<s>
Codd	B-Application
's	I-Application
theorem	I-Application
states	O
that	O
relational	B-Algorithm
algebra	I-Algorithm
and	O
the	O
domain-independent	O
relational	B-Application
calculus	I-Application
queries	O
,	O
two	O
well-known	O
foundational	O
query	O
languages	O
for	O
the	O
relational	B-Architecture
model	I-Architecture
,	O
are	O
precisely	O
equivalent	O
in	O
expressive	B-Language
power	I-Language
.	O
</s>
<s>
The	O
theorem	O
is	O
named	O
after	O
Edgar	O
F	O
.	O
Codd	O
,	O
the	O
father	O
of	O
the	O
relational	B-Architecture
model	I-Architecture
for	O
database	O
management	O
.	O
</s>
<s>
The	O
domain	O
independent	O
relational	B-Application
calculus	I-Application
queries	O
are	O
precisely	O
those	O
relational	B-Application
calculus	I-Application
queries	O
that	O
are	O
invariant	O
under	O
choosing	O
domains	O
of	O
values	O
beyond	O
those	O
appearing	O
in	O
the	O
database	O
itself	O
.	O
</s>
<s>
Codd	B-Application
's	I-Application
Theorem	I-Application
is	O
notable	O
since	O
it	O
establishes	O
the	O
equivalence	O
of	O
two	O
syntactically	O
quite	O
dissimilar	O
languages	O
:	O
relational	B-Algorithm
algebra	I-Algorithm
is	O
a	O
variable-free	O
language	O
,	O
while	O
relational	B-Application
calculus	I-Application
is	O
a	O
logical	O
language	O
with	O
variables	O
and	O
quantification	O
.	O
</s>
<s>
Relational	B-Application
calculus	I-Application
is	O
essentially	O
equivalent	O
to	O
first-order	O
logic	O
,	O
and	O
indeed	O
,	O
Codd	B-Application
's	I-Application
Theorem	I-Application
had	O
been	O
known	O
to	O
logicians	O
since	O
the	O
late	O
1940s	O
.	O
</s>
<s>
Query	O
languages	O
that	O
are	O
equivalent	O
in	O
expressive	B-Language
power	I-Language
to	O
relational	B-Algorithm
algebra	I-Algorithm
were	O
called	O
relationally	O
complete	O
by	O
Codd	O
.	O
</s>
<s>
By	O
Codd	B-Application
's	I-Application
Theorem	I-Application
,	O
this	O
includes	O
relational	B-Application
calculus	I-Application
.	O
</s>
<s>
Well-known	O
examples	O
of	O
inexpressible	O
queries	O
include	O
simple	O
aggregations	O
(	O
counting	O
tuples	O
,	O
or	O
summing	O
up	O
values	O
occurring	O
in	O
tuples	O
,	O
which	O
are	O
operations	O
expressible	O
in	O
SQL	O
but	O
not	O
in	O
relational	B-Algorithm
algebra	I-Algorithm
)	O
and	O
computing	O
the	O
transitive	O
closure	O
of	O
a	O
graph	O
given	O
by	O
its	O
binary	O
edge	O
relation	O
(	O
see	O
also	O
expressive	B-Language
power	I-Language
)	O
.	O
</s>
<s>
Codd	B-Application
's	I-Application
theorem	I-Application
also	O
does	O
n't	O
consider	O
SQL	B-Language
nulls	I-Language
and	O
the	O
three-valued	B-Language
logic	I-Language
they	O
entail	O
;	O
the	O
logical	O
treatment	O
of	O
nulls	B-Language
remains	O
mired	O
in	O
controversy	O
.	O
</s>
<s>
Additionally	O
,	O
SQL	O
has	O
multiset	B-Language
semantics	O
and	O
allows	O
duplicate	O
rows	O
.	O
</s>
<s>
Nevertheless	O
,	O
relational	O
completeness	O
constitutes	O
an	O
important	O
yardstick	O
by	O
which	O
the	O
expressive	B-Language
power	I-Language
of	O
query	O
languages	O
can	O
be	O
compared	O
.	O
</s>
