write symbol, move, new state) Read symbol in current square Based on rule for current state and symbol read: Write a symbol into current square Move left or right one position Change state of mind
16. How many computable numbers? Countable: Natural Numbers Any set that has 1-to-1 mapping to whole numbers Set of strings in {0, 1}* Set of all Turing Machines Not Countable: Real Numbers
17. A number is computable if its decimal can be written down by a machine. All green text: taken directly from Turing’s Paper I give some arguments showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, …, the numbers , e, etc. How many computable numbers?
18. Computable Numbers Countable: Natural Numbers Set of strings in {0, 1}* Set of all Turing Machines Computable numbers Not Countable: Real Numbers Each TM produces at most one new number!
19. Numbers that can be produced by any mechanical computing machine All Numbers
20. Numbers that can be produced by any mechanical computing machine All Numbers What is an example of a number that cannot be produced by a TM?
21. What is the smallest natural number that cannot be described in eleven words?
22. What is the smallest natural number that cannot be described in eleven words? The smallest natural number that cannot be described in eleven words. 1 2 3 4 5 6 7 8 9 10 11
23. Charge • Submit Project 6 before midnight tonight to be eligible to start Red Belt test Monday • Monday: – Computability in theory and practice