turing machines cs 105: introduction to computer science

Turing Machines

CS 105: Introduction to

Computer Science

Some Interesting Questions

What is a computer? What is computation? Are there problems that some computers

can solve but others can’t? Are there problems that no computer can

solve?

What is a Computer?

That’s 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.

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.

First model of computation:Finite State Automata…

A B

1

0

10

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. (We’re just considering decision problems for now.)

Let’s see some examples…

FSA Example

A B

1

0

10Triangle indicates start state.

Double circle indicates accepting state.

What happens on input 100101?

FSA Example

A B

1

0

10

100101

FSA Example

A B

1

0

10

100101

FSA Example

A B

1

0

10

100101

FSA Example

A B

1

0

10

100101

FSA Example

A B

1

0

10

100101

FSA Example

A B

1

0

10

100101

ACCEPT!

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

How does it work?

... Tape

Read/Write Head

Control DeviceCurrent State:Rules:

What can it do?

Write a character to the current tape cell Move Read/Write Head one cell to the left or

right Go into a new state

... Tape

Read/Write Head

Control DeviceCurrent State:Rules:

How does it decide what to do?

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

... Tape

Read/Write Head

Control DeviceCurrent State:Rules:

Let’s see one in action

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

1Rules:

1 0 0 1

What are current state and input?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

1Rules:

1 0 0 1

Which rule will the TM follow?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

1 0 0 1

1 1 | 1 R 1

1

Write to the tape

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

1 0 0 1

1 1 | 1 R 1

1

Move right or left

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

1 0 0 1

1 1 | 1 R 1

1

Go to new state: Done with rule

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

1 0 0 1

1 1 | 1 R 1

1

What are current state and input?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

1Rules:

1 0 0 1

Which rule will the TM follow?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

0 1

1 0 | 1 R 2

1

1 0

Write to the tape

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

0 1

1

1 0 | 1 R 2

1 1

Move right or left

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

0 1

1

1 0 | 1 R 2

1 1

Go to new state: Done with rule

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

0 1

2

1 0 | 1 R 2

1 1

What are current state and input?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

1 0 1

2

1

Which rule will the TM follow?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

1

2 0 | 1 R 2

1 0

2

1

Write to the tape

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

11

2 0 | 1 R 2

1

2

1

Move right or left

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

11

2 0 | 1 R 2

1

2

1

Go to new state: Done with rule

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

11

2 0 | 1 R 2

1

2

1

What are current state and input?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

11 1

2

1

Which rule will the TM follow?

... Tape

Read/Write Head

Control DeviceCurrent State:

State Symbol | Write Move State 1 1 | 1 R 1 1 0 | 1 R 2 2 0 | 1 R 2

Rules:

11 1

2

1

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.

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.

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.

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.

The Important Implication

If there’s anything that we CAN’T do with a Turing Machine

then ...

we can’t do it on any computer now or in the future.

Are there things we can’t compute on a Turing Machine?

YES

An Example: The Halting Problem

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.