<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
random-access	B-Application
machine	I-Application
(	O
RAM	B-Architecture
)	O
is	O
an	O
abstract	B-Application
machine	I-Application
in	O
the	O
general	O
class	O
of	O
register	B-Application
machines	I-Application
.	O
</s>
<s>
The	O
RAM	B-Architecture
is	O
very	O
similar	O
to	O
the	O
counter	B-Application
machine	I-Application
but	O
with	O
the	O
added	O
capability	O
of	O
'	O
indirect	O
addressing	O
 '	O
of	O
its	O
registers	O
.	O
</s>
<s>
Like	O
the	O
counter	B-Application
machine	I-Application
,	O
The	O
RAM	B-Architecture
has	O
its	O
instructions	O
in	O
the	O
finite-state	O
portion	O
of	O
the	O
machine	O
(	O
the	O
so-called	O
Harvard	B-Architecture
architecture	I-Architecture
)	O
.	O
</s>
<s>
The	O
RAM	B-Architecture
's	O
equivalent	O
of	O
the	O
universal	O
Turing	O
machinewith	O
its	O
program	B-Application
in	O
the	O
registers	O
as	O
well	O
as	O
its	O
datais	O
called	O
the	O
random-access	B-Application
stored-program	I-Application
machine	I-Application
or	O
RASP	O
.	O
</s>
<s>
It	O
is	O
an	O
example	O
of	O
the	O
so-called	O
von	B-Architecture
Neumann	I-Architecture
architecture	I-Architecture
and	O
is	O
closest	O
to	O
the	O
common	O
notion	O
of	O
a	O
computer	O
.	O
</s>
<s>
Together	O
with	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
and	O
counter-machine	B-Application
models	I-Application
,	O
the	O
RAM	B-Architecture
and	O
RASP	B-Application
models	I-Application
are	O
used	O
for	O
computational	O
complexity	O
analysis	O
.	O
</s>
<s>
Van	O
Emde	O
Boas	O
(	O
1990	O
)	O
calls	O
these	O
three	O
plus	O
the	O
pointer	B-Application
machine	I-Application
"	O
sequential	O
machine	O
"	O
models	O
,	O
to	O
distinguish	O
them	O
from	O
"	O
parallel	B-Operating_System
random-access	I-Operating_System
machine	I-Operating_System
"	O
models	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
random-access	B-Application
machine	I-Application
(	O
RAM	B-Architecture
)	O
starts	O
with	O
the	O
simplest	O
model	O
of	O
all	O
,	O
the	O
so-called	O
counter	B-Application
machine	I-Application
model	O
.	O
</s>
<s>
Two	O
additions	O
move	O
it	O
away	O
from	O
the	O
counter	B-Application
machine	I-Application
,	O
however	O
.	O
</s>
<s>
A	O
random-access	B-Application
machine	I-Application
(	O
RAM	B-Architecture
)	O
is	O
an	O
abstract	O
computational-machine	O
model	O
identical	O
to	O
a	O
multiple-register	O
counter	B-Application
machine	I-Application
with	O
the	O
addition	O
of	O
indirect	O
addressing	O
.	O
</s>
<s>
At	O
the	O
discretion	O
of	O
instruction	O
from	O
its	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
's	O
TABLE	O
,	O
the	O
machine	O
derives	O
a	O
"	O
target	O
"	O
register	O
's	O
address	O
