<s>
Algorithm	B-Application
characterizations	I-Application
are	O
attempts	O
to	O
formalize	O
the	O
word	O
algorithm	O
.	O
</s>
<s>
The	O
most	O
common	O
number-manipulation	O
schemes	O
—	O
both	O
in	O
formal	O
mathematics	O
and	O
in	O
routine	O
life	O
—	O
are	O
:	O
(	O
1	O
)	O
the	O
recursive	O
functions	O
calculated	O
by	O
a	O
person	O
with	O
paper	O
and	O
pencil	O
,	O
and	O
(	O
2	O
)	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
or	O
its	O
Turing	B-Language
equivalents	O
—	O
the	O
primitive	O
register-machine	B-Application
or	O
"	O
counter-machine	O
"	O
model	O
,	O
the	O
random-access	B-Application
machine	I-Application
model	O
(	O
RAM	O
)	O
,	O
the	O
random-access	B-Application
stored-program	I-Application
machine	I-Application
model	O
(	O
RASP	O
)	O
and	O
its	O
functional	O
equivalent	O
"	O
the	O
computer	O
"	O
.	O
</s>
<s>
But	O
this	O
equivalence	O
together	O
with	O
the	O
thesis	O
(	O
unproven	O
assertion	O
)	O
that	O
this	O
includes	O
every	O
calculation/computation	O
indicates	O
why	O
so	O
much	O
emphasis	O
has	O
been	O
placed	O
upon	O
the	O
use	O
of	O
Turing-equivalent	O
machines	O
in	O
the	O
definition	O
of	O
specific	O
algorithms	O
,	O
and	O
why	O
the	O
definition	O
of	O
"	O
algorithm	O
"	O
itself	O
often	O
refers	O
back	O
to	O
"	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
"	O
.	O
</s>
<s>
It	O
is	O
used	O
for	O
classifying	O
of	O
programming	O
languages	O
and	O
abstract	B-Application
machines	I-Application
.	O
</s>
<s>
Examples	O
:	O
a	O
"	O
general	O
purpose	O
"	O
macro	O
language	O
,	O
like	O
M4	B-Device
is	O
unrestricted	O
(	O
Turing	B-Algorithm
complete	I-Algorithm
)	O
,	O
but	O
the	O
C	B-Language
preprocessor	I-Language
macro	I-Language
language	I-Language
is	O
not	O
,	O
so	O
any	O
algorithm	O
expressed	O
in	O
C	B-Language
preprocessor	I-Language
is	O
a	O
"	O
simple	O
algorithm	O
"	O
.	O
</s>
<s>
being	O
able	O
to	O
handle	O
many	O
inputs	O
)	O
,	O
or	O
determinism	B-General_Concept
.	O
</s>
<s>
This	O
section	O
is	O
longer	O
and	O
more	O
detailed	O
than	O
the	O
others	O
because	O
of	O
its	O
importance	O
to	O
the	O
topic	O
:	O
Kleene	O
was	O
the	O
first	O
to	O
propose	O
that	O
all	O
calculations/computations	O
—	O
of	O
every	O
sort	O
,	O
the	O
totality	O
of	O
—	O
can	O
equivalently	O
be	O
(	O
i	O
)	O
calculated	O
by	O
use	O
of	O
five	O
"	O
primitive	B-Architecture
recursive	I-Architecture
operators	O
"	O
plus	O
one	O
special	O
operator	O
called	O
the	O
mu-operator	O
,	O
or	O
be	O
(	O
ii	O
)	O
computed	O
by	O
the	O
actions	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
or	O
an	O
equivalent	O
model	O
.	O
</s>
<s>
Calculation	O
means	O
done	O
by	O
hand	O
,	O
computation	O
means	O
done	O
by	O
Turing	B-Architecture
machine	I-Architecture
(	O
or	O
equivalent	O
)	O
.	O
</s>
<s>
Think	O
of	O
the	O
"	O
function-box	O
"	O
as	O
a	O
little	O
man	O
