<s>
A	O
Post	B-Application
canonical	I-Application
system	I-Application
,	O
also	O
known	O
as	O
a	O
Post	O
production	O
system	O
,	O
as	O
created	O
by	O
Emil	O
Post	O
,	O
is	O
a	O
string-manipulation	O
system	O
that	O
starts	O
with	O
finitely-many	O
strings	O
and	O
repeatedly	O
transforms	O
them	O
by	O
applying	O
a	O
finite	O
set	O
j	O
of	O
specified	O
rules	O
of	O
a	O
certain	O
form	O
,	O
thus	O
generating	O
a	O
formal	O
language	O
.	O
</s>
<s>
Today	O
they	O
are	O
mainly	O
of	O
historical	O
relevance	O
because	O
every	O
Post	B-Application
canonical	I-Application
system	I-Application
can	O
be	O
reduced	O
to	O
a	O
string	O
rewriting	O
system	O
(	O
semi-Thue	O
system	O
)	O
,	O
which	O
is	O
a	O
simpler	O
formulation	O
.	O
</s>
<s>
Both	O
formalisms	O
are	O
Turing	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
The	O
formal	O
language	O
generated	O
by	O
a	O
Post	B-Application
canonical	I-Application
system	I-Application
is	O
the	O
set	O
whose	O
elements	O
are	O
the	O
initial	O
words	O
together	O
with	O
all	O
words	O
obtainable	O
from	O
them	O
by	O
repeated	O
application	O
of	O
the	O
production	O
rules	O
.	O
</s>
<s>
Post	O
1943	O
proved	O
the	O
remarkable	O
Normal-form	O
Theorem	O
,	O
which	O
applies	O
to	O
the	O
most-general	O
type	O
of	O
Post	B-Application
canonical	I-Application
system	I-Application
:	O
</s>
<s>
Given	O
any	O
Post	B-Application
canonical	I-Application
system	I-Application
on	O
an	O
alphabet	O
A	O
,	O
a	O
Post	B-Application
canonical	I-Application
system	I-Application
in	O
normal	O
form	O
can	O
be	O
constructed	O
from	O
it	O
,	O
possibly	O
enlarging	O
the	O
alphabet	O
,	O
such	O
that	O
the	O
set	O
of	O
words	O
involving	O
only	O
letters	O
of	O
A	O
that	O
are	O
generated	O
by	O
the	O
normal-form	O
system	O
is	O
exactly	O
the	O
set	O
of	O
words	O
generated	O
by	O
the	O
original	O
system	O
.	O
</s>
<s>
Tag	B-Application
systems	I-Application
,	O
which	O
comprise	O
a	O
universal	O
computational	O
model	O
,	O
are	O
notable	O
examples	O
of	O
Post	O
normal-form	O
system	O
,	O
being	O
also	O
monogenic	O
.	O
</s>
<s>
That	O
is	O
,	O
each	O
production	O
rule	O
is	O
a	O
simple	O
substitution	O
rule	O
,	O
often	O
written	O
in	O
the	O
form	O
g	O
h	O
.	O
It	O
has	O
been	O
proved	O
that	O
any	O
Post	B-Application
canonical	I-Application
system	I-Application
is	O
reducible	O
to	O
such	O
a	O
substitution	O
system	O
,	O
which	O
,	O
as	O
a	O
formal	O
grammar	O
,	O
is	O
also	O
called	O
a	O
phrase-structure	O
grammar	O
,	O
or	O
a	O
type-0	O
grammar	O
in	O
the	O
Chomsky	O
hierarchy	O
.	O
</s>