either	O
(	O
i	O
)	O
directly	O
from	O
the	O
instruction	O
itself	O
,	O
or	O
(	O
ii	O
)	O
indirectly	O
from	O
the	O
contents	O
(	O
e.g.	O
</s>
<s>
Example	O
:	O
COPY	O
(	O
d	O
,	O
A	O
,	O
i	O
,	O
N	B-Application
)	O
means	O
directly	O
d	O
get	O
the	O
source	O
register	O
's	O
address	O
(	O
register	O
"	O
A	O
"	O
)	O
from	O
the	O
instruction	O
itself	O
but	O
indirectly	O
i	O
get	O
the	O
destination	O
address	O
from	O
pointer-register	O
N	B-Application
.	O
Suppose	O
[N]=3,	O
then	O
register	O
3	O
is	O
the	O
destination	O
and	O
the	O
instruction	O
will	O
do	O
the	O
following	O
:	O
 [ A ] 	O
→	O
3	O
.	O
</s>
<s>
Melzak	O
(	O
1961	O
)	O
provides	O
an	O
easy	O
visualization	O
of	O
a	O
counter	B-Application
machine	I-Application
:	O
its	O
"	O
registers	O
"	O
are	O
holes	O
in	O
the	O
ground	O
,	O
and	O
these	O
holes	O
hold	O
pebbles	O
.	O
</s>
<s>
Minsky	O
(	O
1961	O
)	O
and	O
Hopcroft-Ullman	O
1979	O
(	O
p	O
.	O
171	O
)	O
offer	O
the	O
visualization	O
of	O
a	O
multi-tape	O
Turing	B-Architecture
machine	I-Architecture
with	O
as	O
many	O
left-ended	O
tapes	O
as	O
"	O
registers	O
"	O
.	O
</s>
<s>
The	O
register	B-Application
machine	I-Application
has	O
,	O
for	O
a	O
memory	O
external	O
to	O
its	O
finite-state	O
machinean	O
unbounded	O
(	O
cf	O
:	O
footnote|countable	O
and	O
unbounded	O
)	O
collection	O
of	O
discrete	O
and	O
uniquely	O
labelled	O
locations	O
with	O
unbounded	O
capacity	O
,	O
called	O
"	O
registers	O
"	O
.	O
</s>
<s>
Per	O
a	O
list	O
of	O
sequential	O
instructions	O
in	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
's	O
TABLE	O
,	O
a	O
few	O
(	O
e.g.	O
</s>
<s>
Finally	O
,	O
a	O
conditional-expression	O
in	O
the	O
form	O
of	O
an	O
IF-THEN-ELSE	O
is	O
available	O
to	O
test	O
the	O
contents	O
of	O
one	O
or	O
two	O
registers	O
and	O
"	O
branch/jump	O
"	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
out	O
of	O
the	O
default	O
instruction-sequence	O
.	O
</s>
<s>
Moreover	O
,	O
from	O
base	O
sets	O
1	O
,	O
2	O
,	O
or	O
3	O
we	O
can	O
create	O
any	O
of	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
(	O
cf	O
Minsky	O
(	O
1967	O
)	O
,	O
Boolos-Burgess-Jeffrey	O
(	O
2002	O
)	O
)	O
.	O
</s>
<s>
However	O
,	O
building	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
is	O
difficult	O
because	O
the	O
instruction	O
sets	O
are	O
so	O
...	O
primitive	O
(	O
tiny	O
)	O
.	O
</s>
<s>
The	O
problem	O
appears	O
most	O
dramatically	O
when	O
one	O
tries	O
to	O
use	O
a	O
counter-machine	B-Application
model	I-Application
to	O
build	O
a	O
RASP	O
that	O
is	O
Turing	B-Algorithm
equivalent	I-Algorithm
and	O
thus	O
compute	O
any	O
partial	O
mu	O
recursive	O
function	O
:	O
</s>
<s>
Melzak	O
(	O
1961	O
)	O
added	O
indirection	O
to	O
his	O
"	O