either	O
calculating	O
by	O
hand	O
using	O
"	O
general	O
recursion	O
"	O
or	O
computing	O
by	O
Turing	B-Architecture
machine	I-Architecture
(	O
or	O
an	O
equivalent	O
machine	O
)	O
.	O
</s>
<s>
"	O
General	O
recursive	O
"	O
was	O
Kleene	O
's	O
way	O
of	O
writing	O
what	O
today	O
is	O
called	O
just	O
"	O
recursion	O
"	O
;	O
however	O
,	O
"	O
primitive	B-Architecture
recursion	I-Architecture
"	O
—	O
calculation	O
by	O
use	O
of	O
the	O
five	O
recursive	O
operators	O
—	O
is	O
a	O
lesser	O
form	O
of	O
recursion	O
that	O
lacks	O
access	O
to	O
the	O
sixth	O
,	O
additional	O
,	O
mu-operator	O
that	O
is	O
needed	O
only	O
in	O
rare	O
instances	O
.	O
</s>
<s>
This	O
is	O
not	O
as	O
daunting	O
as	O
it	O
may	O
sound	O
–	O
"	O
general	O
"	O
recursion	O
is	O
just	O
a	O
way	O
of	O
making	O
our	O
everyday	O
arithmetic	O
operations	O
from	O
the	O
five	O
"	O
operators	O
"	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
together	O
with	O
the	O
additional	O
mu-operator	O
as	O
needed	O
.	O
</s>
<s>
Indeed	O
,	O
Kleene	O
gives	O
13	O
examples	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
and	O
Boolos	O
–	O
Burgess	O
–	O
Jeffrey	O
add	O
some	O
more	O
,	O
most	O
of	O
which	O
will	O
be	O
familiar	O
to	O
the	O
reader	O
—	O
e.g.	O
</s>
<s>
;	O
for	O
a	O
list	O
see	O
Some	B-Architecture
common	I-Architecture
primitive	I-Architecture
recursive	I-Architecture
functions	I-Architecture
.	O
</s>
<s>
"	O
in	O
...	O
a	O
certain	O
sense	O
,	O
the	O
length	O
of	O
the	O
computation	O
algorithm	O
of	O
a	O
recursive	O
function	O
which	O
is	O
not	O
also	O
primitive	B-Architecture
recursive	I-Architecture
grows	O
faster	O
with	O
the	O
arguments	O
than	O
the	O
value	O
of	O
any	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
"	O
(	O
Kleene	O
(	O
1935	O
)	O
reprinted	O
p	O
.	O
246	O
in	O
The	O
Undecidable	O
,	O
plus	O
footnote	O
13	O
with	O
regards	O
to	O
the	O
need	O
for	O
an	O
additional	O
operator	O
,	O
boldface	O
added	O
)	O
.	O
</s>
<s>
As	O
indicated	O
above	O
by	O
Kleene	O
's	O
list	O
of	O
common	O
calculations	O
,	O
a	O
person	O
goes	O
about	O
their	O
life	O
happily	O
computing	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
without	O
fear	O
of	O
encountering	O
the	O
monster	O
numbers	O
created	O
by	O
Ackermann	O
's	O
function	O
(	O
e.g.	O
</s>
<s>
Turing	B-Language
's	O
Thesis	O
hypothesizes	O
the	O
computability	O
of	O
"	O
all	O
computable	O
functions	O
"	O
by	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
model	O
and	O
its	O
equivalents	O
.	O
</s>
<s>
Thus	O
we	O
have	O
arrived	O
at	O
Turing	B-Language
's	O
Thesis	O
:	O
</s>
<s>
For	O
example	O
,	O
Davis	O
(	O
1958	O
)	O
gives	O
Turing	B-Language
tables	O
for	O
the	O
Constant	O
,	O
Successor	O
and	O
Identity	O
functions	O
,	O
three	O
of	O
the	O
five	O
operators	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
:	O
</s>
<s>
