<s>
In	O
rewriting	O
,	O
a	O
reduction	B-Application
strategy	I-Application
or	O
rewriting	O
strategy	O
is	O
a	O
relation	O
specifying	O
a	O
rewrite	O
for	O
each	O
object	O
or	O
term	O
,	O
compatible	O
with	O
a	O
given	O
reduction	O
relation	O
.	O
</s>
<s>
Some	O
authors	O
use	O
the	O
term	O
to	O
refer	O
to	O
an	O
evaluation	B-Language
strategy	I-Language
.	O
</s>
<s>
Formally	O
,	O
for	O
an	O
abstract	O
rewriting	O
system	O
,	O
a	O
reduction	B-Application
strategy	I-Application
is	O
a	O
binary	O
relation	O
on	O
with	O
,	O
where	O
is	O
the	O
transitive	O
closure	O
of	O
(	O
but	O
not	O
the	O
reflexive	O
closure	O
)	O
.	O
</s>
<s>
A	O
one	O
step	O
reduction	B-Application
strategy	I-Application
is	O
one	O
where	O
.	O
</s>
<s>
A	O
deterministic	O
strategy	O
is	O
one	O
where	O
is	O
a	O
partial	B-Algorithm
function	I-Algorithm
,	O
i.e.	O
</s>
<s>
Stratego	O
is	O
a	O
domain-specific	B-Language
language	I-Language
designed	O
specifically	O
for	O
programming	O
term	O
rewriting	O
strategies	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
the	O
lambda	B-Language
calculus	I-Language
,	O
normal-order	O
reduction	O
refers	O
to	O
leftmost-outermost	O
reduction	O
in	O
the	O
sense	O
given	O
above	O
.	O
</s>
<s>
Normal	O
and	O
applicative	O
order	O
reduction	O
are	O
strong	O
in	O
that	O
they	O
allow	O
reduction	O
under	O
lambda	B-Language
abstractions	I-Language
.	O
</s>
<s>
In	O
contrast	O
,	O
weak	O
reduction	O
does	O
not	O
reduce	O
under	O
a	O
lambda	B-Language
abstraction	I-Language
.	O
</s>
<s>
Call-by-name	O
reduction	O
is	O
the	O
weak	O
reduction	B-Application
strategy	I-Application
that	O
reduces	O
the	O
leftmost	O
outermost	O
redex	O
not	O
inside	O
a	O
lambda	B-Language
abstraction	I-Language
,	O
while	O
call-by-value	O
reduction	O
is	O
the	O
weak	O
reduction	B-Application
strategy	I-Application
that	O
reduces	O
the	O
leftmost	O
innermost	O
redex	O
not	O
inside	O
a	O
lambda	B-Language
abstraction	I-Language
.	O
</s>
<s>
These	O
strategies	O
were	O
devised	O
to	O
reflect	O
the	O
call-by-name	O
and	O
call-by-value	O
evaluation	B-Language
strategies	I-Language
.	O
</s>
<s>
Unfortunately	O
,	O
weak	O
reduction	O
is	O
not	O
confluent	O
,	O
and	O
the	O
traditional	O
reduction	O
equations	O
of	O
the	O
lambda	B-Language
calculus	I-Language
are	O
useless	O
,	O
because	O
they	O
suggest	O
relationships	O
that	O
violate	O
the	O
weak	O
evaluation	O
regime	O
.	O
</s>
<s>
For	O
example	O
,	O
is	O
in	O
normal	O
form	O
for	O
a	O
weak	O
reduction	B-Application
strategy	I-Application
because	O
the	O
redex	O
is	O
contained	O
in	O
a	O
lambda	B-Language
abstraction	I-Language
.	O
</s>
<s>
But	O
the	O
term	O
can	O
still	O
be	O
reduced	O
under	O
the	O
extended	O
weak	O
reduction	B-Application
strategy	I-Application
,	O
because	O
the	O
redex	O
does	O
not	O
refer	O
to	O
.	O
</s>
<s>
Optimal	O
reduction	O
is	O
not	O
a	O
reduction	B-Application
strategy	I-Application
for	O
the	O
lambda	B-Language
calculus	I-Language
in	O
a	O
narrow	O
sense	O
because	O
performing	O
β-reduction	O
loses	O
the	O
information	O
about	O
the	O
substituted	O
redexes	O
being	O
shared	O
.	O
</s>
<s>
Instead	O
it	O
is	O
defined	O
for	O
the	O
labelled	O
lambda	B-Language
calculus	I-Language
,	O
an	O
annotated	O
lambda	B-Language
calculus	I-Language
which	O
captures	O
a	O
precise	O
notion	O
of	O
the	O
work	O
that	O
should	O
be	O
shared	O
.	O
</s>
<s>
A	O
labelled	O
term	O
is	O
a	O
lambda	B-Language
calculus	I-Language
term	O
where	O
each	O
subterm	O
has	O
a	O
label	O
.	O
</s>
<s>
The	O
Bologna	O
Optimal	O
Higher-order	O
Machine	O
(	O
BOHM	O
)	O
is	O
a	O
prototype	O
implementation	O
of	O
an	O
extension	O
of	O
the	O
technique	O
to	O
interaction	B-Application
nets	I-Application
.	O
</s>
<s>
Lambdascope	O
is	O
a	O
more	O
recent	O
implementation	O
of	O
optimal	O
reduction	O
,	O
also	O
using	O
interaction	B-Application
nets	I-Application
.	O
</s>
<s>
Call	O
by	O
need	O
reduction	O
can	O
be	O
defined	O
similarly	O
to	O
optimal	O
reduction	O
as	O
weak	O
leftmost-outermost	O
reduction	O
using	O
parallel	O
reduction	O
of	O
redexes	O
with	O
the	O
same	O
label	O
,	O
for	O
a	O
slightly	O
different	O
labelled	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
As	O
with	O
call-by-name	O
and	O
call-by-value	O
,	O
call-by-need	O
reduction	O
was	O
devised	O
to	O
mimic	O
the	O
behavior	O
of	O
the	O
evaluation	B-Language
strategy	I-Language
known	O
as	O
"	O
call-by-need	O
"	O
or	O
lazy	O
evaluation	O
.	O
</s>