hole-and-pebble	O
"	O
model	O
so	O
that	O
his	O
model	O
could	O
modify	O
itself	O
with	O
a	O
"	O
computed	O
goto	O
"	O
and	O
provides	O
two	O
examples	O
of	O
its	O
use	O
(	O
"	O
Decimal	O
representation	O
in	O
the	O
scale	O
of	O
d	O
"	O
and	O
"	O
Sorting	O
by	O
magnitude	O
"	O
,	O
whether	O
these	O
are	O
used	O
in	O
his	O
proof	O
that	O
the	O
model	O
is	O
Turing	B-Algorithm
equivalent	I-Algorithm
is	O
unclear	O
since	O
"	O
the	O
program	B-Application
itself	O
is	O
left	O
to	O
the	O
reader	O
as	O
an	O
exercise	O
"	O
(	O
p	O
.	O
292	O
)	O
)	O
.	O
</s>
<s>
Minsky	O
(	O
1961	O
,	O
1967	O
)	O
was	O
able	O
to	O
demonstrate	O
that	O
,	O
with	O
suitable	O
(	O
but	O
difficult-to-use	O
)	O
Gödel	O
number	O
encoding	O
,	O
the	O
register	O
model	O
did	O
not	O
need	O
indirection	O
to	O
be	O
Turing	B-Algorithm
equivalent	I-Algorithm
;	O
but	O
it	O
did	O
need	O
at	O
least	O
one	O
unbounded	O
register	O
.	O
</s>
<s>
Elgot	O
and	O
Robinson	O
(	O
1964	O
)	O
proved	O
that	O
their	O
RASP	B-Application
model	I-Application
P0it	O
has	O
no	O
indirection	O
capabilitycannot	O
compute	O
all	O
"	O
recursive	O
sequential	O
functions	O
"	O
(	O
ones	O
that	O
have	O
parameters	O
of	O
arbitrary	O
length	O
)	O
if	O
it	O
does	O
not	O
have	O
the	O
capability	O
of	O
modifying	O
its	O
own	O
instructions	O
,	O
but	O
it	O
can	O
via	O
Gödel	O
numbers	O
if	O
it	O
does	O
(	O
p	O
.	O
395-397	O
;	O
in	O
particular	O
figure	O
2	O
and	O
footnote	O
p	O
.	O
395	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
their	O
RASP	B-Application
model	I-Application
P'0	O
equipped	O
with	O
an	O
"	O
index	O
register	O
"	O
(	O
indirect	O
addressing	O
)	O
can	O
compute	O
all	O
the	O
"	O
partial	O
recursive	O
sequential	O
functions	O
"	O
(	O
the	O
mu	O
recursive	O
functions	O
)	O
(	O
p	O
.	O
397-398	O
)	O
.	O
</s>
<s>
The	O
indirect	O
instructions	O
are	O
necessary	O
in	O
order	O
for	O
a	O
fixed	O
program	B-Application
to	O
access	O
an	O
unbounded	O
number	O
of	O
registers	O
as	O
the	O
inputs	O
vary.	O
"	O
</s>
<s>
Unbounded	O
capacities	O
of	O
registers	O
versus	O
bounded	O
capacities	O
of	O
state-machine	B-Architecture
instructions	O
:	O
The	O
so-called	O
finite	B-Architecture
state	I-Architecture
part	O
of	O
the	O
machine	O
is	O
supposed	O
to	O
beby	O
the	O
normal	O
definition	O
of	O
algorithmvery	O
finite	O
both	O
in	O
the	O
number	O
of	O
"	O
states	O
"	O
(	O
instructions	O
)	O
and	O
the	O
instructions	O
 '	O
sizes	O
(	O
their	O
capacity	O
to	O
hold	O
symbols/signs	O
)	O
.	O
</s>
<s>
So	O
how	O
does	O
a	O
state	B-Architecture
machine	I-Architecture
move	O
an	O
arbitrarily	O
large	O
constant	O
directly	O
into	O
a	O
register	O
,	O
e.g.	O
</s>
<s>
If	O
huge	O
constants	O
are	O
necessary	O
they	O
must	O
either	O