Computable	O
by	O
Turing	B-Architecture
machine	I-Architecture
:	O
</s>
<s>
Boolos	O
–	O
Burgess	O
–	O
Jeffrey	O
(	O
2002	O
)	O
give	O
the	O
following	O
as	O
prose	O
descriptions	O
of	O
Turing	B-Architecture
machines	I-Architecture
for	O
:	O
</s>
<s>
With	O
regards	O
to	O
the	O
counter	B-Application
machine	I-Application
,	O
an	O
abstract	B-Application
machine	I-Application
model	O
equivalent	O
to	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
:	O
</s>
<s>
Demonstrations	O
of	O
computability	O
by	O
abacus	B-Application
machine	I-Application
(	O
Boolos	O
–	O
Burgess	O
–	O
Jeffrey	O
(	O
2002	O
)	O
)	O
and	O
by	O
counter	B-Application
machine	I-Application
(	O
Minsky	O
1967	O
)	O
:	O
</s>
<s>
With	O
his	O
Theorem	O
XXX	O
Kleene	O
proves	O
the	O
equivalence	O
of	O
the	O
two	O
"	O
Theses	O
"	O
—	O
the	O
Church	O
Thesis	O
and	O
the	O
Turing	B-Language
Thesis	O
.	O
</s>
<s>
Thus	O
by	O
Kleene	O
's	O
Theorem	O
XXX	O
:	O
either	O
method	O
of	O
making	O
numbers	O
from	O
input-numbers	O
—	O
recursive	O
functions	O
calculated	O
by	O
hand	O
or	O
computated	O
by	O
Turing-machine	O
or	O
equivalent	O
—	O
results	O
in	O
an	O
"	O
effectively	O
calculable/computable	O
function	O
"	O
.	O
</s>
<s>
If	O
we	O
accept	O
the	O
hypothesis	O
that	O
every	O
calculation/computation	O
can	O
be	O
done	O
by	O
either	O
method	O
equivalently	O
we	O
have	O
accepted	O
both	O
Kleene	O
's	O
Theorem	O
XXX	O
(	O
the	O
equivalence	O
)	O
and	O
the	O
Church	O
–	O
Turing	B-Language
Thesis	O
(	O
the	O
hypothesis	O
of	O
"	O
every	O
"	O
)	O
.	O
</s>
<s>
The	O
notion	O
of	O
separating	O
out	O
Church	O
's	O
and	O
Turing	B-Language
's	O
theses	O
from	O
the	O
"	O
Church	O
–	O
Turing	B-Language
thesis	O
"	O
appears	O
not	O
only	O
in	O
Kleene	O
(	O
1952	O
)	O
but	O
in	O
Blass-Gurevich	O
(	O
2003	O
)	O
as	O
well	O
.	O
</s>
<s>
In	O
fact	O
the	O
notion	O
of	O
algorithm	O
is	O
richer	O
these	O
days	O
than	O
it	O
was	O
in	O
Turing	B-Language
's	O
days	O
.	O
</s>
<s>
(	O
Turing	B-Language
(	O
1936	O
)	O
p	O
.	O
136	O
in	O
Davis	O
ed	O
.	O
</s>
<s>
In	O
Turing	B-Language
's	O
analysis	O
the	O
requirement	O
that	O
the	O
action	O
depend	O
only	O
on	O
a	O
bounded	O
portion	O
of	O
the	O
record	O
was	O
based	O
on	O
a	O
human	O
limitiation	O
.	O
</s>
<s>
1936	O
:	O
A	O
rather	O
famous	O
quote	O
from	O
Kurt	O
Gödel	B-Language
appears	O
in	O
a	O
"	O
Remark	O
added	O
in	O
proof	O
[	O
of	O
the	O
original	O
German	O
publication ]	O
in	O
his	O
paper	O
"	O
On	O
the	O
Length	O
of	O
Proofs	O
"	O
translated	O
by	O
Martin	O
Davis	O
appearing	O
on	O
pp.82	O
–	O
83	O
of	O
The	O
Undecidable	O
.	O
</s>
<s>
1963	O
:	O
In	O
a	O
"	O
Note	O
"	O
dated	O
28	O
August	O
1963	O
added	O
to	O
his	O
famous	O
paper	O
On	O
Formally	O
Undecidable	O
Propositions	O
(	O
1931	O
)	O
Gödel	B-Language
states	O
(	O
in	O
a	O
footnote	O
)	O
his	O
belief	O
that	O
"	O
formal	O
systems	O
