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The Turing Machine

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Title: The Turing Machine


1
The Turing Machine
  • A definition of computability
  • by Noah Richards

2
Computability
  • In the 1930s, mathematicians were looking for a
    definition of what it means to be able to
    compute a function. They came up with
  • A function is computable if it can be computed
    by a Turing machine.

3
What is a Turing Machine?
  • Invented by Alonzo Church and Alan Turing, it is
    an incredibly simple machine with all of the
    power of any digital computer.

4
Parts of a Turing Machine
  1. An infinite tape that is bi-directional and made
    up of squares containing a symbol from a finite
    alphabet. It contains only finitely many
    non-blank squares on the tape.

5
Parts, cont.
  1. A read/write head positioned at some square on
    the tape.
  2. A finite number of internal states at which it
    can be at exactly one at any given time.

6
Using a Turing Machine
  • The general form of instruction for a touring
    machine is
  • (current_state, current_symbol, new_state,
    new_symbol, left/right)

7
Using, cont.
  • An example of this is
  • (s1, 1, s2, 0, R)
  • This means when the machine is in state s1 and
    reads a 1, it transfers to state s2, writes a 0
    (to the same cell), and moves to the right one
    cell

8
Multiply
  • Lets assume we want to write a program to
    multiply two numbers. We define this function
    as
  • multiply(n1, n2) n1 n2
  • As per common Turing convention, we will be
    representing numbers in unary (strings of 1s)
    separated by spaces (_s)

9
Heres the code
Multiply function Turing Machine code
Step by step output for the multiply function
10
Virtual Turing Machine
  • This code was created and run on the Virtual
    Turing Machine

11
Other sources of information
  • The Turing Machine, written in JavaScript with
    examples in unary addition, divisibility, and
    primality testing
  • You guessed it a Turing Machine written in scheme

12
Moral of this story
  • Enjoy writing code in Scheme and Haskell. After
    all, you could be writing it for a Turing Machine.
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