start	O
out	O
in	O
the	O
registers	O
themselves	O
or	O
be	O
created	O
by	O
the	O
state	B-Architecture
machine	I-Architecture
using	O
a	O
finite	O
number	O
of	O
instructions	O
e.g.	O
</s>
<s>
The	O
problem	O
arises	O
when	O
the	O
number	O
to	O
be	O
created	O
exhausts	O
the	O
number	O
of	O
instructions	O
available	O
to	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
;	O
there	O
is	O
always	O
a	O
bigger	O
constant	O
than	O
the	O
number	O
of	O
instructions	O
available	O
to	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
.	O
</s>
<s>
Unbounded	O
numbers	O
of	O
registers	O
versus	O
bounded	O
state-machine	B-Architecture
instructions	O
:	O
This	O
is	O
more	O
severe	O
than	O
the	O
first	O
problem	O
.	O
</s>
<s>
In	O
particular	O
,	O
this	O
problem	O
arises	O
when	O
we	O
attempt	O
to	O
build	O
a	O
so-called	O
RASP	O
,	O
a	O
"	O
universal	O
machine	O
"	O
(	O
see	O
more	O
at	O
Universal	O
Turing	B-Architecture
machine	I-Architecture
)	O
that	O
uses	O
its	O
finite-state	B-Architecture
machine	I-Architecture
to	O
interpret	O
a	O
"	O
program	B-Application
of	O
instructions	O
"	O
located	O
in	O
its	O
registersi.e.	O
</s>
<s>
we	O
are	O
building	O
what	O
is	O
nowadays	O
called	O
a	O
computer	O
with	O
the	O
von	B-Architecture
Neumann	I-Architecture
architecture	I-Architecture
.	O
</s>
<s>
Observe	O
that	O
the	O
counter	B-Application
machine	I-Application
's	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
must	O
call	O
out	O
a	O
register	O
explicitly	O
(	O
directly	O
)	O
by	O
its	O
name/number	O
:	O
INC	O
(	O
65,356	O
)	O
calls	O
out	O
register	O
number	O
"	O
65,365	O
"	O
explicitly	O
.	O
</s>
<s>
If	O
the	O
number	O
of	O
registers	O
exceeds	O
the	O
capability	O
of	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
to	O
address	O
them	O
,	O
then	O
registers	O
outside	O
the	O
bounds	O
will	O
be	O
unreachable	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
can	O
only	O
reach	O
65,536	O
=	O
216	O
registers	O
then	O
how	O
can	O
it	O
reach	O
the	O
65,537	O
th	O
?	O
</s>
<s>
So	O
how	O
do	O
we	O
address	O
a	O
register	O
beyond	O
the	O
bounds	O
of	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
?	O
</s>
<s>
One	O
approach	O
would	O
be	O
to	O
modify	O
the	O
program-instructions	O
(	O
the	O
ones	O
stored	O
in	O
the	O
registers	O
)	O
so	O
that	O
they	O
contain	O
more	O
than	O
one	O
command	O
.	O
</s>
<s>
So	O
why	O
not	O
use	O
just	O
one	O
"	O
über-instruction	O
"	O
one	O
really	O
really	O
big	O
numberthat	O
contains	O
all	O
the	O
program	B-Application
instructions	O
encoded	O
into	O
it	O
!	O
</s>
<s>
This	O
is	O
how	O
Minsky	O
solves	O
the	O
problem	O
,	O
but	O
the	O
Gödel	O
numbering	O
he	O
uses	O
represents	O
a	O
great	O
inconvenience	O
