<s>
A	O
Turing	B-Architecture
machine	I-Architecture
is	O
a	O
mathematical	O
model	O
of	O
computation	O
describing	O
an	O
abstract	B-Application
machine	I-Application
that	O
manipulates	O
symbols	O
on	O
a	O
strip	O
of	O
tape	O
according	O
to	O
a	O
table	O
of	O
rules	O
.	O
</s>
<s>
The	B-Architecture
Turing	I-Architecture
machine	I-Architecture
was	O
invented	O
in	O
1936	O
by	O
Alan	O
Turing	O
,	O
who	O
called	O
it	O
an	O
"	O
a-machine	B-Architecture
"	O
(	O
automatic	O
machine	O
)	O
.	O
</s>
<s>
It	O
was	O
Turing	O
's	O
Doctoral	O
advisor	O
,	O
Alonzo	O
Church	O
,	O
who	O
later	O
coined	O
the	O
term	O
"	O
Turing	B-Architecture
machine	I-Architecture
"	O
in	O
a	O
review	O
.	O
</s>
<s>
Turing	B-Architecture
machines	I-Architecture
proved	O
the	O
existence	O
of	O
fundamental	O
limitations	O
on	O
the	O
power	O
of	O
mechanical	O
computation	O
.	O
</s>
<s>
While	O
they	O
can	O
express	O
arbitrary	O
computations	O
,	O
their	O
minimalist	O
design	O
makes	O
them	O
unsuitable	O
for	O
computation	O
in	O
practice	O
:	O
real-world	O
computers	O
are	O
based	O
on	O
different	O
designs	O
that	O
,	O
unlike	O
Turing	B-Architecture
machines	I-Architecture
,	O
use	O
random-access	B-Architecture
memory	I-Architecture
.	O
</s>
<s>
Turing	B-Algorithm
completeness	I-Algorithm
is	O
the	O
ability	O
for	O
a	O
system	O
of	O
instructions	O
to	O
simulate	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
A	O
programming	O
language	O
that	O
is	O
Turing	B-Algorithm
complete	I-Algorithm
is	O
theoretically	O
capable	O
of	O
expressing	O
all	O
tasks	O
accomplishable	O
by	O
computers	O
;	O
nearly	O
all	O
programming	O
languages	O
are	O
Turing	B-Algorithm
complete	I-Algorithm
if	O
the	O
limitations	O
of	O
finite	O
memory	O
are	O
ignored	O
.	O
</s>
<s>
A	O
Turing	B-Architecture
machine	I-Architecture
is	O
a	O
general	O
example	O
of	O
a	O
central	B-General_Concept
processing	I-General_Concept
unit	I-General_Concept
(	O
CPU	O
)	O
that	O
controls	O
all	O
data	O
manipulation	O
done	O
by	O
a	O
computer	O
,	O
with	O
the	O
canonical	O
machine	O
using	O
sequential	O
memory	O
to	O
store	O
data	O
.	O
</s>
<s>
More	O
specifically	O
,	O
it	O
is	O
a	O
machine	O
(	O
automaton	O
)	O
capable	O
of	O
enumerating	B-Language
some	O
arbitrary	O
subset	O
of	O
valid	O
strings	O
of	O
an	O
alphabet	O
;	O
these	O
strings	O
are	O
part	O
of	O
a	O
recursively	O
enumerable	O
set	O
.	O
</s>
<s>
A	O
Turing	B-Architecture
machine	I-Architecture
has	O
a	O
tape	O
of	O
infinite	O
length	O
on	O
which	O
it	O
can	O
perform	O
read	O
and	O
write	O
operations	O
.	O
</s>
<s>
Assuming	O
a	O
black	B-Device
box	I-Device
,	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
cannot	O
know	O
whether	O
it	O
will	O
eventually	O
enumerate	O
any	O
one	O
specific	O
string	O
of	O
the	O
subset	O
with	O
a	O
given	O
program	O
.	O
</s>
<s>
The	B-Architecture
Turing	I-Architecture
machine	I-Architecture
is	O
capable	O
of	O
processing	O
an	O
unrestricted	O
grammar	O
,	O
which	O
further	O
implies	O
that	O
it	O
is	O
capable	O
of	O
robustly	O
evaluating	O
first-order	O
logic	O
in	O
an	O
infinite	O
number	O
of	O
ways	O
.	O
</s>
<s>
This	O
is	O
famously	O
demonstrated	O
through	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
A	O
Turing	B-Architecture
machine	I-Architecture
that	O
is	O
able	O
to	O
simulate	O
any	O
other	O
Turing	B-Architecture
machine	I-Architecture
is	O
called	O
a	O
universal	O
Turing	B-Architecture
machine	I-Architecture
(	O
UTM	O
,	O
or	O
simply	O
a	O
universal	O
machine	O
)	O
.	O
</s>
<s>
A	O
more	O
mathematically	O
oriented	O
definition	O
