<s>
In	O
the	O
study	O
of	O
denotational	B-Application
semantics	I-Application
of	O
the	O
lambda	B-Language
calculus	I-Language
,	O
Böhm	B-Application
trees	I-Application
,	O
Lévy-Longo	O
trees	O
,	O
and	O
Berarducci	O
trees	O
are	O
(	O
potentially	O
infinite	O
)	O
tree-like	O
mathematical	O
objects	O
that	O
capture	O
the	O
"	O
meaning	O
"	O
of	O
a	O
term	O
up	O
to	O
some	O
set	O
of	O
"	O
meaningless	O
"	O
terms	O
.	O
</s>
<s>
In	O
particular	O
,	O
considering	O
the	O
lambda	B-Language
calculus	I-Language
as	O
a	O
rewriting	O
system	O
,	O
each	O
beta	O
reduction	O
step	O
is	O
a	O
rewrite	O
step	O
,	O
and	O
once	O
there	O
are	O
no	O
further	O
beta	O
reductions	O
the	O
term	O
is	O
in	O
normal	O
form	O
.	O
</s>
<s>
For	O
example	O
the	O
meanings	O
of	O
I	O
=	O
λx.x	O
and	O
I	O
I	O
are	O
both	O
I	O
.	O
</s>
<s>
This	O
works	O
for	O
any	O
strongly	O
normalizing	O
subset	O
of	O
the	O
lambda	B-Language
calculus	I-Language
,	O
such	O
as	O
a	O
typed	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
This	O
naive	O
assignment	O
of	O
meaning	O
is	O
however	O
inadequate	O
for	O
the	O
full	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
The	O
term	O
Ω	O
=(	O
λx.x	O
x	O
)	O
(	O
λx.x	O
x	O
)	O
does	O
not	O
have	O
a	O
normal	O
form	O
,	O
and	O
similarly	O
the	O
term	O
X	O
=	O
λx.xΩ	O
does	O
not	O
have	O
a	O
normal	O
form	O
.	O
</s>
<s>
But	O
the	O
application	O
Ω	O
(	O
K	O
I	O
)	O
,	O
where	O
K	O
denotes	O
the	O
standard	O
lambda	O
term	O
λx.λy.x	O
,	O
reduces	O
only	O
to	O
itself	O
,	O
whereas	O
the	O
application	O
X	O
(	O
K	O
I	O
)	O
reduces	O
with	O
normal	O
order	O
reduction	O
to	O
I	O
,	O
hence	O
has	O
a	O
meaning	O
.	O
</s>
<s>
In	O
the	O
infinitary	O
lambda	B-Language
calculus	I-Language
,	O
the	O
term	O
N	O
N	O
,	O
where	O
N	O
=	O
λx.I(xx )	O
,	O
reduces	O
to	O
both	O
to	O
I	O
(	O
I	O
(	O
...	O
)	O
)	O
and	O
Ω	O
.	O
</s>
<s>
This	O
corresponds	O
to	O
the	O
standard	O
infinitary	O
lambda	B-Language
calculus	I-Language
plus	O
terms	O
containing	O
.	O
</s>
<s>
Beta-reduction	B-Language
on	O
this	O
set	O
is	O
defined	O
in	O
the	O
standard	O
way	O
.	O
</s>
<s>
The	O
Böhm	B-Application
trees	I-Application
are	O
obtained	O
by	O
considering	O
the	O
λ⊥	O
-terms	O
where	O
the	O
set	O
of	O
meaningless	O
terms	O
consists	O
of	O
those	O
without	O
head	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
More	O
explicitly	O
,	O
the	O
Böhm	B-Application
tree	I-Application
BT(M )	O
of	O
a	O
lambda	O
term	O
M	O
can	O
be	O
computed	O
as	O
follows	O
:	O
</s>
<s>
For	O
example	O
,	O
BT(Ω )	O
=	O
⊥	O
,	O
BT(I )	O
=	O
I	O
,	O
and	O
BT( 	O
λx.xΩ	O
)	O
=	O
λx.x	O
⊥	O
.	O
</s>
<s>
Determining	O
whether	O
a	O
term	O
has	O
a	O
head	B-Application
normal	I-Application
form	I-Application
is	O
an	O
undecidable	O
problem	O
.	O
</s>
<s>
Barendregt	O
introduced	O
a	O
notion	O
of	O
an	O
"	O
effective	O
"	O
Böhm	B-Application
tree	I-Application
that	O
is	O
computable	O
,	O
with	O
the	O
only	O
difference	O
being	O
that	O
terms	O
with	O
no	O
head	B-Application
normal	I-Application
form	I-Application
are	O
not	O
marked	O
with	O
.	O
</s>
<s>
Note	O
that	O
computing	O
the	O
Böhm	B-Application
tree	I-Application
is	O
similar	O
to	O
finding	O
a	O
normal	O
form	O
for	O
M	O
.	O
If	O
M	O
has	O
a	O
normal	O
form	O
,	O
the	O
Böhm	B-Application
tree	I-Application
is	O
finite	O
and	O
has	O
a	O
simple	O
correspondence	O
to	O
the	O
normal	O
form	O
.	O
</s>
<s>
Since	O
the	O
Böhm	B-Application
tree	I-Application
may	O
be	O
infinite	O
the	O
procedure	O
should	O
be	O
understood	O
as	O
being	O
applied	O
co-recursively	O
or	O
as	O
taking	O
the	O
limit	O
of	O
an	O
infinite	O
series	O
of	O
approximations	O
.	O
</s>
<s>
The	O
Lévy-Longo	O
trees	O
are	O
obtained	O
by	O
considering	O
the	O
λ⊥	O
-terms	O
where	O
the	O
set	O
of	O
meaningless	O
terms	O
consists	O
of	O
those	O
without	O
weak	O
head	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
LLT(M )	O
is	O
,	O
if	O
M	O
has	O
no	O
weak	O
head	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
If	O
reduces	O
to	O
the	O
weak	O
head	B-Application
normal	I-Application
form	I-Application
,	O
then	O
.	O
</s>