to	O
the	O
model	O
,	O
and	O
the	O
result	O
is	O
nothing	O
at	O
all	O
like	O
our	O
intuitive	O
notion	O
of	O
a	O
"	O
stored	O
program	B-Application
computer	O
"	O
.	O
</s>
<s>
to	O
fetch	O
instructions	O
from	O
them	O
)	O
but	O
only	O
if	O
the	O
RASP	O
allows	O
"	O
self	O
modification	O
"	O
of	O
its	O
program	B-Application
instructions	O
,	O
and	O
has	O
encoded	O
its	O
"	O
data	O
"	O
in	O
a	O
Gödel	O
number	O
(	O
Fig	O
.	O
</s>
<s>
He	O
offers	O
us	O
a	O
bounded	O
RPT	O
that	O
together	O
with	O
CLR	O
(	O
r	O
)	O
and	O
INC	O
(	O
r	O
)	O
can	O
compute	O
any	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
,	O
and	O
he	O
offers	O
the	O
unbounded	O
RPT	O
quoted	O
above	O
that	O
as	O
playing	O
the	O
role	O
of	O
μ	O
operator	O
;	O
it	O
together	O
with	O
CLR	O
(	O
r	O
)	O
and	O
INC	O
(	O
r	O
)	O
can	O
compute	O
the	O
mu	O
recursive	O
functions	O
.	O
</s>
<s>
But	O
he	O
does	O
not	O
discuss	O
"	O
indirection	O
"	O
or	O
the	O
RAM	B-Application
model	I-Application
per	O
se	O
.	O
</s>
<s>
Hartmanis	O
 '	O
modelquite	O
similar	O
to	O
Melzak	O
's	O
(	O
1961	O
)	O
modeluses	O
two	O
and	O
three-register	O
adds	O
and	O
subtracts	O
and	O
two	O
parameter	O
copies	O
;	O
Cook	O
and	O
Reckhow	O
's	O
model	O
reduce	O
the	O
number	O
of	O
parameters	O
(	O
registers	O
called	O
out	O
in	O
the	O
program	B-Application
instructions	O
)	O
to	O
one	O
call-out	O
by	O
use	O
of	O
an	O
accumulator	O
"	O
AC	O
"	O
.	O
</s>
<s>
The	O
pointer	O
register	O
is	O
exactly	O
like	O
any	O
other	O
register	O
with	O
one	O
exception	O
:	O
under	O
the	O
circumstances	O
called	O
"	O
indirect	O
addressing	O
"	O
it	O
provides	O
its	O
contents	O
,	O
rather	O
than	O
the	O
address-operand	O
in	O
the	O
state	B-Architecture
machine	I-Architecture
's	O
TABLE	O
,	O
to	O
be	O
the	O
address	O
of	O
the	O
target	O
register	O
(	O
including	O
possibly	O
itself	O
!	O
</s>
<s>
Our	O
simpler	O
counter-machine	B-Application
model	I-Application
can	O
do	O
a	O
"	O
bounded	O
"	O
form	O
of	O
indirectionand	O
thereby	O
compute	O
the	O
sub-class	O
of	O
primitive	B-Architecture
recursive	I-Architecture
functionsby	O
using	O
a	O
primitive	B-Architecture
recursive	I-Architecture
"	O
operator	O
"	O
called	O
"	O
definition	O
by	O
cases	O
"	O
(	O
defined	O
in	O
Kleene	O
(	O
1952	O
)	O
p.229	O
and	O
Boolos-Burgess-Jeffrey	O
p.74	O
)	O
.	O
</s>
<s>
Is	O
the	O
number	O
in	O
register	O
N	B-Application
equal	O
to	O
0	O
?	O
</s>
<s>
To	O
be	O
Turing	B-Algorithm
equivalent	I-Algorithm
the	O
counter	B-Application
machine	I-Application
needs	O
to	O
either	O
use	O
the	O
unfortunate	O
single-register	O
Minsky	O
Gödel	O
number	O
method	O
,	O
or	O
be	O
augmented	O
with	O
an	O
ability	O
to	O
explore	O
the	O
ends	O
of	O
its	O
