<s>
A	O
tag	B-Application
system	I-Application
is	O
a	O
deterministic	O
computational	B-Application
model	I-Application
published	O
by	O
Emil	O
Leon	O
Post	O
in	O
1943	O
as	O
a	O
simple	O
form	O
of	O
a	O
Post	B-Application
canonical	I-Application
system	I-Application
.	O
</s>
<s>
A	O
tag	B-Application
system	I-Application
may	O
also	O
be	O
viewed	O
as	O
an	O
abstract	O
machine	O
,	O
called	O
a	O
Post	O
tag	O
machine	O
(	O
not	O
to	O
be	O
confused	O
with	O
Post	B-Application
–	I-Application
Turing	I-Application
machines	I-Application
)	O
—	O
briefly	O
,	O
a	O
finite-state	B-Architecture
machine	I-Architecture
whose	O
only	O
tape	O
is	O
a	O
FIFO	B-Operating_System
queue	B-Application
of	O
unbounded	O
length	O
,	O
such	O
that	O
in	O
each	O
transition	O
the	O
machine	O
reads	O
the	O
symbol	O
at	O
the	O
head	O
of	O
the	O
queue	B-Application
,	O
deletes	O
a	O
constant	O
number	O
of	O
symbols	O
from	O
the	O
head	O
,	O
and	O
appends	O
to	O
the	O
tail	O
a	O
symbol-string	O
that	O
depends	O
solely	O
on	O
the	O
first	O
symbol	O
read	O
in	O
this	O
transition	O
.	O
</s>
<s>
A	O
computation	O
by	O
a	O
tag	B-Application
system	I-Application
is	O
a	O
finite	O
sequence	O
of	O
words	O
produced	O
by	O
iterating	O
the	O
transformation	O
t	O
,	O
starting	O
with	O
an	O
initially	O
given	O
word	O
and	O
halting	O
when	O
a	O
halting	O
word	O
is	O
produced	O
.	O
</s>
<s>
In	O
the	O
sequence	O
computed	O
by	O
the	O
tag	B-Application
system	I-Application
below	O
we	O
skip	O
this	O
intermediate	O
step	O
,	O
hence	O
the	O
successor	O
of	O
n	O
is	O
for	O
oddn	O
.	O
</s>
<s>
In	O
this	O
tag	B-Application
system	I-Application
,	O
a	O
positive	O
integer	O
n	O
is	O
represented	O
by	O
the	O
word	O
aa	O
...	O
a	O
with	O
n	O
a	O
's	O
.	O
</s>
<s>
For	O
each	O
m	O
>	O
1	O
,	O
the	O
set	O
of	O
m-tag	O
systems	O
is	O
Turing-complete	O
;	O
i.e.	O
,	O
for	O
each	O
m	O
>	O
1	O
,	O
it	O
is	O
the	O
case	O
that	O
for	O
any	O
given	O
Turing	O
machine	O
T	O
,	O
there	O
is	O
an	O
m-tag	O
system	O
that	O
emulates	B-Algorithm
T	O
.	O
In	O
particular	O
,	O
a	O
2-tag	O
system	O
can	O
be	O
constructed	O
to	O
emulate	O
a	O
Universal	O
Turing	O
machine	O
,	O
as	O
was	O
done	O
by	O
Wang	O
1963	O
and	O
by	O
Cocke	O
&	O
Minsky	O
1964	O
.	O
</s>
<s>
For	O
example	O
,	O
Rogozhin	O
1996	O
proved	O
the	O
universality	O
of	O
the	O
class	O
of	O
2-tag	O
systems	O
with	O
alphabet	O
{	O
a1	O
,	O
...	O
,	O
an	O
,	O
}	O
and	O
corresponding	O
productions	O
{	O
ananW1	O
,	O
...	O
,	O
ananWn-1	O
,	O
anan	O
,	O
}	O
,	O
where	O
the	O
Wk	O
are	O
nonempty	O
words	O
;	O
he	O
then	O
proved	O
the	O
universality	O
of	O
a	O
very	O
small	O
(	O
4-state	O
,	O
