<s>
In	O
computer	B-General_Concept
programming	I-General_Concept
,	O
bidirectional	B-Algorithm
transformations	I-Algorithm
(	O
bx	O
)	O
are	O
programs	O
in	O
which	O
a	O
single	O
piece	O
of	O
code	O
can	O
be	O
run	O
in	O
several	O
ways	O
,	O
such	O
that	O
the	O
same	O
data	O
are	O
sometimes	O
considered	O
as	O
input	O
,	O
and	O
sometimes	O
as	O
output	O
.	O
</s>
<s>
Some	O
bidirectional	B-Algorithm
languages	I-Algorithm
are	O
bijective	B-Algorithm
.	O
</s>
<s>
The	O
bijectivity	B-Algorithm
of	O
a	O
language	O
is	O
a	O
severe	O
restriction	O
of	O
its	O
power	O
,	O
because	O
a	O
bijective	B-Algorithm
language	O
is	O
merely	O
relating	O
two	O
different	O
ways	O
to	O
present	O
the	O
very	O
same	O
information	O
.	O
</s>
<s>
More	O
general	O
is	O
a	O
lens	B-Algorithm
language	I-Algorithm
,	O
in	O
which	O
there	O
is	O
a	O
distinguished	O
forward	O
direction	O
(	O
"	O
get	O
"	O
)	O
that	O
takes	O
a	O
concrete	O
input	O
to	O
an	O
abstract	O
output	O
,	O
discarding	O
some	O
information	O
in	O
the	O
process	O
:	O
the	O
concrete	O
state	O
includes	O
all	O
the	O
information	O
that	O
is	O
in	O
the	O
abstract	O
state	O
,	O
and	O
usually	O
some	O
more	O
.	O
</s>
<s>
The	O
most	O
general	O
case	O
is	O
that	O
of	O
symmetric	O
bidirectional	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
Bidirectional	B-Algorithm
transformations	I-Algorithm
can	O
be	O
used	O
to	O
:	O
</s>
<s>
Bidirectional	B-Algorithm
transformations	I-Algorithm
fall	O
into	O
various	O
well-studied	O
categories	O
.	O
</s>
<s>
If	O
lenses	O
allow	O
"	O
focusing	O
"	O
(	O
viewing	O
,	O
updating	O
)	O
on	O
a	O
part	O
of	O
a	O
product	O
type	O
,	O
prisms	O
allow	O
focusing	O
(	O
possible	O
viewing	O
,	O
building	O
)	O
on	O
a	O
part	O
of	O
a	O
sum	B-Language
type	I-Language
.	O
</s>
<s>
Both	O
lenses	O
and	O
prisms	O
,	O
as	O
well	O
as	O
other	O
constructions	O
such	O
as	O
traversals	O
,	O
are	O
more	O
general	O
notion	O
of	O
bidirectional	B-Algorithm
transformations	I-Algorithm
known	O
as	O
optics	O
.	O
</s>
