<s>
P′′	B-Application
(	O
P	O
double	O
prime	O
)	O
is	O
a	O
primitive	O
computer	O
programming	O
language	O
created	O
by	O
Corrado	O
Böhm	O
in	O
1964	O
to	O
describe	O
a	O
family	O
of	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
(	O
hereinafter	O
written	O
P′′	B-Application
)	O
is	O
formally	O
defined	O
as	O
a	O
set	O
of	O
words	O
on	O
the	O
four-instruction	O
alphabet	O
,	O
as	O
follows	O
:	O
</s>
<s>
and	O
are	O
words	O
in	O
P′′	B-Application
.	O
</s>
<s>
If	O
and	O
are	O
words	O
in	O
P′′	B-Application
,	O
then	O
is	O
a	O
word	O
in	O
P′′	B-Application
.	O
</s>
<s>
If	O
is	O
a	O
word	O
in	O
P′′	B-Application
,	O
then	O
is	O
a	O
word	O
in	O
P′′	B-Application
.	O
</s>
<s>
Only	O
words	O
derivable	O
from	O
the	O
previous	O
three	O
rules	O
are	O
words	O
in	O
P′′	B-Application
.	O
</s>
<s>
is	O
the	O
tape-alphabet	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
with	O
left-infinite	O
tape	O
,	O
being	O
the	O
blank	O
symbol	O
,	O
equivalent	O
to	O
.	O
</s>
<s>
All	O
instructions	O
in	O
P′′	B-Application
are	O
permutations	B-Algorithm
of	O
the	O
set	O
of	O
all	O
possible	O
tape	O
configurations	O
;	O
that	O
is	O
,	O
all	O
possible	O
configurations	O
of	O
both	O
the	O
contents	O
of	O
the	O
tape	O
and	O
the	O
position	O
of	O
the	O
tape-head	O
.	O
</s>
<s>
is	O
a	O
predicate	B-Algorithm
saying	O
that	O
the	O
current	O
symbol	O
is	O
not	O
.	O
</s>
<s>
means	O
the	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
means	O
iterate	B-Algorithm
in	O
a	O
while	O
loop	O
,	O
with	O
the	O
condition	O
.	O
</s>
<s>
The	O
Brainfuck	B-Language
language	I-Language
(	O
apart	O
from	O
its	O
I/O	O
commands	O
)	O
is	O
a	O
minor	O
informal	O
variation	O
of	O
P′′	B-Application
.	O
</s>
<s>
Böhm	O
gives	O
explicit	O
P′′	B-Application
programs	O
for	O
each	O
of	O
a	O
set	O
of	O
basic	O
functions	O
sufficient	O
to	O
compute	O
any	O
computable	O
function	O
,	O
using	O
only	O
,	O
and	O
the	O
four	O
words	O
where	O
with	O
denoting	O
the	O
th	O
iterate	B-Algorithm
of	O
,	O
and	O
.	O
</s>
<s>
These	O
are	O
the	O
equivalents	O
of	O
the	O
six	O
respective	O
Brainfuck	B-Language
commands	O
,	O
,	O
,	O
,	O
,	O
.	O
</s>
<s>
which	O
translates	O
directly	O
to	O
the	O
equivalent	O
Brainfuck	B-Language
program	O
:	O
</s>
