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Class 35: On Computable Numbers

David Evans
April 22, 2016

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.

David Evans

April 22, 2016
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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. 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?
  3. 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.
  4. How many real numbers? “Je le vois, mais je ne

    le crois pas.” Georg Cantor
  5. 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}*
  6. 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
  7. 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
  8. 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?
  9. 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!
  10. 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?
  11. 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
  12. Charge • Submit Project 6 before midnight tonight to be

    eligible to start Red Belt test Monday • Monday: – Computability in theory and practice