with	O
a	O
similar	O
"	O
universal	O
"	O
nature	O
was	O
introduced	O
by	O
Alonzo	O
Church	O
,	O
whose	O
work	O
on	O
lambda	B-Language
calculus	I-Language
intertwined	O
with	O
Turing	O
's	O
in	O
a	O
formal	O
theory	O
of	O
computation	O
known	O
as	O
the	O
Church-Turing	O
thesis	O
.	O
</s>
<s>
The	O
thesis	O
states	O
that	O
Turing	B-Architecture
machines	I-Architecture
indeed	O
capture	O
the	O
informal	O
notion	O
of	O
effective	O
methods	O
in	O
logic	O
and	O
mathematics	O
,	O
and	O
provide	O
a	O
model	O
through	O
which	O
one	O
can	O
reason	O
about	O
an	O
algorithm	O
or	O
"	O
mechanical	O
procedure	O
"	O
.	O
</s>
<s>
Studying	O
their	O
abstract	B-Application
properties	I-Application
yields	O
many	O
insights	O
into	O
computer	B-General_Concept
science	I-General_Concept
and	O
complexity	O
theory	O
.	O
</s>
<s>
The	B-Architecture
Turing	I-Architecture
machine	I-Architecture
mathematically	O
models	O
a	O
machine	O
that	O
mechanically	O
operates	O
on	O
a	O
tape	O
.	O
</s>
<s>
More	O
explicitly	O
,	O
a	O
Turing	B-Architecture
machine	I-Architecture
consists	O
of	O
:	O
</s>
<s>
The	O
tape	O
is	O
assumed	O
to	O
be	O
arbitrarily	O
extendable	O
to	O
the	O
left	O
and	O
to	O
the	O
right	O
,	O
so	O
that	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
is	O
always	O
supplied	O
with	O
as	O
much	O
tape	O
as	O
it	O
needs	O
for	O
its	O
computation	O
.	O
</s>
<s>
A	O
state	O
register	O
that	O
stores	O
the	O
state	O
of	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
,	O
one	O
of	O
finitely	O
many	O
.	O
</s>
<s>
Among	O
these	O
is	O
the	O
special	O
start	B-Architecture
state	I-Architecture
with	O
which	O
the	O
state	O
register	O
is	O
initialized	O
.	O
</s>
<s>
A	O
finite	O
table	O
of	O
instructions	O
that	O
,	O
given	O
the	O
state(qi )	O
the	O
machine	O
is	O
currently	O
in	O
and	O
the	O
symbol(aj )	O
it	O
is	O
reading	O
on	O
the	O
tape	O
(	O
symbol	O
currently	O
under	O
the	O
head	O
)	O
,	O
tells	O
the	O
machine	O
to	O
do	O
the	O
following	O
in	O
sequence	O
(	O
for	O
the	O
5-tuple	O
models	O
)	O
:	O
</s>
<s>
In	O
the	O
4-tuple	O
models	O
,	O
erasing	O
or	O
writing	O
a	O
symbol	O
(	O
aj1	O
)	O
and	O
moving	O
the	O
head	O
left	O
or	O
right	O
(	O
dk	O
)	O
are	O
specified	O
as	O
separate	O
instructions	O
.	O
</s>
<s>
its	O
state	O
,	O
symbol-collections	O
,	O
and	O
used	O
tape	O
at	O
any	O
given	O
time	O
)	O
and	O
its	O
actions	O
(	O
such	O
as	O
printing	O
,	O
erasing	O
and	O
tape	O
motion	O
)	O
is	O
finite	O
,	O
discrete	O
and	O
distinguishable	O
;	O
it	O
is	O
the	O
unlimited	O
amount	O
of	O
tape	O
and	O
runtime	O
that	O
gives	O
it	O
an	O
unbounded	O
amount	O
of	O
storage	B-General_Concept
space	I-General_Concept
.	O
</s>
<s>
is	O
the	O
set	O
of	O
final	O
states	O
or	O
accepting	B-Architecture
states	I-Architecture
.	O
</s>
<s>
is	O
a	O
partial	B-Algorithm
function	I-Algorithm
called	O
the	O
transition	B-Application
function	I-Application
,	O
where	O
L	O
is	O
left	O
shift	O
,	O
R	O
is	O
right	O
shift	O
.	O
</s>
<s>
If	O
is	O
not	O
defined	O
on	O
the	O
current	O
state	O
and	O
the	O
current	O
tape	O
symbol	O
,	O
then	O
the	O
machine	O
halts	O
;	O
intuitively	O
,	O
the	O
transition	B-Application
function	I-Application
specifies	O
the	O
next	O
state	O
transited	O
from	O
the	O
current	O
state	O
,	O
which	O
symbol	O
to	O
overwrite	O
the	O
current	O
symbol	O
pointed	O
by	O
the	O
head	O
,	O
and	O
the	O
next	O
head	O
movement	O
.	O
</s>
<s>
The	O
7-tuple	O
for	O
the	O
3-state	O
busy	O
beaver	O
looks	O
like	O
this	O
(	O
see	O
more	O
about	O
this	O
busy	O
beaver	O
at	O
Turing	B-Architecture
