<s>
It	O
plays	O
important	O
roles	O
in	O
database	B-General_Concept
theory	I-General_Concept
as	O
well	O
as	O
in	O
practice	O
.	O
</s>
<s>
In	O
its	O
simplest	O
application	O
the	O
chase	O
is	O
used	O
for	O
testing	O
whether	O
the	O
projection	B-Algorithm
of	O
a	O
relation	B-Language
schema	I-Language
constrained	O
by	O
some	O
functional	B-Application
dependencies	I-Application
onto	O
a	O
given	O
decomposition	O
can	O
be	O
recovered	O
by	O
rejoining	O
the	O
projections	B-Algorithm
.	O
</s>
<s>
Let	O
t	O
be	O
a	O
tuple	O
in	O
where	O
R	O
is	O
a	O
relation	B-Language
and	O
F	O
is	O
a	O
set	O
of	O
functional	B-Application
dependencies	I-Application
(	O
FD	O
)	O
.	O
</s>
<s>
If	O
tuples	O
in	O
R	O
are	O
represented	O
as	O
t1	O
,	O
...	O
,	O
tk	O
,	O
the	O
join	O
of	O
the	O
projections	B-Algorithm
of	O
each	O
ti	O
should	O
agree	O
with	O
t	O
on	O
where	O
i	O
=	O
1	O
,	O
2	O
,	O
...	O
,	O
k	O
.	O
If	O
ti	O
is	O
not	O
on	O
,	O
the	O
value	O
is	O
unknown	O
.	O
</s>
<s>
Suppose	O
R	O
has	O
attributes	B-General_Concept
A	O
,	O
B	O
,	O
...	O
and	O
components	O
of	O
t	O
are	O
a	O
,	O
b	O
,	O
....	O
For	O
ti	O
use	O
the	O
same	O
letter	O
as	O
t	O
in	O
the	O
components	O
that	O
are	O
in	O
Si	O
but	O
subscript	O
the	O
letter	O
with	O
i	O
if	O
the	O
component	O
is	O
not	O
in	O
Si	O
.	O
</s>
<s>
Let	O
R(A, B, C, D )	O
be	O
a	O
relation	B-Language
schema	I-Language
known	O
to	O
obey	O
the	O
set	O
of	O
functional	B-Application
dependencies	I-Application
F	O
=	O
{	O
A	O
→	O
B	O
,	O
B	O
→	O
C	O
,	O
CD	O
→	O
A}	O
.	O
</s>
<s>
Suppose	O
R	O
is	O
decomposed	O
into	O
three	O
relation	B-Language
schemas	I-Language
S1	O
=	O
{	O
A	O
,	O
D}	O
,	O
S2	O
=	O
{	O
A	O
,	O
C}	O
and	O
S3	O
=	O
{	O
B	O
,	O
C	O
,	O
D}	O
.	O
</s>
<s>
The	O
components	O
for	O
attributes	B-General_Concept
A	O
and	O
D	O
are	O
unsubscripted	O
and	O
those	O
for	O
attributes	B-General_Concept
B	O
and	O
C	O
are	O
subscripted	O
with	O
i	O
=	O
1	O
.	O
</s>
<s>
A	O
final	O
tableau	O
with	O
a	O
row	O
that	O
is	O
the	O
same	O
as	O
t	O
implies	O
that	O
any	O
tuple	O
t	O
in	O
the	O
join	O
of	O
the	O
projections	B-Algorithm
is	O
actually	O
a	O
tuple	O
of	O
R	O
.	O
</s>