"	O
have	O
"	O
the	O
characteristic	O
property	O
that	O
reasoning	O
in	O
them	O
,	O
in	O
principle	O
,	O
can	O
be	O
completely	O
replaced	O
by	O
mechanical	O
devices	O
"	O
(	O
p.616	O
in	O
van	O
Heijenoort	O
)	O
.	O
</s>
<s>
due	O
to	O
"	O
A	O
.	O
M	O
.	O
Turing	B-Language
's	O
work	O
a	O
precise	O
and	O
unquestionably	O
adequate	O
definition	O
of	O
the	O
general	O
notion	O
of	O
formal	O
system	O
can	O
now	O
be	O
given	O
 [ and ] 	O
a	O
completely	O
general	O
version	O
of	O
Theorems	O
VI	O
and	O
XI	O
is	O
now	O
possible.	O
"	O
</s>
<s>
1964	O
:	O
In	O
a	O
Postscriptum	O
,	O
dated	O
1964	O
,	O
to	O
a	O
paper	O
presented	O
to	O
the	O
Institute	O
for	O
Advanced	O
Study	O
in	O
spring	O
1934	O
,	O
Gödel	B-Language
amplified	O
his	O
conviction	O
that	O
"	O
formal	O
systems	O
"	O
are	O
those	O
that	O
can	O
be	O
mechanized	O
:	O
</s>
<s>
"	O
In	O
consequence	O
of	O
later	O
advances	O
,	O
in	O
particular	O
of	O
the	O
fact	O
that	O
,	O
due	O
to	O
A	O
.	O
M	O
.	O
Turing	B-Language
's	O
work	O
,	O
a	O
precise	O
and	O
unquestionably	O
adequate	O
definition	O
of	O
the	O
general	O
concept	O
of	O
formal	O
system	O
can	O
now	O
be	O
given	O
.	O
</s>
<s>
Turing	B-Language
's	O
work	O
gives	O
an	O
analysis	O
of	O
the	O
concept	O
of	O
"	O
mechanical	O
procedure	O
"	O
(	O
alias	O
"	O
algorithm	O
"	O
or	O
"	O
computational	O
procedure	O
"	O
or	O
"	O
finite	O
combinatorial	O
procedure	O
"	O
)	O
.	O
</s>
<s>
This	O
concept	O
is	O
shown	O
to	O
be	O
equivalent	O
with	O
that	O
of	O
a	O
"	O
Turing	B-Architecture
machine	I-Architecture
"	O
.	O
</s>
<s>
The	O
*	O
indicates	O
a	O
footnote	O
in	O
which	O
Gödel	B-Language
cites	O
the	O
papers	O
by	O
Alan	O
Turing	B-Language
(	O
1937	O
)	O
and	O
Emil	O
Post	O
(	O
1936	O
)	O
and	O
then	O
goes	O
on	O
to	O
make	O
the	O
following	O
intriguing	O
statement	O
:	O
</s>
<s>
Church	O
's	O
definitions	O
encompass	O
so-called	O
"	O
recursion	O
"	O
and	O
the	O
"	O
lambda	B-Language
calculus	I-Language
"	O
(	O
i.e.	O
</s>
<s>
His	O
footnote	O
18	O
says	O
that	O
he	O
discussed	O
the	O
relationship	O
of	O
"	O
effective	O
calculatibility	O
"	O
and	O
"	O
recursiveness	O
"	O
with	O
Gödel	B-Language
but	O
that	O
he	O
independently	O
questioned	O
"	O
effectively	O
calculability	O
"	O
and	O
"	O
λ-definability	O
"	O
:	O
</s>
<s>
It	O
would	O
appear	O
from	O
this	O
,	O
and	O
the	O
following	O
,	O
that	O
far	O
as	O
Gödel	B-Language
was	O
concerned	O
,	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
was	O
sufficient	O
and	O
the	O
lambda	B-Language
calculus	I-Language
was	O
"	O
much	O
less	O
suitable.	O
"	O
</s>
<s>
He	O
immediately	O
introduces	O
"	O
Turing	B-Language
's	O
Analysis	O
of	O
Computation	O
Process	O
"	O
(	O
his	O
chapter	O
5.2	O
)	O
.	O
</s>
<s>
"	O
Any	O
process	O
which	O
could	O
naturally	O
be	O
called	O
an	O
effective	O
procedure	O
can	O
be	O