machine	I-Architecture
examples	O
)	O
:	O
</s>
<s>
see	O
state-table	O
below	O
(	O
transition	B-Application
function	I-Application
)	O
.	O
</s>
<s>
In	O
the	O
words	O
of	O
van	O
Emde	O
Boas	O
(	O
1990	O
)	O
,	O
p.6	O
:	O
"	O
The	O
set-theoretical	O
object	O
[	O
his	O
formal	O
seven-tuple	O
description	O
similar	O
to	O
the	O
above ]	O
provides	O
only	O
partial	O
information	O
on	O
how	O
the	O
machine	O
will	O
behave	O
and	O
what	O
its	O
computations	O
will	O
look	O
like.	O
"	O
</s>
<s>
The	O
shift	O
left	O
and	O
shift	O
right	O
operations	O
may	O
shift	O
the	O
tape	O
head	O
across	O
the	O
tape	O
,	O
but	O
when	O
actually	O
building	O
a	O
Turing	B-Architecture
machine	I-Architecture
it	O
is	O
more	O
practical	O
to	O
make	O
the	O
tape	O
slide	O
back	O
and	O
forth	O
under	O
the	O
head	O
instead	O
.	O
</s>
<s>
The	O
tape	O
cannot	O
be	O
fixed	O
in	O
length	O
,	O
since	O
that	O
would	O
not	O
correspond	O
to	O
the	O
given	O
definition	O
and	O
would	O
seriously	O
limit	O
the	O
range	O
of	O
computations	O
the	O
machine	O
can	O
perform	O
to	O
those	O
of	O
a	O
linear	B-Application
bounded	I-Application
automaton	I-Application
if	O
the	O
tape	O
was	O
proportional	O
to	O
the	O
input	O
size	O
,	O
or	O
finite-state	B-Architecture
machine	I-Architecture
if	O
it	O
was	O
strictly	O
fixed-length	O
.	O
</s>
<s>
The	O
most	O
common	O
convention	O
represents	O
each	O
"	O
Turing	O
instruction	O
"	O
in	O
a	O
"	O
Turing	O
table	O
"	O
by	O
one	O
of	O
nine	O
5-tuples	O
,	O
per	O
the	O
convention	O
of	O
Turing/Davis	O
(	O
Turing	O
(	O
1936	O
)	O
in	O
The	O
Undecidable	O
,	O
p.126	O
–	O
127	O
and	O
Davis	O
(	O
2000	O
)	O
p.152	O
)	O
:	O
</s>
<s>
+	O
Example	O
:	O
state	O
table	O
for	O
the	O
3-state	O
2-symbol	O
busy	O
beaver	O
reduced	O
to	O
5-tuples	O
Current	O
state	O
Scanned	O
symbol	O
Print	O
symbol	O
Move	O
tape	O
Final	O
(	O
i.e.	O
</s>
<s>
Subsequent	O
to	O
Turing	O
's	O
original	O
paper	O
in	O
1936	O
–	O
1937	O
,	O
machine-models	O
have	O
allowed	O
all	O
nine	O
possible	O
types	O
of	O
five-tuples	O
:	O
</s>
<s>
Current	O
m-configuration(Turing state )	O
Tape	O
symbol	O
Print-operation	O
Tape-motion	O
Final	O
m-configuration(Turing state )	O
5-tuple	O
5-tuple	O
comments	O
4-tuple	O
N1	O
qi	O
Sj	O
Print(Sk )	O
Left	O
L	O
qm	O
(	O
qi	O
,	O
Sj	O
,	O
Sk	O
,	O
L	O
,	O
qm	O
)	O
"	O
blank	O
"	O
=	O
S0	O
,	O
1	O
=	O
S1	O
,	O
etc	O
.	O
</s>
<s>
Any	O
Turing	O
table	O
(	O
list	O
of	O
instructions	O
)	O
can	O
be	O
constructed	O
from	O
the	O
above	O
nine	O
5-tuples	O
.	O
</s>
<s>
For	O
examples	O
see	O
Turing	B-Architecture
machine	I-Architecture
examples	O
.	O
</s>
<s>
Less	O
frequently	O
the	O
use	O
of	O
4-tuples	O
are	O
encountered	O
:	O
these	O
represent	O
a	O
further	O
atomization	O
of	O
the	O
Turing	O
instructions	O
(	O
cf	O
.	O
</s>
<s>
Post	O
(	O
1947	O
)	O
,	O
Boolos	O
&	O
Jeffrey	O
(	O
1974	O
,	O
1999	O
)	O
,	O
Davis-Sigal-Weyuker	O
(	O
1994	O
)	O
)	O
;	O
also	O
see	O
more	O
at	O
Post	B-Application
–	I-Application
Turing	I-Application
machine	I-Application
.	O
</s>
<s>
The	O
word	O
"	O
state	O
"	O
used	O
in	O
context	O
of	O
Turing	B-Architecture
machines	I-Architecture
can	O
be	O
a	O
source	O
of	O
confusion	O
,	O
as	O
it	O
can	O
mean	O
two	O
things	O
.	O
</s>
<s>
"	O
State	O
"	O
in	O
the	O
context	O
of	O
Turing	B-Architecture
machines	I-Architecture
should	O
be	O
clarified	O
as	O
to	O
which	O
is	O
being	O
described	O
:	O
the	O
current	O
instruction	O
,	O
or	O
the	O
list	O
of	O
symbols	O
on	O
the	O
tape	O
together	O
with	O
the	O
current	O
instruction	O
,	O
or	O
the	O
list	O
of	O
symbols	O
on	O
the	O
