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
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 of
states, and does not depend
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
Time Hierarchy Theorem
Def:
f:N
N is
time

constructible
, if there exists
a TM M
Time(T(n))
s.t
. M(x)=[T(x)].
Thm
:
If
•
f,g
: N
N are time

constructible, and
•
f(n)log f(n) = o(g(n))
Then
: Time(f(n
))
Time(g(n))
37
Growth Rate: rough classification
10
n
n
2
2
n
n! =
2
O(n lg n)
input length
time
38
complexity

Basic split in time
Name the Class
39
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.“
"A probabilistic Turing machine can
efficiently
simulate any physical
process.“
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?
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