<s>
Although	O
some	O
authors	O
use	O
the	O
name	O
"	O
register	B-Application
machine	I-Application
"	O
synonymously	O
with	O
"	O
counter	B-Application
machine	I-Application
"	O
,	O
this	O
article	O
will	O
give	O
details	O
and	O
examples	O
of	O
only	O
of	O
the	O
most	O
primitive	O
speciesthe	O
"	O
counter	B-Application
machine	I-Application
"	O
of	O
the	O
genus	O
"	O
register	O
machine.	O
"	O
</s>
<s>
Within	O
the	O
species	O
"	O
counter	B-Application
machine	I-Application
"	O
there	O
are	O
a	O
number	O
of	O
varieties	O
:	O
the	O
models	O
of	O
Hermes	O
(	O
1954	O
)	O
,	O
Kaphengst	O
(	O
1957	O
)	O
,	O
Ershov	O
(	O
1958	O
)	O
,	O
Péter	O
(	O
1958	O
)	O
,	O
Minsky	O
(	O
1961	O
)	O
and	O
Minsky	O
(	O
1967	O
)	O
,	O
Melzak	O
(	O
1961	O
)	O
,	O
Lambek	O
(	O
1961	O
)	O
,	O
Shepherdson	O
and	O
Sturgis	O
(	O
1963	O
)	O
,	O
and	O
Schönhage	O
(	O
1980	O
)	O
.	O
</s>
<s>
Shepherdson	O
and	O
Sturgis	O
observe	O
that	O
"	O
the	O
proof	O
of	O
this	O
universality	O
[	O
of	O
digital	O
computers	O
to	O
Turing	O
machines ]	O
...	O
seems	O
to	O
have	O
been	O
first	O
written	O
down	O
by	O
Hermes	O
,	O
who	O
showed	O
in	O
[	O
7--their	O
reference	O
number ]	O
how	O
an	O
idealized	O
computer	O
could	O
be	O
programmed	O
to	O
duplicate	O
the	O
behavior	O
of	O
any	O
Turing	B-Architecture
machine	I-Architecture
"	O
(	O
Shepherdson	O
and	O
Sturgis	O
,	O
p.219	O
)	O
.	O
</s>
<s>
Thus	O
this	O
model	O
is	O
actually	O
a	O
random-access	B-Application
machine	I-Application
.	O
</s>
<s>
"	O
from	O
the	O
point	O
of	O
view	O
of	O
proving	O
as	O
quickly	O
as	O
possible	O
the	O
computability	O
of	O
all	O
partial	O
recursive	O
functions	O
Péter	O
's	O
is	O
perhaps	O
the	O
best	O
;	O
for	O
proving	O
their	O
computability	O
by	O
Turing	B-Architecture
machines	I-Architecture
a	O
further	O
analysis	O
of	O
the	O
copying	O
operation	O
is	O
necessary	O
along	O
the	O
lines	O
we	O
have	O
taken	O
above.	O
"	O
</s>
<s>
In	O
his	O
inquiry	O
into	O
problems	O
of	O
Emil	O
Post	O
(	O
the	O
tag	B-Application
system	I-Application
)	O
and	O
Hilbert	O
's	O
10th	O
problem	O
(	O
Hilbert	O
's	O
problems	O
,	O
Diophantine	O
equation	O
)	O
led	O
Minsky	O
to	O
the	O
following	O
definition	O
of	O
:	O
</s>
<s>
Their	O
model	O
is	O
strongly	O
influenced	O
by	O
the	O
model	O
and	O
the	O
spirit	O
of	O
Hao	O
Wang	O
(	O
1957	O
)	O
and	O
his	O
Wang	O
B-machine	O
(	O
also	O
see	O
Post	B-Application
–	I-Application
Turing	I-Application
machine	I-Application
)	O
.	O
</s>
<s>
Unlimited	O
Register	B-Application
Machine	I-Application
URM	O
:	O
This	O
,	O
their	O
"	O
most	O
flexible	O
machine	O
...	O
consists	O
of	O
a	O
denumerable	O
sequence	O
of	O
registers	O
numbered	O
1	O
,	O
2	O
,	O
3	O
,	O
...	O
,	O
each	O
of	O
which	O
can	O
store	O
any	O
natural	O
number	O
...	O
Each	O
particular	O
program	O
,	O
however	O
involves	O
only	O
a	O
finite	O
number	O
of	O
these	O
registers	O
"	O
(	O
p.219	O
)	O
.	O
</s>
<s>
Limited	O
Register	B-Application
Machine	I-Application
LRM	O
:	O
Here	O
they	O
restrict	O
the	O
machine	O
to	O
a	O
finite	O
number	O
of	O
registers	O
N	O
,	O
but	O
they	O
also	O
allow	O
more	O
registers	O
to	O
"	O
be	O
brought	O
in	O
"	O
or	O
removed	O
if	O
empty	O
(	O
cf	O
.	O
</s>
<s>
Single-Register	O
Machine	O
SRM	O
:	O
Here	O
they	O
are	O
implementing	O
the	O
tag	B-Application
system	I-Application
of	O
Emil	O
Post	O
and	O
thereby	O
allow	O
only	O
writing	O
to	O
the	O
end	O
of	O
the	O
string	O
and	O
erasing	O
from	O
the	O
beginning	O
.	O
</s>
<s>
And	O
then	O
he	O
defines	O
his	O
"	O
REPEAT	O
"	O
 [ RPT ] 	O
and	O
shows	O
that	O
we	O
can	O
define	O
any	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
by	O
the	O
"	O
successor-repeat	O
"	O
set	O
{	O
[	O
0	O
]	O
,	O
[	O
'	O
]	O
,	O
 [ RPT ] 	O
}	O
(	O
where	O
the	O
range	O
of	O
the	O
[	O
RPT	O
]	O
cannot	O
include	O
itself	O
.	O
</s>
<s>
Schönhage	O
(	O
1980	O
)	O
developed	O
his	O
computational	O
model	O
in	O
context	O
of	O
a	O
"	O
new	O
"	O
model	O
he	O
called	O
the	O
Storage	O
Machine	O
Modification	O
model	O
(	O
SMM	O
)	O
,	O
his	O
variety	O
of	O
pointer	B-Application
machine	I-Application
.	O
</s>
<s>
His	O
development	O
described	O
a	O
RAM	O
(	O
random-access	B-Application
machine	I-Application
)	O
model	O
with	O
a	O
remarkable	O
instruction	O
set	O
requiring	O
no	O
operands	O
at	O
all	O
,	O
excepting	O
,	O
perhaps	O
,	O
the	O
"	O
conditional	O
jump	O
"	O
(	O
and	O
even	O
that	O
could	O
be	O
achieved	O
without	O
an	O
operand	O
)	O
:	O
</s>
<s>
Primarily	O
for	O
referencethis	O
is	O
a	O
RAM	B-Application
model	I-Application
,	O
not	O
a	O
counter-machine	O
modelthe	O
following	O
is	O
the	O
Schönhage	O
RAM0	O
instruction	O
set	O
:	O
</s>
<s>
Again	O
,	O
the	O
above	O
instruction	O
set	O
is	O
for	O
a	O
random-access	B-Application
machine	I-Application
,	O
a	O
RAMa	O
counter	B-Application
machine	I-Application
with	O
indirect	O
addressing	O
;	O
instruction	O
"	O
N	O
"	O
allows	O
for	O
indirect	O
storage	O
of	O
the	O
accumulator	O
,	O
and	O
instruction	O
"	O
L	O
"	O
allows	O
for	O
indirect	O
load	O
of	O
the	O
accumulator	O
.	O
</s>
