1. Theory of Computation
Turing Machines

2. Turing Machines: Origin
In the 40s, a mathematician called David Hilbert proposed a question
Known as his 10th problem
This was a decision problem
The solution was either yes or no
Alan Turing was intrigued by this problem
Is it true that 𝑥 2 + 𝑦 2 +1=0 has roots which are whole numbers?
Theory of Computation: Turing Machines

3. Turing Machines: Origin
From this question, Turing devised an abstract model of computation
Originally aimed solely at solving this problem
He defined an effective procedure for solving this problem
A more rigorous definition of an algorithm
He based this procedure on how human computers carried out algorithms
People which calculated values
Theory of Computation: Turing Machines

4. Turing Machines: Origin
The procedure involves a few things
An infinitely long one-dimensional piece of tape (with equally-sized spaces on it)
An instrument for writing/reading spaces on the tape
A mechanism for moving to different spaces on the tape
Theory of Computation: Turing Machines

5. Turing Machines: Origin
Each space on the tape can contain
Either nothing at all (an empty space)
OR a specific symbol (chosen from an Alphabet)
These symbols are usually restricted (i.e. 0 and 1)
As we can easily create any other symbol as a combination of base symbols
As shown in the table N
Series 1 2 11 3
111 4
1111 5
11111
Theory of Computation: Turing Machines

6. Turing Machine: Origins
Finally, the behaviour of the machine was determined solely by
The current state-of-mind (or just ‘state’) of the machine, and
The observed symbols in the current space on the tape
While there may by infinite spaces on the tape, the machine can only observe one at a time
The same also applies to the different state the machine can be in
The machine has a finite number of them, and can only be in one at a time
Theory of Computation: Turing Machines

7. Turing Machines: Origins
Turing devised this idea for human computers
Turing also thought that anything that is computable can be broken into its smallest computations
And represented in one of these machines
For any computer/program we’ve made so far, that is correct!
Theory of Computation: Turing Machines

8. Worked Example Here is the problem we need to ‘solve’
This is another decision problem
Which we are going to solve using a Turing machine
To ‘set up’ this machine, we need
To put any starting symbols on the tape
To work out what are states are (and how to transition between them)
Can we test if an input string has 𝑛 0s, followed by 𝑛 1s?
Theory of Computation: Turing Machines

9. Theory of Computation: Turing Machines
Worked Example
Let’s start by getting the tape set up
The tape is infinite
Giving us as much space as we want to work with
We’re going to have three symbols in our alphabet for this machine
#, 0, 1
Then let’s start with an example input like so: # 1
Theory of Computation: Turing Machines

10. Theory of Computation: Turing Machines
Worked Example
For this machine, we either accept or reject the input based on
The symbols between the two #s
If there are no symbols between these #s, then the input is accepted
If there are symbols left over, the input is rejected # 1
Theory of Computation: Turing Machines

11. Theory of Computation: Turing Machines
Worked Example
With the state of the tape set up, the next thing we need to work out is a bit trickier
What states and transitions will the machine use?
You may recognise both these terms from Finite-State Machines
That’s because a Turing Machine uses the same concepts
It has states that it is currently on
And can transition between states based off of an input
Theory of Computation: Turing Machines

12. Theory of Computation: Turing Machines
Worked Example
We need to create states and transitions so that the following happens
Start at the left-most #
Remove the first 0 that is found, then move to the right-most #
Remove the first 1 that is found, then move to the left-most #
Repeat this until no 0 or 1 is found, at which point the machine stops
The transitions that go with these states need
An input (the possible symbol we could read under the ‘read head’)
An output (where the ‘read head’ will move, and what it could write in the current space)
Theory of Computation: Turing Machines

13. Theory of Computation: Turing Machines
Worked Example
Here is the state transition diagram for this Turing Machine
Each transition is a directional arrow
From one state to another
Each label on a transition is the input/output
Input goes on the left of the bar (|)
Output goes on the right
Theory of Computation: Turing Machines

14. Theory of Computation: Turing Machines
Worked Example
Any input for a transition is what must be read from the tape in the current position
Any output for a transition is what will be written to on that space
The arrows show the direction of movement for the read head
Theory of Computation: Turing Machines

15. Can we test if an input string only contains 0s (and no 1s at all)?
Create a Turing Machine that answers the following problem
You will need to show the starting tape (including any symbols)
And you will need to create the state transition diagram
Once done, state what possible tape symbols your Turing Machine may end with
And what those mean
Can we test if an input string only contains 0s (and no 1s at all)?
Theory of Computation: Turing Machines

16. Transition Rules Reminder
Remember (from our look at Finite State Machines) that transition rules have different representations
They are commonly seen in state transition diagrams
As labels on directed arrows
However, we can represent them mathematically using a specific symbol
Can you remember what that symbol is?
Theory of Computation: Turing Machines

