TuringMachines3x

Turing,

Machines, and,

Turing machines.

[AB, Ch
1
,
3
]

The Digital Era

.
Internet
A Visualization of the various routes through a portion of the

The Jacquard loom (
1801
)

The
Jacquard loom
, on display at the
Museum of
Science and Industry in Manchester
,
England
, was
one of the first programmable devices.

Bomba (
1938
)

Cryptologic

bomb
. Diagram from
Marian
Rejewski
's

papers
.

1
: Rotors (for clarity, only one
3
-
rotor set is shown).

2
: Electric motor.

3
: Switches.

The Columbia supercomputer

The Weather

Eye of Hurricane Isabel from the International
Space Station, September
15
,
2003
.

Cup of tea

What is a computer?


1.
A
computer

is a programmable
machine

that
receives input, stores and manipulates
data
,
and provides output in a useful format.


2.
A
computer

is any physical process that acts
on input and changes its configuration.


Alan Turing (
1912
-
1954
)

Alan Turing

1918
-
1930
–

School

1930
-

Turing's hopes and ambitions at school were raised
by the close friendship he developed with a slightly
older fellow student,
Christopher
Morcom
, who was
Turing's first love interest.
Morcom

died suddenly only
a few weeks into their last term at
Sherborne
, from
complications of
bovine tuberculosis
, contracted after
drinking infected cow's milk as a boy.
[
13
]

Turing's
religious faith was shattered and he became an atheist.
He adopted the conviction that all phenomena,
including the workings of the human brain, must be
materialistic.
[
14
]

Alan Turing

1931
–

1934
Kings College

1936
-

In his momentous paper "On Computable Numbers,
with an Application to the
Entscheidungsproblem
"
[
15
]

(submitted on
28
May
1936
), Turing reformulated
Kurt
Gödel
's
1931
results on the limits of proof and
computation, replacing Gödel's universal arithmetic
-
based
formal language with what are now called
Turing machines
,
formal and simple devices. He proved that some such
machine would be capable of performing any conceivable
mathematical computation if it were
representable

as an
algorithm
.


… the
halting problem

for Turing machines is
undecidable

Alan Turing

1936
–

1938
Princeton. Working with Church.

his dissertation introduced the notion of relative
computing, where Turing machines are
augmented with so
-
called
oracles
, allowing a
study of problems that cannot be solved by a
Turing machine.

Alan Turing

1938
–

1945
Cryptanalysis. UK, USA.


•
There should be no question in anyone's mind that Turing's work was the
biggest factor in Hut
8
's success. In the early days he was the only
cryptographer who thought the problem worth tackling and not only was
he primarily responsible for the main theoretical work within the Hut but
he also shared with
Welchman

and Keen the chief credit for the invention
of the Bombe. It is always difficult to say that anyone is absolutely
indispensable but if anyone was indispensable to Hut
8
it was Turing. The
pioneer's work always tends to be forgotten when experience and routine
later make everything seem easy and many of us in Hut
8
felt that the
magnitude of Turing's contribution was never fully realized by the outside
world.

—
Alexander, Sir C. Hugh O'D. Cryptographic History of Work on the German
Naval Enigma. The National Archives, Kew, Reference HW
25
/
1
.


Alan Turing

In
1945
, Turing was awarded the
OBE

for his wartime services

Alan Turing

1945
-
1947
,
National Physical Laboratory
, the first
detailed design of a
stored
-
program computer
.


1948
-

.. proposed an experiment now known as
the
Turing test
, an attempt to define a standard
for a machine to be called "intelligent". The idea
was that a computer could be said to "think" if it
could fool an interrogator into thinking that the
conversation was with a human.

Alan Turing

In January
1952
Turing picked up
19
-
year
-
old Arnold Murray outside a cinema in
Manchester. After a lunch date, Turing invited Murray to spend the weekend with
him at his house, an invitation which Murray accepted although he did not show
up. The pair met again in Manchester the following Monday, when Murray agreed
to accompany Turing to the latter's house. A few weeks later Murray visited
Turing's house again, and apparently spent the night there.
[
36
]

After Murray helped an accomplice to break into his house, Turing reported the crime
to the police. During the investigation, Turing acknowledged a sexual relationship
with Murray.
Homosexual

acts were illegal in the United Kingdom at that time,
[
6
]

and so both were charged with gross indecency under
Section
11

of the
Criminal
Law Amendment Act
1885
, the same crime that
Oscar Wilde

had been convicted
of more than fifty years earlier.
[
37
]

Turing was given a choice between imprisonment or probation conditional on his
agreement to undergo
hormonal

treatment

designed to reduce
libido
. He
accepted
chemical castration

via
oestrogen

hormone injections,
[
38
]

one of the side
effects of which was that he grew breasts.
[
37
]


Alan Turing

On
8
June
1954
, Turing's cleaner found him dead;
he had died the previous day. A
post
-
mortem

examination established that the cause of death
was cyanide poisoning. When his body was
discovered an apple lay half
-
eaten beside his bed,
and although the apple was not tested for
cyanide,
[
41
]

it is speculated that this was the
means by which a fatal dose was delivered. An
inquest

determined that he had committed
suicide, and he was cremated at
Woking

crematorium on
12

June
1954
.

