<s>
Lambda	O
calculus	O
is	O
a	O
formal	O
mathematical	O
system	O
based	O
on	O
lambda	O
abstraction	O
and	O
function	B-Algorithm
application	I-Algorithm
.	O
</s>
<s>
Variables	O
that	O
fall	O
within	O
the	O
scope	B-Language
of	O
an	O
abstraction	O
are	O
said	O
to	O
be	O
bound	O
.	O
</s>
<s>
Closed	O
lambda	O
expressions	O
are	O
also	O
known	O
as	O
combinators	B-Application
and	O
are	O
equivalent	O
to	O
terms	O
in	O
combinatory	B-Application
logic	I-Application
.	O
</s>
<s>
In	O
programming	O
languages	O
with	O
static	B-Language
scope	I-Language
,	O
alpha-conversion	O
can	O
be	O
used	O
to	O
make	O
name	B-Application
resolution	I-Application
simpler	O
by	O
ensuring	O
that	O
no	O
variable	O
name	O
masks	B-Language
a	O
name	O
in	O
a	O
containing	O
scope	B-Language
(	O
see	O
alpha	O
renaming	O
to	O
make	O
name	B-Application
resolution	I-Application
trivial	O
)	O
.	O
</s>
<s>
β-reduction	O
captures	O
the	O
idea	O
of	O
function	B-Algorithm
application	I-Algorithm
.	O
</s>
<s>
A	O
lambda	O
expression	O
that	O
cannot	O
be	O
reduced	O
further	O
,	O
by	O
either	O
β-redex	O
,	O
or	O
η-redex	O
is	O
in	O
normal	B-Application
form	I-Application
.	O
</s>
<s>
See	O
the	O
main	O
article	O
on	O
Beta	B-Application
normal	I-Application
form	I-Application
for	O
details	O
.	O
</s>
<s>
Normal	B-Application
Form	I-Application
Type	O
Definition	O
.	O
</s>
<s>
Normal	B-Application
Form	I-Application
No	O
β	O
-	O
or	O
η-reductions	O
are	O
possible	O
.	O
</s>
<s>
Head	B-Application
Normal	I-Application
Form	I-Application
In	O
the	O
form	O
of	O
a	O
lambda	O
abstraction	O
whose	O
body	O
is	O
not	O
reducible	O
.	O
</s>
<s>
Weak	O
Head	B-Application
Normal	I-Application
Form	I-Application
In	O
the	O
form	O
of	O
a	O
lambda	O
abstraction	O
.	O
</s>
<s>
Execution	O
is	O
performing	O
β-reductions	O
and	O
η-reductions	O
on	O
subexpressions	O
in	O
the	O
canonym	O
of	O
a	O
lambda	O
expression	O
until	O
the	O
result	O
is	O
a	O
lambda	O
function	O
(	O
abstraction	O
)	O
in	O
the	O
normal	B-Application
form	I-Application
.	O
</s>
<s>
Eager	O
evaluation	O
will	O
apply	O
all	O
reductions	O
possible	O
,	O
leaving	O
the	O
result	O
in	O
normal	B-Application
form	I-Application
,	O
while	O
lazy	O
evaluation	O
will	O
omit	O
some	O
reductions	O
in	O
parameters	O
,	O
leaving	O
the	O
result	O
in	O
"	O
weak	O
head	B-Application
normal	I-Application
form	I-Application
"	O
.	O
</s>
<s>
β-reduction	O
captures	O
the	O
idea	O
of	O
function	B-Algorithm
application	I-Algorithm
(	O
also	O
called	O
a	O
function	O
call	O
)	O
,	O
and	O
implements	O
the	O
substitution	O
of	O
the	O
actual	O
parameter	O
expression	O
for	O
the	O
formal	O
parameter	O
variable	O
.	O
</s>