realized	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
"	O
(	O
p	O
.	O
108	O
.	O
</s>
<s>
he	O
observes	O
that	O
"	O
equivalence	O
of	O
many	O
intuitive	O
formulations	O
"	O
of	O
Turing	B-Language
,	O
Church	O
,	O
Kleene	O
,	O
Post	O
,	O
and	O
Smullyan	O
"	O
leads	O
...	O
us	O
to	O
suppose	O
that	O
there	O
is	O
really	O
here	O
an	O
'	O
objective	O
 '	O
or	O
'	O
absolute	O
 '	O
notion	O
.	O
</s>
<s>
"	O
In	O
this	O
sense	O
,	O
the	O
notion	O
of	O
effectively	O
computable	O
function	O
is	O
one	O
of	O
the	O
few	O
'	O
absolute	O
 '	O
concepts	O
produced	O
by	O
modern	O
work	O
in	O
the	O
foundations	O
of	O
mathematics	O
 '	O
"	O
(	O
Minsky	O
p	O
.	O
111	O
quoting	O
Rogers	O
,	O
Hartley	O
Jr	O
(	O
1959	O
)	O
The	O
present	O
theory	O
of	O
Turing	B-Architecture
machine	I-Architecture
computability	O
,	O
J	O
.	O
SIAM	O
7	O
,	O
114-130	O
.	O
)	O
</s>
<s>
9	O
A	O
fixed	O
finite	O
bound	O
on	O
the	O
capacity	O
or	O
ability	O
of	O
the	O
computing	O
agent	O
(	O
Rogers	O
illustrates	O
with	O
example	O
simple	O
mechanisms	O
similar	O
to	O
a	O
Post	B-Application
–	I-Application
Turing	I-Application
machine	I-Application
or	O
a	O
counter	B-Application
machine	I-Application
)	O
,	O
</s>
<s>
He	O
makes	O
an	O
effort	O
in	O
this	O
direction	O
in	O
his	O
first	O
volume	O
where	O
he	O
defines	O
in	O
detail	O
what	O
he	O
calls	O
the	O
"	O
machine	B-Language
language	I-Language
"	O
for	O
his	O
"	O
mythical	O
MIX	O
...	O
the	O
world	O
's	O
first	O
polyunsaturated	O
computer	O
"	O
(	O
pp.120ff	O
)	O
.	O
</s>
<s>
He	O
also	O
uses	O
tree	B-Algorithm
diagrams	I-Algorithm
,	O
flow	O
diagrams	O
and	O
state	B-Application
diagrams	I-Application
.	O
</s>
<s>
After	O
providing	O
us	O
with	O
his	O
definition	O
,	O
Stone	O
introduces	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
model	O
and	O
states	O
that	O
the	O
set	O
of	O
five-tuples	O
that	O
are	O
the	O
machine	O
’s	O
instructions	O
are	O
“	O
an	O
algorithm	O
...	O
known	O
as	O
a	O
Turing	B-Architecture
machine	I-Architecture
program	O
”	O
(	O
p.9	O
)	O
.	O
</s>
<s>
Immediately	O
thereafter	O
he	O
goes	O
on	O
say	O
that	O
a	O
“	O
computation	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
is	O
described	O
by	O
stating	O
:	O
</s>
<s>
that	O
he	O
believes	O
that	O
"	O
an	O
algorithm	O
"	O
is	O
actually	O
"	O
a	O
Turing	B-Architecture
machine	I-Architecture
"	O
or	O
"	O
a	O
pointer	B-Application
machine	I-Application
"	O
doing	O
a	O
computation	O
.	O
</s>
<s>
every	O
algorithm	O
can	O
be	O
simulated	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
"	O
It	O
is	O
often	O
thought	O
that	O
the	O
problem	O
of	O
formalizing	O
the	O
notion	O
of	O
sequential	O
algorithm	O
was	O
solved	O
by	O
Church	O
 [ 1936 ] 	O
and	O
Turing	B-Language
 [ 1936 ] 	O
.	O
</s>
<s>
For	O
example	O
,	O
according	O
to	O
Savage	O
[1987],	O
an	O
algorithm	O
is	O
a	O
computational	O
process	O