tape	O
together	O
with	O
the	O
current	O
instruction	O
placed	O
to	O
the	O
left	O
of	O
the	O
scanned	O
symbol	O
or	O
to	O
the	O
right	O
of	O
the	O
scanned	O
symbol	O
.	O
</s>
<s>
Many	O
machines	O
that	O
might	O
be	O
thought	O
to	O
have	O
more	O
computational	O
capability	O
than	O
a	O
simple	O
universal	O
Turing	B-Architecture
machine	I-Architecture
can	O
be	O
shown	O
to	O
have	O
no	O
more	O
power	O
(	O
Hopcroft	O
and	O
Ullman	O
p.159	O
,	O
cf	O
.	O
</s>
<s>
(	O
The	O
Church	O
–	O
Turing	O
thesis	O
hypothesizes	O
this	O
to	O
be	O
true	O
for	O
any	O
kind	O
of	O
machine	O
:	O
that	O
anything	O
that	O
can	O
be	O
"	O
computed	O
"	O
can	O
be	O
computed	O
by	O
some	O
Turing	B-Architecture
machine	I-Architecture
.	O
)	O
</s>
<s>
A	O
Turing	B-Architecture
machine	I-Architecture
is	O
equivalent	O
to	O
a	O
single-stack	O
pushdown	B-Application
automaton	I-Application
(	O
PDA	O
)	O
that	O
has	O
been	O
made	O
more	O
flexible	O
and	O
concise	O
by	O
relaxing	O
the	O
last-in-first-out	B-Application
(	O
LIFO	B-Application
)	O
requirement	O
of	O
its	O
stack	B-Application
.	O
</s>
<s>
In	O
addition	O
,	O
a	O
Turing	B-Architecture
machine	I-Architecture
is	O
also	O
equivalent	O
to	O
a	O
two-stack	O
PDA	O
with	O
standard	O
LIFO	B-Application
semantics	O
,	O
by	O
using	O
one	O
stack	B-Application
to	O
model	O
the	O
tape	O
left	O
of	O
the	O
head	O
and	O
the	O
other	O
stack	B-Application
for	O
the	O
tape	O
to	O
the	O
right	O
.	O
</s>
<s>
At	O
the	O
other	O
extreme	O
,	O
some	O
very	O
simple	O
models	O
turn	O
out	O
to	O
be	O
Turing-equivalent	B-Algorithm
,	O
i.e.	O
</s>
<s>
to	O
have	O
the	O
same	O
computational	O
power	O
as	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
model	O
.	O
</s>
<s>
Common	O
equivalent	O
models	O
are	O
the	O
multi-tape	B-Application
Turing	I-Application
machine	I-Application
,	O
multi-track	O
Turing	B-Architecture
machine	I-Architecture
,	O
machines	O
with	O
input	O
and	O
output	O
,	O
and	O
the	O
non-deterministic	O
Turing	B-Architecture
machine	I-Architecture
(	O
NDTM	O
)	O
as	O
opposed	O
to	O
the	O
deterministic	B-Architecture
Turing	I-Architecture
machine	I-Architecture
(	O
DTM	O
)	O
for	O
which	O
the	O
action	O
table	O
has	O
at	O
most	O
one	O
entry	O
for	O
each	O
combination	O
of	O
symbol	O
and	O
state	O
.	O
</s>
<s>
Read-only	O
,	O
right-moving	O
Turing	B-Architecture
machines	I-Architecture
are	O
equivalent	O
to	O
DFAs	B-General_Concept
(	O
as	O
well	O
as	O
NFAs	B-General_Concept
by	O
conversion	O
using	O
the	O
NDFA	B-General_Concept
to	I-General_Concept
DFA	I-General_Concept
conversion	I-General_Concept
algorithm	I-General_Concept
)	O
.	O
</s>
<s>
For	O
practical	O
and	O
didactical	O
intentions	O
the	O
equivalent	O
register	B-Application
machine	I-Application
can	O
be	O
used	O
as	O
a	O
usual	O
assembly	B-Language
programming	I-Language
language	I-Language
.	O
</s>
<s>
While	O
the	O
computation	O
of	O
a	O
real	O
computer	O
is	O
based	O
on	O
finite	B-Architecture
states	I-Architecture
and	O
thus	O
not	O
capable	O
to	O
simulate	O
a	O
Turing	B-Architecture
machine	I-Architecture
,	O
programming	O
languages	O
themselves	O
do	O
not	O
necessarily	O
have	O
this	O
limitation	O
.	O
</s>
<s>
Kirner	O
et	O
al.	O
,	O
2009	O
have	O
shown	O
that	O
among	O
the	O
general-purpose	O
programming	O
languages	O
some	O
are	O
Turing	B-Algorithm
complete	I-Algorithm
while	O
others	O
are	O
not	O
.	O
</s>
<s>
For	O
example	O
,	O
ANSI	O
C	O
is	O
not	O
Turing-equivalent	B-Algorithm
,	O
as	O
all	O
instantiations	O
of	O
ANSI	O
C	O
(	O
different	O
instantiations	O
are	O
possible	O
as	O