6-symbol	O
)	O
Turing	O
machine	O
by	O
showing	O
that	O
it	O
can	O
simulate	O
this	O
class	O
of	O
tag	B-Application
systems	I-Application
.	O
</s>
<s>
The	O
above	O
definition	O
differs	O
from	O
that	O
of	O
Post	O
1943	O
,	O
whose	O
tag	B-Application
systems	I-Application
use	O
no	O
halting	O
symbol	O
,	O
but	O
rather	O
halt	O
only	O
on	O
the	O
empty	O
word	O
,	O
with	O
the	O
tag	O
operation	O
t	O
being	O
defined	O
as	O
follows	O
:	O
</s>
<s>
The	O
above	O
remark	O
concerning	O
the	O
Turing-completeness	O
of	O
the	O
set	O
of	O
m-tag	O
systems	O
,	O
for	O
any	O
m	O
1	O
,	O
applies	O
also	O
to	O
these	O
tag	B-Application
systems	I-Application
as	O
originally	O
defined	O
by	O
Post	O
.	O
</s>
<s>
A	O
cyclic	O
tag	B-Application
system	I-Application
is	O
a	O
modification	O
of	O
the	O
original	O
tag	B-Application
system	I-Application
.	O
</s>
<s>
Cyclic	O
tag	B-Application
systems	I-Application
were	O
created	O
by	O
Matthew	O
Cook	O
and	O
were	O
used	O
in	O
Cook	O
's	O
demonstration	O
that	O
the	O
Rule	O
110	O
cellular	O
automaton	O
is	O
universal	O
.	O
</s>
<s>
A	O
key	O
part	O
of	O
the	O
demonstration	O
was	O
that	O
cyclic	O
tag	B-Application
systems	I-Application
can	O
emulate	O
a	O
Turing-complete	O
class	O
of	O
tag	B-Application
systems	I-Application
.	O
</s>
<s>
An	O
m-tag	O
system	O
with	O
alphabet	O
{	O
a1	O
,	O
...	O
,	O
an}	O
and	O
corresponding	O
productions	O
{	O
P1	O
,	O
...	O
,	O
Pn}	O
is	O
emulated	O
by	O
a	O
cyclic	O
tag	B-Application
system	I-Application
with	O
m*n	O
productions	O
(	O
Q1	O
,	O
...	O
,	O
Qn	O
,	O
-	O
,	O
-	O
,	O
...	O
,	O
-	O
)	O
,	O
where	O
all	O
but	O
the	O
first	O
n	O
productions	O
are	O
the	O
empty	O
string	O
(	O
denoted	O
by	O
''	O
)	O
.	O
</s>
<s>
The	O
Qk	O
are	O
encodings	O
of	O
the	O
respective	O
Pk	O
,	O
obtained	O
by	O
replacing	O
each	O
symbol	O
of	O
the	O
tag	B-Application
system	I-Application
alphabet	O
by	O
a	O
length-n	O
binary	O
string	O
as	O
follows	O
(	O
these	O
are	O
to	O
be	O
applied	O
also	O
to	O
the	O
initial	O
word	O
of	O
a	O
tag	B-Application
system	I-Application
computation	O
)	O
:	O
</s>
<s>
Successive	O
lines	O
of	O
a	O
tag	B-Application
system	I-Application
computation	O
will	O
then	O
occur	O
encoded	O
as	O
every	O
(	O
m*n	O
)	O
th	O
line	O
of	O
its	O
emulation	O
by	O
the	O
cyclic	O
tag	B-Application
system	I-Application
.	O
</s>
<s>
Every	O
sixth	O
line	O
(	O
marked	O
by	O
''	O
)	O
produced	O
by	O
the	O
cyclic	O
tag	B-Application
system	I-Application
is	O
the	O
encoding	O
of	O
a	O
corresponding	O
line	O
of	O
the	O
tag	B-Application
system	I-Application
computation	O
,	O
until	O
the	O
emulated	O
halt	O
is	O
reached	O
.	O
</s>