register	O
string	O
,	O
ad	O
infinitum	O
if	O
necessary	O
.	O
</s>
<s>
Once	O
we	O
make	O
this	O
change	O
the	O
model	O
is	O
no	O
longer	O
a	O
counter	B-Application
machine	I-Application
,	O
but	O
rather	O
a	O
random-access	B-Application
machine	I-Application
.	O
</s>
<s>
INC	O
is	O
specified	O
,	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
's	O
instruction	O
will	O
have	O
to	O
specify	O
where	O
the	O
address	O
of	O
the	O
register	O
of	O
interest	O
will	O
come	O
from	O
.	O
</s>
<s>
This	O
where	O
can	O
be	O
either	O
(	O
i	O
)	O
the	O
state	B-Architecture
machine	I-Architecture
's	O
instruction	O
that	O
provides	O
an	O
explicit	O
label	O
,	O
or	O
(	O
ii	O
)	O
the	O
pointer-register	O
whose	O
contents	O
is	O
the	O
address	O
of	O
interest	O
.	O
</s>
<s>
Another	O
approach	O
(	O
Schönhage	O
does	O
this	O
too	O
)	O
is	O
to	O
declare	O
a	O
specific	O
register	O
the	O
"	O
indirect	O
address	O
register	O
"	O
and	O
confine	O
indirection	O
relative	O
to	O
this	O
register	O
(	O
Schonhage	O
's	O
RAM0	O
model	O
uses	O
both	O
A	O
and	O
N	B-Application
registers	O
for	O
indirect	O
as	O
well	O
as	O
direct	O
instructions	O
)	O
.	O
</s>
<s>
Again	O
our	O
new	O
register	O
has	O
no	O
conventional	O
nameperhaps	O
"	O
N	B-Application
"	O
from	O
"	O
iNdex	O
"	O
,	O
or	O
"	O
iNdirect	O
"	O
or	O
"	O
address	O
Number	O
"	O
.	O
</s>
<s>
For	O
maximum	O
flexibility	O
,	O
as	O
we	O
have	O
done	O
for	O
the	O
accumulator	O
Awe	O
will	O
consider	O
N	B-Application
just	O
another	O
register	O
subject	O
to	O
increment	O
,	O
decrement	O
,	O
clear	O
,	O
test	O
,	O
direct	O
copy	O
,	O
etc	O
.	O
</s>
<s>
STAN	O
(	O
i/d	O
)	O
=	O
CPY	O
(	O
d	O
,	O
A	O
,	O
i/d	O
,	O
N	B-Application
)	O
.	O
</s>
<s>
(	O
2	O
)	O
Reduce	O
a	O
RAM	B-Architecture
to	O
a	O
Post-Turing	B-Application
machine	I-Application
:	O
</s>
<s>
Posing	O
as	O
minimalists	O
,	O
we	O
reduce	O
all	O
the	O
registers	O
excepting	O
the	O
accumulator	O
A	O
and	O
indirection	O
register	O
N	B-Application
e.g.	O
</s>
<s>
We	O
restrict	O
any	O
arithmetic	O
to	O
the	O
registers	O
{	O
A	O
,	O
N	B-Application
}	O
,	O
use	O
indirect	O
operations	O
to	O
pull	O
the	O
contents	O
of	O
registers	O
into	O
the	O
accumulator	O
and	O
write	O
0	O
or	O
1	O
from	O
the	O
accumulator	O
to	O
a	O
register	O
:	O
</s>
<s>
Rename	O
the	O
COPY	O
instructions	O
and	O
call	O
INC	O
(	O
N	B-Application
)	O
=	O
RIGHT	O
,	O
DEC	O
(	O
N	B-Application
)	O
=	O
LEFT	O
and	O
we	O
have	O
the	O
same	O
instructions	O
as	O
the	O
Post-Turing	B-Application
machine	I-Application
,	O
plus	O
an	O
extra	O
CLRN	O
:	O
</s>
<s>
In	O
the	O
section	O
above	O
we	O
informally	O
showed	O
that	O
a	O
RAM	B-Architecture
with	O
an	O
unbounded	O
indirection	O
capability	O
produces	O