the	O
standard	O
deliberately	O
leaves	O
certain	O
behaviour	O
undefined	O
for	O
legacy	O
reasons	O
)	O
imply	O
a	O
finite-space	O
memory	O
.	O
</s>
<s>
However	O
,	O
other	O
programming	O
languages	O
like	O
Pascal	B-Application
do	O
not	O
have	O
this	O
feature	O
,	O
which	O
allows	O
them	O
to	O
be	O
Turing	B-Algorithm
complete	I-Algorithm
in	O
principle	O
.	O
</s>
<s>
It	O
is	O
just	O
Turing	B-Algorithm
complete	I-Algorithm
in	O
principle	O
,	O
as	O
memory	B-General_Concept
allocation	I-General_Concept
in	O
a	O
programming	O
language	O
is	O
allowed	O
to	O
fail	O
,	O
which	O
means	O
the	O
programming	O
language	O
can	O
be	O
Turing	B-Algorithm
complete	I-Algorithm
when	O
ignoring	O
failed	O
memory	B-General_Concept
allocations	I-General_Concept
,	O
but	O
the	O
compiled	O
programs	O
executable	O
on	O
a	O
real	O
computer	O
cannot	O
.	O
</s>
<s>
Turing	O
(	O
1936	O
)	O
does	O
not	O
elaborate	O
further	O
except	O
in	O
a	O
footnote	O
in	O
which	O
he	O
describes	O
how	O
to	O
use	O
an	O
a-machine	B-Architecture
to	O
"	O
find	O
all	O
the	O
provable	O
formulae	O
of	O
the	O
 [ Hilbert ] 	O
calculus	O
"	O
rather	O
than	O
use	O
a	O
choice	O
machine	O
.	O
</s>
<s>
This	O
is	O
indeed	O
the	O
technique	O
by	O
which	O
a	O
deterministic	O
(	O
i.e.	O
,	O
a	O
-	O
)	O
Turing	B-Architecture
machine	I-Architecture
can	O
be	O
used	O
to	O
mimic	O
the	O
action	O
of	O
a	O
nondeterministic	O
Turing	B-Architecture
machine	I-Architecture
;	O
Turing	O
solved	O
the	O
matter	O
in	O
a	O
footnote	O
and	O
appears	O
to	O
dismiss	O
it	O
from	O
further	O
consideration	O
.	O
</s>
<s>
An	O
oracle	O
machine	O
or	O
o-machine	O
is	O
a	O
Turing	O
a-machine	B-Architecture
that	O
pauses	O
its	O
computation	O
at	O
state	O
"	O
o	O
"	O
while	O
,	O
to	O
complete	O
its	O
calculation	O
,	O
it	O
"	O
awaits	O
the	O
decision	O
"	O
of	O
"	O
the	O
oracle	O
"	O
—	O
an	O
unspecified	O
entity	O
"	O
apart	O
from	O
saying	O
that	O
it	O
cannot	O
be	O
a	O
machine	O
"	O
(	O
Turing	O
(	O
1939	O
)	O
,	O
The	O
Undecidable	O
,	O
p.166	O
–	O
168	O
)	O
.	O
</s>
<s>
In	O
terms	O
of	O
computational	O
complexity	O
,	O
a	O
multi-tape	O
universal	O
Turing	B-Architecture
machine	I-Architecture
need	O
only	O
be	O
slower	O
by	O
logarithmic	O
factor	O
compared	O
to	O
the	O
machines	O
it	O
simulates	O
.	O
</s>
<s>
It	O
is	O
often	O
believed	O
that	O
Turing	B-Architecture
machines	I-Architecture
,	O
unlike	O
simpler	O
automata	O
,	O
are	O
as	O
powerful	O
as	O
real	O
machines	O
,	O
and	O
are	O
able	O
to	O
execute	O
any	O
operation	O
that	O
a	O
real	O
program	O
can	O
.	O
</s>
<s>
What	O
is	O
neglected	O
in	O
this	O
statement	O
is	O
that	O
,	O
because	O
a	O
real	O
machine	O
can	O
only	O
have	O
a	O
finite	O
number	O
of	O
configurations	O
,	O
it	O
is	O
nothing	O
but	O
a	O
finite-state	B-Architecture
machine	I-Architecture
,	O
whereas	O
a	O
Turing	B-Architecture
machine	I-Architecture
has	O
an	O
unlimited	O
amount	O
of	O
storage	B-General_Concept
space	I-General_Concept
available	O
for	O
its	O
computations	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
ways	O
to	O
explain	O
why	O
Turing	B-Architecture
machines	I-Architecture
are	O
useful	O
models	O
of	O
real	O
computers	O
:	O
</s>
<s>
Anything	O
a	O
real	O
computer	O
can	O
compute	O
,	O
a	O
Turing	B-Architecture
machine	I-Architecture
can	O
also	O
compute	O
.	O
</s>
<s>
For	O
example	O
:	O
"	O
A	O
Turing	B-Architecture
machine	I-Architecture
can	O
simulate	O
any	O
type	O
of	O
subroutine	O
found	O
in	O
programming	O
languages	O
,	O
including	O
recursive	O
procedures	O
and	O
any	O
of	O
the	O
known	O