17. Transition Rules Reminder
We use the symbol 𝛿 (delta)!
The input goes inside brackets after it
The output goes after the equal sign
Here is an example: 𝛿 1, 0 =☐, →
This means “if 1 or 0 is in the current space under the read/write head, replace it with a blank space and move the read/write head to the right”
Theory of Computation: Turing Machines

18. Tracking Turing Machine Execution
A handy thing to be able to do is track the execution of a Turing Machine
This means keep track of all the states of the tape as the Turing Machine executes its states/transitions
We can then show all these tape states in something like a table
Theory of Computation: Turing Machines

19. Tracking Turing Machine Execution
However, we first need to know the notation we’re going to use
We’ll consider the tape as three different sections
Everything to the left of the read/write head
The cell under the read/write head
Everything to the right of the read/write head
We will show that using something like this: ☐, 1, 00☐1
Theory of Computation: Turing Machines

20. Tracking Turing Machine Execution
Staying on this example: #, 1, 00☐1
The first value in the brackets means there is only one cell to the left of the read/write head
And it contains #
Then the read/write head is currently looking at the symbol 1
Finally, there are four spaces to the right of the read/write head
And the order of them in the brackets is their order, from left-to-right
The tape is infinite, meaning we ignore any blank spaces to the left/right of any written symbols
Theory of Computation: Turing Machines

21. Tracking Turing Machine Execution
Let’s use the same 0s as 1s machine from the worked example earlier
With a tape of #0011#, here is the machine’s execution
Execution of Machine
(, #, 0011#)
(#, 0, 011#)
(#, ☐, 011#)
(#☐, 0, 11#)
(#☐0, 1, 1#)
(#☐01, 1, #)
(#☐011, #, )
Execution of Machine
(#☐01, 1, #)
(#☐01, ☐, #)
(#☐0, 1, ☐#)
(#☐, 0, 1☐#)
(#, ☐, 01☐#)
(#☐, ☐, 1☐#)
Execution of Machine
(#☐☐, 1, ☐#)
(#☐☐1, ☐, #)
(#☐☐, ☐, ☐#)
(#☐, ☐, ☐☐#)
Theory of Computation: Turing Machines

22. Theory of Computation: Turing Machines
Copy down the execution of the machine you made for an earlier exercise
Which tests if an input string only contains 0s (and no 1s)
Make two executions for these inputs
0000
0100
Feel free to add any other symbols your Turing Machine requires
And lay out the inputs at any position on the tape
Theory of Computation: Turing Machines

23. Turing Machines and Computation
Turing Machines are very important to the idea of computation
And whether a solution is computable
It is enough to know that a specific result can be found by following a specific algorithm
However, we cannot show that a result is not computable
We can show the ‘computability’ of a solution by seeing if we can make a Turing Machine for it
If one exists, then an algorithm for its solution exists too
If not, then no computable solution exists
Theory of Computation: Turing Machines

24. Turing Machines and Computation
A program can be represented by a Turing Machine
Saying that a solution to a problem is computable only if a Turing Machine exists is correct
However, we can reverse that logic
We can better say if a Turing Machine exists, there exists a computable solution
This means every algorithm we can create/think of can also exist as a Turing Machine
This concept is known as the Church-Turing thesis
Theory of Computation: Turing Machines

25. Turing Machines and Computation
This ultimately means that anything a computer can compute, a Turing Machine can compute as well !
So, any limitations a Turing Machine has, a computer has as well
A Turing Machine is the most basic of operations
It cannot be broken down further
However, other computers can be reduced
Which means no computer can be more powerful than a Turing Machine
Theory of Computation: Turing Machines

26. Turing Machines and Computation
We can simulate more complex calculations by connecting together multiple Turing Machines
Which use the result as one as the input of another
When thinking about the computability of solutions, using larger computers/devices would be bad
Take a processor, which changes every few years
If we base our idea on computability on one of these processors, we would need to re-evaluate these thoughts for every new processor
However, basing our thoughts on a Turing Machine, which doesn’t change, means we don’t need to re-evaluate our ideas of computability
Theory of Computation: Turing Machines

27. Universal Turing Machine
The idea of connecting multiple Turing Machines together gave Turing an idea
Can a universal machine be created that can ‘simulate’ other machines?
This machine is known as a Universal Turing Machine
This machine would have three bits of information for the machine it is simulating
A basic description of the machine
The contents of the machine’s tape
The internal state of the machine
Theory of Computation: Turing Machines

28. Universal Turing Machine
The universal machine would simulate the machine in question by looking at the input on the tape
And the state of the machine
It would then control the machine by changing its state based off the input
This leads to the idea of a “computer running other computers”
Theory of Computation: Turing Machines

29. Universal Turing Machine
This concept of computers running other computers actually gave John von Neumann the idea of the stored program
And even gave way to the idea of interpreters
As they are programs that run other programs
Without this idea of a Universal Turing Machine, modern computers wouldn’t be a thing
Or they would be drastically different
Theory of Computation: Turing Machines