a	O
Post	B-Application
–	I-Application
Turing	I-Application
machine	I-Application
.	O
</s>
<s>
The	O
Post	B-Application
–	I-Application
Turing	I-Application
machine	I-Application
is	O
Turing	B-Algorithm
equivalent	I-Algorithm
,	O
so	O
we	O
have	O
shown	O
that	O
the	O
RAM	B-Architecture
with	O
indirection	O
is	O
Turing	B-Algorithm
equivalent	I-Algorithm
.	O
</s>
<s>
Begin	O
by	O
designing	O
our	O
model	O
with	O
three	O
reserved	O
registers	O
"	O
E	O
"	O
,	O
"	O
P	O
"	O
,	O
and	O
"	O
N	B-Application
"	O
,	O
plus	O
an	O
unbounded	O
set	O
of	O
registers	O
1	O
,	O
2	O
,	O
...	O
,	O
n	B-Application
to	O
the	O
right	O
.	O
</s>
<s>
The	O
registers	O
1	O
,	O
2	O
,	O
...	O
,	O
n	B-Application
will	O
be	O
considered	O
"	O
the	O
squares	O
of	O
the	O
tape	O
"	O
.	O
</s>
<s>
Register	O
"	O
N	B-Application
"	O
points	O
to	O
"	O
the	O
scanned	O
square	O
"	O
that	O
"	O
the	O
head	O
"	O
is	O
currently	O
observing	O
.	O
</s>
<s>
As	O
we	O
decrement	O
or	O
increment	O
"	O
N	B-Application
"	O
the	O
(	O
apparent	O
)	O
head	O
will	O
"	O
move	O
left	O
"	O
or	O
"	O
right	O
"	O
along	O
the	O
squares	O
.	O
</s>
<s>
We	O
will	O
move	O
the	O
contents	O
of	O
"	O
E	O
"	O
=	O
0	O
or	O
"	O
P	O
"	O
=	O
1	O
to	O
the	O
"	O
scanned	O
square	O
"	O
as	O
pointed	O
to	O
by	O
N	B-Application
,	O
using	O
the	O
indirect	O
CPY	O
.	O
</s>
<s>
The	O
fact	O
that	O
our	O
tape	O
is	O
left-ended	O
presents	O
us	O
with	O
a	O
minor	O
problem	O
:	O
Whenever	O
LEFT	O
occurs	O
our	O
instructions	O
will	O
have	O
to	O
test	O
to	O
determine	O
whether	O
or	O
not	O
the	O
contents	O
of	O
"	O
N	B-Application
"	O
is	O
zero	O
;	O
if	O
so	O
we	O
should	O
leave	O
its	O
count	O
at	O
"	O
0	O
"	O
(	O
this	O
is	O
our	O
choice	O
as	O
designersfor	O
example	O
we	O
might	O
have	O
the	O
machine/model	O
"	O
trigger	O
an	O
event	O
"	O
of	O
our	O
choosing	O
)	O
.	O
</s>
<s>
The	O
following	O
table	O
both	O
defines	O
the	O
Post-Turing	O
instructions	O
in	O
terms	O
of	O
their	O
RAM	B-Architecture
equivalent	O
instructions	O
and	O
gives	O
an	O
example	O
of	O
their	O
functioning	O
.	O
</s>
<s>
Mnemonic	O
label	O
:	O
E	O
P	O
N	B-Application
r0	O
r1	O
r2	O
r3	O
r4	O
r5	O
etc	O
.	O
</s>
<s>
Action	O
on	O
registers	O
Action	O
on	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
Instruction	O
Register	O
IR	O
start	O
:	O
</s>
<s>
R	O
right	O
:	O
INC	O
(	O
N	B-Application
)	O
0	O
1	O
4	O
1	O
0	O
 [ N ] 	O
+1	O
→	O
N	B-Application
 [ IR ] 	O
+1	O
→	O
IR	O
P	O
print	O
:	O
CPY	O
(	O
d	O
,	O
P	O
,	O
i	O
,	O
N	B-Application
)	O
0	O
1	O
4	O
1	O
1	O
[P]=1	O
→	O
[N]=r4	O
 [ IR ] 	O
+1	O
→	O
IR	O
E	O
erase	O
:	O
CPY	O
(	O
d	O
,	O
E	O
,	O
i	O
,	O
N	B-Application
)	O
0	O
1	O
4	O
1	O
0	O
[E]=0	O
→	O
[N]=r4	O
 [ IR ] 	O
+1	O
→	O
IR	O
L	O
left	O