2009
-

Apology

Thousands of people have come together to
demand justice for Alan Turing and recognition of
the appalling way he was treated. While Turing
was dealt with under the law of the time and we
can't put the clock back, his treatment was of
course utterly unfair and I am pleased to have the
chance to say how deeply sorry I and we all are
for what happened to him

... So on behalf of the
British government, and all those who live freely
thanks to Alan's work I am very proud to say:
we're sorry, you deserved so much better.

TM:
Formally


20

Turing Machine: Schematic

q
8

q
6

21

TM:
Formally


23

Computations

q
0

24

Computation Step

q
0


(
q
0
,a)=(q
6
,b,R)

q
6

•
specified next...

25

My first TM

26

The Transitions Function

Complexity

©D.Moshkovitz

q
0

q
3

q
2

q
1

q
4

q
ac

a

a, R

c

Z,L

b

Y,R

b

b, R

Z

Z, L

X

X, R

Y

Y, R
Z

Z, R

_

_, R

_

_, R

Y

Y, R

a

X,R

Y

Y, R

Z

Z, R

b

b, L
a

a, L

Y

Y, L

q
0

q
3

q
2

q
1

q
4

q
ac

a

a
, R

c

Z,L

b

Y,R

b

b, R

Z

Z, L

X

X, R

Y

Y, R
Z

Z, R

_

_, R

Y

Y, R

a

X,R

Y

Y, R

Z

Z, R

b

b, L
a

a, L

Y

Y, L

The thing to remember

Turing machines (algorithms) are objects :


•

with
constant
description size, and yet,


•
They work on inputs of
arbitrary size
.


A
uniform

model of computation.


28

Multi
-
Tape TMs





29

Multi
-
Tape Turing Machines

q
8

The universal TM

Thm
:

There exists a TM U, such that


U(
x,k
)=M
k
(x)


If M
k

halts on x within T steps, then

U(
x,k
) halts within O(T log T) steps.



The hidden constant depends only on
M
k
’s

alphabet size,

number of tapes and number on states, and does not

on the input length.

Languages

Def:

A language is some subset S

{
0
,
1
}
*

Example:

S={a
n
b
n
c
n
| n

N}


Def:

A Turing machine (TM) accepts L if it
accepts every
x
L

and rejects every
x
L
.


In particular it must terminate.

Time complexity

Def:

The running time of T on input x is the
number of
δ

transitions from the initial state
to an accept/reject state.




33

Time
-
Complexity


34


35

Name the Class

36

The Church
-
Turing thesis

“Everything computable is computable

by a Turing machine."


Though not formally proven, today the thesis has
near
-
universal acceptance.

The Physical Church
-
Turing thesis

All physically computable functions are

Turing computable

The Strong Church
-
Turing thesis

"
A
probabilistic Turing machine

can
efficiently

simulate any realistic
model of computation.
“


New Evidence that Quantum
Mechanics is Hard to Simulate on
Classical Computers


I'll discuss new types of evidence that quantum mechanics is hard to simulate
classically
--

evidence that's more complexity
-
theoretic in character than
(say) Shor's factoring algorithm, and that also corresponds to experiments
that seem easier than building a universal quantum computer. Specifically:


(
1
) I'll show that, by using linear optics (that is, systems of non
-
interacting
bosonic

particles), one can generate probability distributions that can't be
efficiently sampled by a classical computer, unless P^#P = BPP^NP and
hence the polynomial hierarchy collapses. The proof exploits an old
observation: that computing the amplitude for n bosons to evolve to a
given configuration involves taking the Permanent of an n
-
by
-
n matrix. I'll
also discuss an extension of this result to samplers that only approximate
the boson distribution. (Based on recent joint work with Alex
Arkhipov
)


(
2
) Time permitting, I'll also discuss new oracle evidence that BQP has
capabilities outside the entire polynomial hierarchy. (arXiv:
0910.4698
)

“Can machines Think?”

Turing (
1950
):
I PROPOSE to consider the
question, 'Can
machi


The question of whether it is possible for machines to think has a long history, which is
views of the

materialist
and

dualist
firmly entrenched in the distinction between
(or, at the very

physical
-
non
is

mind
mind. From the perspective of dualism, the
) and, therefore, cannot be explained in purely
]
6
[
physical properties
-
non
least, has
physical terms. The materialist perspective argues that the mind can be explained
physically, and thus leaves open the possibility of minds that are artificially
]
7
[
produced.


Are there imaginable digital computers which
would do as well as human beings?

What are we?

•
Was Alan Turing a computer mistreated by
other computers?

•
Will there ever be a computer passing Turing’s
test?

•
Can everything in our universe be captured as
computation?

•
Is there free choice?