defined	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
Church	O
and	O
Turing	B-Language
did	O
not	O
solve	O
the	O
problem	O
of	O
formalizing	O
the	O
notion	O
of	O
sequential	O
algorithm	O
.	O
</s>
<s>
First	O
,	O
there	O
are	O
some	O
things	O
that	O
are	O
obviously	O
algorithms	O
by	O
anyone	O
's	O
definition	O
--	O
Turing	B-Architecture
machines	I-Architecture
,	O
sequential-time	O
ASMs	O
[	O
Abstract	O
State	O
Machines ]	O
,	O
and	O
the	O
like	O
.	O
</s>
<s>
Blass	O
and	O
Gurevich	O
describe	O
their	O
work	O
as	O
evolved	O
from	O
consideration	O
of	O
Turing	B-Architecture
machines	I-Architecture
and	O
pointer	B-Application
machines	I-Application
,	O
specifically	O
Kolmogorov-Uspensky	O
machines	O
(	O
KU	O
machines	O
)	O
,	O
Schönhage	O
Storage	O
Modification	O
Machines	O
(	O
SMM	O
)	O
,	O
and	O
linking	O
automata	O
as	O
defined	O
by	O
Knuth	O
.	O
</s>
<s>
"	O
Turing	B-Language
...	O
'	O
s	O
informal	O
argument	O
in	O
favor	O
of	O
his	O
thesis	O
justifies	O
a	O
stronger	O
thesis	O
:	O
every	O
algorithm	O
can	O
be	O
simulated	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
....	O
In	O
practice	O
,	O
it	O
would	O
be	O
ridiculous	O
...	O
[c]an	O
one	O
generalize	O
Turing	B-Architecture
machines	I-Architecture
so	O
that	O
any	O
algorithm	O
,	O
never	O
mind	O
how	O
abstract	O
,	O
can	O
be	O
modeled	O
by	O
a	O
generalized	O
machine	O
?...	O
But	O
suppose	O
such	O
generalized	O
Turing	B-Architecture
machines	I-Architecture
exist	O
.	O
</s>
<s>
The	O
above	O
phrase	O
computation	O
as	O
an	O
evolution	O
of	O
the	O
state	O
differs	O
markedly	O
from	O
the	O
definition	O
of	O
Knuth	O
and	O
Stone	O
—	O
the	O
"	O
algorithm	O
"	O
as	O
a	O
Turing	B-Architecture
machine	I-Architecture
program	O
.	O
</s>
<s>
Rather	O
,	O
it	O
corresponds	O
to	O
what	O
Turing	B-Language
called	O
the	O
complete	O
configuration	O
(	O
cf	O
Turing	B-Language
's	O
definition	O
in	O
Undecidable	O
,	O
p.118	O
)	O
--	O
and	O
includes	O
both	O
the	O
current	O
instruction	O
(	O
state	O
)	O
and	O
the	O
status	O
of	O
the	O
tape	O
.	O
</s>
<s>
John	O
Searle	O
,	O
the	O
creator	O
of	O
the	O
Chinese	O
room	O
thought	O
experiment	O
,	O
claims	O
that	O
"	O
syntax	B-Application
[	O
that	O
is	O
,	O
logical	O
structure ]	O
is	O
by	O
itself	O
not	O
sufficient	O
for	O
semantic	O
content	O
[	O
that	O
is	O
,	O
meaning ]	O
"	O
.	O
</s>
<s>
(	O
ii	O
)	O
At	O
the	O
outset	O
of	O
their	O
example	O
they	O
specify	O
the	O
machine	O
to	O
be	O
used	O
in	O
the	O
computation	O
as	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
They	O
have	O
previously	O
specified	O
(	O
p.26	O
)	O
that	O
the	O
Turing-machine	O
will	O
be	O
of	O
the	O
4-tuple	O
,	O
rather	O
than	O
5-tuple	O
,	O
variety	O
.	O
</s>
<s>
For	O
more	O
on	O
this	O
convention	O
see	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
For	O
more	O
on	O
this	O
convention	O