:	O
JZ	O
(	O
i	O
,	O
N	B-Application
,	O
end	O
)	O
0	O
1	O
4	O
1	O
0	O
IF	O
N	B-Application
=	O
r4 ]	O
=	O
0	O
THEN	O
"	O
end	O
"	O
→	O
IR	O
else	O
[IR]+1	O
→	O
IR	O
DEC	O
(	O
N	B-Application
)	O
0	O
1	O
3	O
1	O
0	O
 [ N ] 	O
-1	O
→	O
N	B-Application
J0	O
(	O
halt	O
)	O
jump_if_blank	O
:	O
JZ	O
(	O
i	O
,	O
N	B-Application
,	O
end	O
)	O
0	O
1	O
3	O
1	O
0	O
IF	O
N	B-Application
=	O
r3 ]	O
=	O
0	O
THEN	O
"	O
end	O
"	O
→	O
IR	O
else	O
[IR]+1	O
→	O
IR	O
J1	O
(	O
halt	O
)	O
jump_if_mark	O
:	O
JZ	O
(	O
i	O
,	O
N	B-Application
,	O
halt	O
)	O
0	O
1	O
3	O
1	O
0	O
N	B-Application
=	O
r3 ]	O
→	O
A	O
IF	O
N	B-Application
=	O
r3 ]	O
=	O
0	O
THEN	O
"	O
end	O
"	O
→	O
IR	O
else	O
[IR]+1	O
→	O
IR	O
end	O
.	O
</s>
<s>
Throughout	O
this	O
demonstration	O
we	O
have	O
to	O
keep	O
in	O
mind	O
that	O
the	O
instructions	O
in	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
's	O
TABLE	O
is	O
bounded	O
,	O
i.e.	O
</s>
<s>
However	O
,	O
note	O
that	O
because	O
this	O
indirect	O
CPY	O
is	O
"	O
bounded	O
"	O
by	O
the	O
size/extent	O
of	O
the	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
,	O
a	O
RASP	O
using	O
this	O
indirect	O
CPY	O
can	O
only	O
calculate	O
the	O
primitive	B-Architecture
recursive	I-Architecture
functions	I-Architecture
,	O
not	O
the	O
full	O
suite	O
of	O
mu	O
recursive	O
functions	O
.	O
</s>
<s>
case_2	O
through	O
case	O
n	B-Application
:	O
IF	O
.	O
</s>
<s>
Case_n	O
(	O
the	O
induction	O
step	O
)	O
looks	O
like	O
this	O
;	O
remember	O
,	O
each	O
instance	O
of	O
"	O
n	B-Application
"	O
,	O
"	O
n+1	O
"	O
,	O
...	O
,	O
"	O
last	O
"	O
must	O
be	O
an	O
explicit	O
natural	O
number	O
:	O
</s>
<s>
But	O
it	O
can'tits	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
's	O
"	O
state	O
register	O
"	O
has	O
reached	O
its	O
maximum	O
count	O
(	O
e.g.	O
</s>
<s>
Schönhage	O
(	O
1980	O
)	O
describes	O
a	O
very	O
primitive	O
,	O
atomized	O
model	O
chosen	O
for	O
his	O
proof	O
of	O
the	O
equivalence	O
of	O
his	O
SMM	O
pointer	B-Application
machine	I-Application
model	O
:	O
</s>
<s>
RAM1	O
model	O
:	O
Schönhage	O
demonstrates	O
how	O
his	O
construction	O
can	O
be	O
used	O
to	O
form	O
the	O
more	O
common	O
,	O
usable	O
form	O
of	O
"	O
successor	O
"	O
-like	O
RAM	B-Architecture
(	O
using	O
this	O
article	O
's	O
mnemonics	O
)	O
:	O
</s>
<s>
Schönhage	O
designated	O
the	O
accumulator	O
with	O
"	O
z	O
"	O
,	O
"	O
N	B-Application
"	O
with	O
"	O
n	B-Application
"	O
,	O
etc	O
.	O
</s>
<s>
Indirection	O
comes	O
(	O
i	O
)	O
from	O
CPYAN	O
(	O
copy/transfer	O
contents	O
A	O
to	O
N	B-Application
)	O
working	O
with	O
store_A_via_N	O
STAN	O
,	O
and	O
from	O
(	O
ii	O
)	O
the	O
peculiar	O
indirection	O
instruction	O
LDAA	O
(	O
 [  [ A ]  ] 	O
→	O
 [ A ] 	O
)	O
.	O
</s>