parameter-passing	O
mechanisms	O
"	O
(	O
Hopcroft	O
and	O
Ullman	O
p.157	O
)	O
.	O
</s>
<s>
Thus	O
,	O
a	O
statement	O
about	O
the	O
limitations	O
of	O
Turing	B-Architecture
machines	I-Architecture
will	O
also	O
apply	O
to	O
real	O
computers	O
.	O
</s>
<s>
The	O
difference	O
lies	O
only	O
with	O
the	O
ability	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
to	O
manipulate	O
an	O
unbounded	O
amount	O
of	O
data	O
.	O
</s>
<s>
However	O
,	O
given	O
a	O
finite	O
amount	O
of	O
time	O
,	O
a	O
Turing	B-Architecture
machine	I-Architecture
(	O
like	O
a	O
real	O
machine	O
)	O
can	O
only	O
manipulate	O
a	O
finite	O
amount	O
of	O
data	O
.	O
</s>
<s>
Like	O
a	O
Turing	B-Architecture
machine	I-Architecture
,	O
a	O
real	O
machine	O
can	O
have	O
its	O
storage	B-General_Concept
space	I-General_Concept
enlarged	O
as	O
needed	O
,	O
by	O
acquiring	O
more	O
disks	O
or	O
other	O
storage	O
media	O
.	O
</s>
<s>
Descriptions	O
of	O
real	O
machine	O
programs	O
using	O
simpler	O
abstract	O
models	O
are	O
often	O
much	O
more	O
complex	O
than	O
descriptions	O
using	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
a	O
Turing	B-Architecture
machine	I-Architecture
describing	O
an	O
algorithm	O
may	O
have	O
a	O
few	O
hundred	O
states	O
,	O
while	O
the	O
equivalent	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
DFA	B-General_Concept
)	O
on	O
a	O
given	O
real	O
machine	O
has	O
quadrillions	O
.	O
</s>
<s>
This	O
makes	O
the	O
DFA	B-General_Concept
representation	O
infeasible	O
to	O
analyze	O
.	O
</s>
<s>
Turing	B-Architecture
machines	I-Architecture
describe	O
algorithms	O
independent	O
of	O
how	O
much	O
memory	O
they	O
use	O
.	O
</s>
<s>
Turing	B-Architecture
machines	I-Architecture
allow	O
us	O
to	O
make	O
statements	O
about	O
algorithms	O
which	O
will	O
(	O
theoretically	O
)	O
hold	O
forever	O
,	O
regardless	O
of	O
advances	O
in	O
conventional	O
computing	O
machine	O
architecture	O
.	O
</s>
<s>
Turing	B-Architecture
machines	I-Architecture
simplify	O
the	O
statement	O
of	O
algorithms	O
.	O
</s>
<s>
Algorithms	O
running	O
on	O
Turing-equivalent	B-Algorithm
abstract	B-Application
machines	I-Application
are	O
usually	O
more	O
general	O
than	O
their	O
counterparts	O
running	O
on	O
real	O
machines	O
,	O
because	O
they	O
have	O
arbitrary-precision	O
data	O
types	O
available	O
and	O
never	O
have	O
to	O
deal	O
with	O
unexpected	O
conditions	O
(	O
including	O
,	O
but	O
not	O
limited	O
to	O
,	O
running	O
out	B-General_Concept
of	I-General_Concept
memory	I-General_Concept
)	O
.	O
</s>
<s>
A	O
limitation	O
of	O
Turing	B-Architecture
machines	I-Architecture
is	O
that	O
they	O
do	O
not	O
model	O
the	O
strengths	O
of	O
a	O
particular	O
arrangement	O
well	O
.	O
</s>
<s>
For	O
instance	O
,	O
modern	O
stored-program	O
computers	O
are	O
actually	O
instances	O
of	O
a	O
more	O
specific	O
form	O
of	O
abstract	B-Application
machine	I-Application
known	O
as	O
the	O
random-access	B-Application
stored-program	I-Application
machine	I-Application
or	O
RASP	B-Application
machine	I-Application
model	O
.	O
</s>
<s>
Like	O
the	O
universal	O
Turing	B-Architecture
machine	I-Architecture
,	O
the	O
RASP	O
stores	O
its	O
"	O
program	O
"	O
in	O
"	O
memory	O
"	O
external	O
to	O
its	O
finite-state	B-Architecture
machine	I-Architecture
's	O
"	O
instructions	O
"	O
.	O
</s>
<s>
Unlike	O
the	O
universal	O
Turing	B-Architecture
machine	I-Architecture
,	O
the	O
RASP	O
has	O
an	O
infinite	O
number	O
of	O
distinguishable	O
,	O
numbered	O
but	O
unbounded	O
"	O
registers	O
"	O
—	O
memory	O
"	O
cells	O
"	O
that	O
can	O
contain	O
any	O