see	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
"	O
We	O
have	O
not	O
given	O
an	O
official	O
definition	O
of	O
what	O
it	O
is	O
for	O
a	O
numerical	O
function	O
to	O
be	O
computable	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
,	O
specifying	O
how	O
inputs	O
or	O
arguments	O
are	O
to	O
be	O
represented	O
on	O
the	O
machine	O
,	O
and	O
how	O
outputs	O
or	O
values	O
represented	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
example	O
Turing	B-Architecture
machine	I-Architecture
computation	O
required	O
only	O
the	O
4	O
columns	O
as	O
shown	O
in	O
the	O
table	O
below	O
(	O
but	O
note	O
:	O
these	O
were	O
presented	O
to	O
the	O
machine	O
in	O
rows	O
)	O
:	O
</s>
<s>
Does	O
Sipser	O
mean	O
that	O
"	O
algorithm	O
"	O
is	O
just	O
"	O
instructions	O
"	O
for	O
a	O
Turing	B-Architecture
machine	I-Architecture
,	O
or	O
is	O
the	O
combination	O
of	O
"	O
instructions	O
+	O
a	O
(	O
specific	O
variety	O
of	O
)	O
Turing	B-Architecture
machine	I-Architecture
"	O
?	O
</s>
<s>
For	O
example	O
,	O
he	O
defines	O
the	O
two	O
standard	O
variants	O
(	O
multi-tape	O
and	O
non-deterministic	O
)	O
of	O
his	O
particular	O
variant	O
(	O
not	O
the	O
same	O
as	O
Turing	B-Language
's	O
original	O
)	O
and	O
goes	O
on	O
,	O
in	O
his	O
Problems	O
(	O
pages	O
160-161	O
)	O
,	O
to	O
describe	O
four	O
more	O
variants	O
(	O
write-once	O
,	O
doubly	O
infinite	O
tape	O
(	O
i.e.	O
</s>
<s>
Van	O
Emde	O
Boas	O
comments	O
on	O
a	O
similar	O
problem	O
with	O
respect	O
to	O
the	O
random-access	B-Application
machine	I-Application
(	O
RAM	O
)	O
abstract	O
model	O
of	O
computation	O
sometimes	O
used	O
in	O
place	O
of	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
when	O
doing	O
"	O
analysis	O
of	O
algorithms	O
"	O
:	O
</s>
<s>
[T]here	O
hardly	O
exists	O
such	O
as	O
a	O
thing	O
as	O
an	O
"	O
innocent	O
"	O
extension	O
of	O
the	O
standard	O
RAM	B-Application
model	I-Application
in	O
the	O
uniform	O
time	O
measures	O
;	O
either	O
one	O
only	O
has	O
additive	O
arithmetic	O
or	O
one	O
might	O
as	O
well	O
include	O
all	O
reasonable	O
multiplicative	O
and/or	O
bitwise	O
Boolean	O
instructions	O
on	O
small	O
operands.	O
"	O
</s>
<s>
He	O
offers	O
us	O
three	O
levels	O
of	O
description	O
of	O
Turing	B-Architecture
machine	I-Architecture
algorithms	O
(	O
p.157	O
)	O
:	O
</s>
<s>
Implementation	O
description	O
:	O
"	O
in	O
which	O
we	O
use	O
...	O
prose	O
to	O
describe	O
the	O
way	O
that	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
moves	O
its	O
head	O
and	O
the	O
way	O
that	O
it	O
stores	O
data	O
on	O
its	O
tape	O
.	O
</s>
<s>
Formal	O
description	O
:	O
"	O
...	O
the	O
lowest	O
,	O
most	O
detailed	O
,	O
level	O
of	O
description	O
...	O
that	O
spells	O
out	O
in	O
full	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
's	O
states	O
,	O
transition	O
function	O
,	O
and	O
so	O
on.	O
"	O
</s>
