Turing Machines

1
Turing Machines

• CS 105 Introduction to
• Computer Science

2
Some Interesting Questions

• What is a computer? What is computation?
• Are there problems that some computers can solve
  but others cant?
• Are there problems that no computer can solve?

3
What is a Computer?

• Thats a tough question.
• We want to talk about the fundamental properties
  of computation.
• We need a model that captures the essential and
  throws out the rest.

4
What is Computation?

• Mapping an input to an output.
• For the sake of concreteness mapping a binary
  input to a binary output.
• Problems describe a particular mapping we are
  interested in.
• Decision problems are a subset of all problems -
  map from inputs to 0 or 1.

5
First model of computationFinite State Automata

1
0
1
A
B
0
6
Finite State Automata

• A fixed number of states.
• The machine moves from one state to another in
  response to inputs.
• If it ends in an ACCEPT state it accepts,
  otherwise it rejects. (Were just considering
  decision problems for now.)
• Lets see some examples

7
FSA Example
1
0
Triangle indicates start state.
1
Double circle indicates accepting state.
A
B
0
What happens on input 100101?
8
FSA Example
1
0
1
A
B
0
100101
9
FSA Example
1
0
1
A
B
0
100101
10
FSA Example
1
0
1
A
B
0
100101
11
FSA Example
1
0
1
A
B
0
100101
12
FSA Example
1
0
1
A
B
0
100101
13
FSA Example
1
0
ACCEPT!
1
A
B
0
100101
14
The Turing Machine

• Problems with the FSA as a model of computation??
• Does an input string have matched parentheses?
  E.g., 01(0(10)101)110(1
• With a slight modification we get the Turing
  machine.
• First described in 1937 by Alan Turing

15
How does it work?
...

Tape
Read/Write Head
Current State
Control Device
Rules

16
What can it do?
...
Tape
Read/Write Head
Current State
Control Device
Rules

• Write a character to the current tape cell
• Move Read/Write Head one cell to the left or
  right
• Go into a new state

17
How does it decide what to do?
...
Tape
Read/Write Head
Current State
Control Device
Rules

• Behavior is based on
• Input the character in the current tape cell
• State the current state

18
Lets see one in action
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
19
What are current state and input?
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
20
Which rule will the TM follow?
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 1 1 R 1
21
Write to the tape
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 1 1 R 1
22
Move right or left
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 1 1 R 1
23
Go to new state Done with rule
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 1 1 R 1
24
What are current state and input?
...

Tape
1
0
0
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
25
Which rule will the TM follow?
...

Tape
0
1
1
0
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 0 1 R 2
26
Write to the tape
...

Tape
0
1
1
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 0 1 R 2
27
Move right or left
...

Tape
0
1
1
1
Read/Write Head
Current State
1
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 0 1 R 2
28
Go to new state Done with rule
...

Tape
0
1
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
1 0 1 R 2
29
What are current state and input?
...

Tape
1
0
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
30
Which rule will the TM follow?
...

Tape
1
1
0
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
2 0 1 R 2
31
Write to the tape
...

Tape
1
1
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
2 0 1 R 2
32
Move right or left
...

Tape
1
1
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
2 0 1 R 2
33
Go to new state Done with rule
...

Tape
1
1
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
2 0 1 R 2
34
What are current state and input?
...

Tape
1
1
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
35
Which rule will the TM follow?
...

Tape
1
1
1
1
Read/Write Head
Current State
2
Control Device
Rules
State Symbol Write Move State 1
1 1 R 1 1
0 1 R 2 2
0 1 R 2
No Rule Done!
36
Is the Turing Machine The Right Model?

• The Church Turing Thesis
• Any reasonable model of computation is equivalent
  to the Turing machine.
• Equivalence here refers to what can be
  computed, not how fast it can be done.
• Not something that can be proved.

37
Why do we believe it?

• Every TM extension to make it more powerful
  (e.g., multiple-tape TMs) has been shown to be
  equivalent to the basic Turing Machine.
• Other models of computation have been shown to be
  weaker (e.g., Finite State Machines) or
  equivalent.

38
What is the significance of the Church-Turing
Thesis?

• Church-Turing Thesis
• Anything you can do on any computer
• you can do with a Turing Machine.
• So, no computer is more powerful (can compute
  more) than a TM.

39
Is my computer LESS powerful than a Turing
Machine?

• No.
• Given an infinite input device (infinite tape,
  infinite number of floppies, infinite keyboard
  input), you can simulate a TM.
• Therefore, anything that can be computed on a TM
  can be computed on your computer.

40
The Important Implication

• If theres anything that we CANT do with a
  Turing Machine
• then ...
• we cant do it on any computer now or in the
  future.

41
Are there things we cant compute on a Turing
Machine?
YES

An Example The Halting Problem
42
Last Words.

• Turing machines are
• Hard to program.
• Easy to prove things about.
• If we can prove something about the capabilities
  of Turing machines, we can immediately apply it
  to all computers.

43
Regular Expressions

• Another formalism for selecting strings to accept
  or reject.
• Intimately related to FSAs
• Practical applications in searching through
  strings.

44
Three Operators

• Union (01)
• accepts/matches either 0 or 1.
• Concatenation 01
• matches 0 followed by 1
• Repetition 0
• Matches any number of zeros (including 0)

45
Examples

• These operators can be applied to regular
  expressions
• (01)1 (matches any string that ends in a 1)
• (01)(01)) (matches any even length string)