<s>
In	O
database	B-General_Concept
theory	I-General_Concept
,	O
Imieliński	B-Application
–	I-Application
Lipski	I-Application
algebra	I-Application
is	O
an	O
extension	O
of	O
relational	B-Algorithm
algebra	I-Algorithm
onto	O
tables	O
with	O
different	O
types	O
of	O
null	B-Language
values	I-Language
.	O
</s>
<s>
Imieliński	B-Application
–	I-Application
Lipski	I-Application
algebras	I-Application
are	O
defined	O
to	O
satisfy	O
precise	O
conditions	O
for	O
semantically	O
meaningful	O
extension	O
of	O
the	O
usual	O
relational	O
operators	O
,	O
such	O
as	O
projection	O
,	O
selection	O
,	O
union	O
,	O
and	O
join	O
,	O
from	O
operators	O
on	O
relations	O
to	O
operators	O
on	O
relations	O
with	O
various	O
kinds	O
of	O
"	O
null	B-Language
values	I-Language
"	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
well	O
known	O
that	O
the	O
three-valued	B-Language
logic	I-Language
approach	O
to	O
deal	O
with	O
null	B-Language
values	I-Language
,	O
supported	O
treatment	O
of	O
nulls	B-Language
values	O
by	O
SQL	B-Language
is	O
not	O
complete	O
,	O
see	O
Ullman	O
book	O
.	O
</s>
<s>
SQL	B-Language
query	O
Q	O
will	O
return	O
empty	O
set	O
(	O
no	O
results	O
)	O
under	O
3-valued	O
semantics	O
currently	O
adopted	O
by	O
all	O
variants	O
of	O
SQL	B-Language
.	O
</s>
<s>
This	O
is	O
the	O
case	O
because	O
in	O
SQL	B-Language
,	O
NULL	B-Language
is	O
never	O
equal	O
to	O
any	O
constant	O
–	O
in	O
this	O
case	O
,	O
neither	O
to	O
“	O
Spring	O
”	O
nor	O
“	O
Fall	O
”	O
nor	O
“	O
Winter	O
”	O
(	O
if	O
there	O
is	O
Winter	O
semester	O
in	O
this	O
school	O
)	O
.	O
</s>
<s>
NULL='Spring'	O
will	O
evaluate	O
to	O
MAYBE	O
and	O
so	O
will	O
NULL='Fall'	O
.	O
</s>
<s>
Indeed	O
,	O
regardless	O
what	O
semester	O
Igor	O
took	O
the	O
Networks	O
class	O
(	O
no	O
matter	O
what	O
was	O
the	O
unknown	O
value	O
of	O
NULL	B-Language
)	O
,	O
the	O
selection	O
condition	O
will	O
be	O
true	O
.	O
</s>
<s>
This	O
“	O
Igor	O
”	O
will	O
be	O
missed	O
by	O
SQL	B-Language
and	O
the	O
SQL	B-Language
answer	O
wo	O
n’t	O
be	O
complete	O
according	O
to	O
completeness	O
requirements	O
specified	O
in	O
Tomasz	O
Imieliński	O
,	O
Witold	O
Lipski	O
,	O
'	O
Incomplete	O
Information	O
in	O
Relational	O
Databases	O
 '	O
.	O
</s>
<s>
It	O
is	O
also	O
argued	O
there	O
that	O
3-valued	B-Language
logic	I-Language
(	O
TRUE	O
,	O
FALSE	O
,	O
MAYBE	O
)	O
can	O
never	O
provide	O
guarantee	O
of	O
complete	O
answer	O
for	O
tables	O
with	O
incomplete	O
information	O
.	O
</s>
<s>
Three	O
algebras	O
which	O
satisfy	O
conditions	O
of	O
safety	O
and	O
completeness	O
are	O
defined	O
as	O
Imielinski	O
–	O
Lipski	O
algebras	O
:	O
the	O
Codd-Tables	B-Algorithm
algebra	O
,	O
the	O
V-tables	O
algebra	O
and	O
the	O
Conditional	O
tables	O
(	O
C-tables	B-Language
)	O
algebra	O
.	O
</s>
<s>
Codd-tables	B-Algorithm
algebra	O
is	O
based	O
on	O
the	O
usual	O
Codd	O
's	O
singe	O
NULL	B-Language
values	I-Language
.	O
</s>
<s>
It	O
is	O
also	O
demonstrated	O
in	O
[	O
IL84	O
that	O
it	O
is	O
not	O
possible	O
to	O
correctly	O
extend	O
more	O
relational	O
operators	O
over	O
Codd-Tables	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
such	O
basic	O
operation	O
as	O
join	O
is	O
not	O
extendable	O
over	O
Codd-tables	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
be	O
able	O
to	O
extend	O
more	O
relational	O
operators	O
,	O
more	O
expressive	O
form	O
of	O
null	B-Language
value	O
representation	O
is	O
needed	O
in	O
tables	O
which	O
are	O
called	O
V-table	O
.	O
</s>
<s>
V-tables	O
algebra	O
is	O
based	O
on	O
many	O
different	O
(	O
"	O
marked	O
"	O
)	O
null	B-Language
values	I-Language
or	O
variables	O
allowed	O
to	O
appear	O
in	O
a	O
table	O
.	O
</s>
<s>
Example	O
of	O
conditional	O
table	O
(	O
c-table	B-Language
)	O
is	O
shown	O
below	O
.	O
</s>
<s>
It	O
has	O
additional	O
column	O
“	O
con	O
”	O
which	O
is	O
a	O
Boolean	O
condition	O
involving	O
variables	O
,	O
null	B-Language
values	I-Language
–	O
same	O
as	O
in	O
V-tables	O
.	O
</s>
<s>
Under	O
closed-world	B-Application
assumption	I-Application
,	O
it	O
can	O
also	O
handle	O
the	O
operator	O
of	O
difference	O
,	O
thus	O
it	O
can	O
support	O
all	O
relational	O
operators	O
.	O
</s>
<s>
Imieliński	B-Application
–	I-Application
Lipski	I-Application
algebras	I-Application
were	O
introduced	O
by	O
Tomasz	O
Imieliński	O
and	O
Witold	O
Lipski	O
Jr.	O
in	O
Incomplete	O
Information	O
in	O
Relational	O
Databases	O
.	O
</s>