integer	O
(	O
cf	O
.	O
</s>
<s>
Elgot	O
and	O
Robinson	O
(	O
1964	O
)	O
,	O
Hartmanis	O
(	O
1971	O
)	O
,	O
and	O
in	O
particular	O
Cook-Rechow	O
(	O
1973	O
)	O
;	O
references	O
at	O
random-access	B-Application
machine	I-Application
)	O
.	O
</s>
<s>
The	O
RASP	O
's	O
finite-state	B-Architecture
machine	I-Architecture
is	O
equipped	O
with	O
the	O
capability	O
for	O
indirect	O
addressing	O
(	O
e.g.	O
,	O
the	O
contents	O
of	O
one	O
register	O
can	O
be	O
used	O
as	O
an	O
address	O
to	O
specify	O
another	O
register	O
)	O
;	O
thus	O
the	O
RASP	O
's	O
"	O
program	O
"	O
can	O
address	O
any	O
register	O
in	O
the	O
register-sequence	O
.	O
</s>
<s>
The	O
upshot	O
of	O
this	O
distinction	O
is	O
that	O
there	O
are	O
computational	O
optimizations	O
that	O
can	O
be	O
performed	O
based	O
on	O
the	O
memory	O
indices	O
,	O
which	O
are	O
not	O
possible	O
in	O
a	O
general	O
Turing	B-Architecture
machine	I-Architecture
;	O
thus	O
when	O
Turing	B-Architecture
machines	I-Architecture
are	O
used	O
as	O
the	O
basis	O
for	O
bounding	O
running	O
times	O
,	O
a	O
"	O
false	O
lower	O
bound	O
"	O
can	O
be	O
proven	O
on	O
certain	O
algorithms	O
 '	O
running	O
times	O
(	O
due	O
to	O
the	O
false	O
simplifying	O
assumption	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
)	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
is	O
binary	O
search	O
,	O
an	O
algorithm	O
that	O
can	O
be	O
shown	O
to	O
perform	O
more	O
quickly	O
when	O
using	O
the	O
RASP	B-Application
model	I-Application
of	O
computation	O
rather	O
than	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
model	O
.	O
</s>
<s>
Another	O
limitation	O
of	O
Turing	B-Architecture
machines	I-Architecture
is	O
that	O
they	O
do	O
not	O
model	O
concurrency	O
well	O
.	O
</s>
<s>
For	O
example	O
,	O
there	O
is	O
a	O
bound	O
on	O
the	O
size	O
of	O
integer	O
that	O
can	O
be	O
computed	O
by	O
an	O
always-halting	O
nondeterministic	O
Turing	B-Architecture
machine	I-Architecture
starting	O
on	O
a	O
blank	O
tape	O
.	O
</s>
<s>
(	O
See	O
article	O
on	O
unbounded	B-Application
nondeterminism	I-Application
.	O
)	O
</s>
<s>
In	O
the	O
early	O
days	O
of	O
computing	O
,	O
computer	O
use	O
was	O
typically	O
limited	O
to	O
batch	B-General_Concept
processing	I-General_Concept
,	O
i.e.	O
,	O
non-interactive	B-General_Concept
tasks	O
,	O
each	O
producing	O
output	O
data	O
from	O
given	O
input	O
data	O
.	O
</s>
<s>
Computability	O
theory	O
,	O
which	O
studies	O
computability	O
of	O
functions	O
from	O
inputs	O
to	O
outputs	O
,	O
and	O
for	O
which	O
Turing	B-Architecture
machines	I-Architecture
were	O
invented	O
,	O
reflects	O
this	O
practice	O
.	O
</s>
<s>
Since	O
the	O
1970s	O
,	O
interactive	B-General_Concept
use	O
of	O
computers	O
became	O
much	O
more	O
common	O
.	O
</s>
<s>
In	O
principle	O
,	O
it	O
is	O
possible	O
to	O
model	O
this	O
by	O
having	O
an	O
external	O
agent	O
read	O
from	O
the	O
tape	O
and	O
write	O
to	O
it	O
at	O
the	O
same	O
time	O
as	O
a	O
Turing	B-Architecture
machine	I-Architecture
,	O
but	O
this	O
rarely	O
matches	O
how	O
interaction	O
actually	O
happens	O
;	O
therefore	O
,	O
when	O
describing	O
interactivity	B-General_Concept
,	O
alternatives	O
such	O
as	O
I/O	B-Operating_System
automata	I-Operating_System
are	O
usually	O
preferred	O
.	O
</s>
<s>
Gandy	O
's	O
analysis	O
of	O
Babbage	O
's	O
analytical	B-Device
engine	I-Device
describes	O
the	O
following	O
five	O
operations	O
(	O
cf	O
.	O
</s>
<s>
Conditional	O
transfer	O
(	O
i.e.	O
,	O
conditional	O
"	O
goto	B-Application
"	O
)	O
.	O
</s>
<s>
B	O
.	O
Rosser	O
by	O
use	O
of	O
Church	O
's	O
lambda-calculus	B-Language
and	O
Gödel	O
's	O
recursion	O
theory	O
(	O
1934	O
)	O
.	O
</s>
<s>
While	O
Church	O
refereed	O
Turing	O
's	O
paper	O
,	O
Turing	O
had	O
time	O
to	O
study	O
Church	O
's	O
paper	O
and	O
add	O
an	O
Appendix	O
where	O
he	O
sketched	O
a	O
proof	O
that	O
Church	O
's	O
lambda-calculus	B-Language
and	O
his	O
machines	O
would	O
compute	O
the	O
same	O
functions	O
.	O
</s>
<s>
Alan	O
Turing	O
invented	O
the	O
"	O
a-machine	B-Architecture
"	O
(	O
automatic	O
machine	O
)	O
in	O
1936	O
.	O
</s>
<s>
Hodges	O
1983:129	O
)	O
It	O
was	O
Turing	O
's	O
doctoral	O
advisor	O
,	O
Alonzo	O
Church	O
,	O
who	O
later	O
coined	O
the	O
term	O
"	O
Turing	B-Architecture
machine	I-Architecture
"	O
in	O
a	O
review	O
.	O
</s>
<s>
When	O
Turing	O
returned	O
to	O
the	O
UK	O
he	O
ultimately	O
became	O
jointly	O
responsible	O
for	O
breaking	O
the	O
German	O
secret	O
codes	O
created	O
by	O
encryption	O
machines	O
called	O
"	O
The	O
Enigma	O
"	O
;	O
he	O
also	O
became	O
involved	O
in	O
the	O
design	O
of	O
the	O
ACE	B-Device
(	O
Automatic	B-Device
Computing	I-Device
Engine	I-Device
)	O
,	O
"[Turing's]	O
ACE	B-Device
proposal	O
was	O
effectively	O
self-contained	O
,	O
and	O
its	O
roots	O
lay	O
not	O
in	O
the	O
EDVAC	B-Device
[	O
the	O
USA	O
's	O
initiative ]	O
,	O
but	O
in	O
his	O
own	O
universal	O
machine	O
"	O
(	O
Hodges	O
p.318	O
)	O
.	O
</s>
<s>
"	O
Alan	O
's	O
task	O
was	O
to	O
embody	O
the	O
logical	O
design	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
in	O
a	O
network	O
of	O
relay-operated	O
switches	O
...	O
"	O
(	O
Hodges	O
p.138	O
)	O
.	O
</s>
<s>
In	O
the	O
early	O
to	O
mid-1950s	O
Hao	O
Wang	O
and	O
Marvin	O
Minsky	O
reduced	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
to	O
a	O
simpler	O
form	O
(	O
a	O
precursor	O
to	O
the	O
Post	B-Application
–	I-Application
Turing	I-Application
machine	I-Application
of	O
Martin	O
Davis	O
)	O
;	O
simultaneously	O
European	O
researchers	O
were	O
reducing	O
the	O
new-fangled	O
electronic	O
computer	O
to	O
a	O
computer-like	O
theoretical	O
object	O
equivalent	O
to	O
what	O
was	O
now	O
being	O
called	O
a	O
"	O
Turing	B-Architecture
machine	I-Architecture
"	O
.	O
</s>
<s>
In	O
the	O
late	O
1950s	O
and	O
early	O
1960s	O
,	O
the	O
coincidentally	O
parallel	O
developments	O
of	O
Melzak	O
and	O
Lambek	O
(	O
1961	O
)	O
,	O
Minsky	O
(	O
1961	O
)	O
,	O
and	O
Shepherdson	O
and	O
Sturgis	O
(	O
1961	O
)	O
carried	O
the	O
European	O
work	O
further	O
and	O
reduced	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
to	O
a	O
more	O
friendly	O
,	O
computer-like	O
abstract	O
model	O
called	O
the	O
counter	B-Application
machine	I-Application
;	O
Elgot	O
and	O
Robinson	O
(	O
1964	O
)	O
,	O
Hartmanis	O
(	O
1971	O
)	O
,	O
Cook	O
and	O
Reckhow	O
(	O
1973	O
)	O
carried	O
this	O
work	O
even	O
further	O
with	O
the	O
register	B-Application
machine	I-Application
and	O
random-access	B-Application
machine	I-Application
models	O
—	O
but	O
basically	O
all	O
are	O
just	O
multi-tape	B-Application
Turing	I-Application
machines	I-Application
with	O
an	O
arithmetic-like	O
instruction	O
set	O
.	O
</s>
<s>
Today	O
,	O
the	O
counter	O
,	O
register	O
and	O
random-access	B-Application
machines	I-Application
and	O
their	O
sire	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
continue	O
to	O
be	O
the	O
models	O
of	O
choice	O
for	O
theorists	O
investigating	O
questions	O
in	O
the	O
theory	O
of	O
computation	O
.	O
</s>
<s>
In	O
particular	O
,	O
computational	O
complexity	O
theory	O
makes	O
use	O
of	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
:	O
</s>
