<s>
A	O
logical	B-Language
spreadsheet	I-Language
is	O
a	O
spreadsheet	B-Application
in	O
which	O
formulas	O
take	O
the	O
form	O
of	O
logical	O
constraints	O
rather	O
than	O
function	O
definitions	O
.	O
</s>
<s>
In	O
traditional	O
spreadsheet	B-Application
systems	O
,	O
such	O
as	O
Excel	B-Application
,	O
cells	O
are	O
partitioned	O
into	O
"	O
directly	O
specified	O
"	O
cells	O
and	O
"	O
computed	O
"	O
cells	O
and	O
the	O
formulas	O
used	O
to	O
specify	O
the	O
values	O
of	O
computed	O
cells	O
are	O
"	O
functional	O
"	O
,	O
i.e.	O
</s>
<s>
Logical	B-Language
Spreadsheets	I-Language
relax	O
these	O
restrictions	O
by	O
dispensing	O
with	O
the	O
distinction	O
between	O
directly	O
specified	O
cells	O
and	O
computed	O
cells	O
and	O
generalizing	O
from	O
functional	O
definitions	O
to	O
logical	O
constraints	O
.	O
</s>
<s>
As	O
an	O
illustration	O
of	O
the	O
difference	O
between	O
traditional	O
spreadsheets	B-Application
and	O
logical	B-Language
spreadsheets	I-Language
,	O
consider	O
a	O
simple	O
numerical	O
spreadsheet	B-Application
with	O
three	O
cells	O
a	O
,	O
b	O
,	O
and	O
c	O
.	O
Each	O
cell	O
accepts	O
a	O
single	O
integer	O
as	O
value	O
;	O
and	O
there	O
is	O
a	O
formula	O
stating	O
that	O
the	O
value	O
of	O
the	O
third	O
cell	O
is	O
the	O
sum	O
of	O
the	O
values	O
of	O
the	O
other	O
two	O
cells	O
.	O
</s>
<s>
Implemented	O
as	O
a	O
traditional	O
spreadsheet	B-Application
,	O
this	O
spreadsheet	B-Application
would	O
allow	O
the	O
user	O
to	O
enter	O
values	O
into	O
cells	O
a	O
and	O
b	O
,	O
and	O
it	O
would	O
automatically	O
compute	O
cell	O
c	O
.	O
For	O
example	O
,	O
if	O
the	O
user	O
were	O
to	O
type	O
1	O
into	O
a	O
and	O
2	O
into	O
b	O
,	O
it	O
would	O
compute	O
the	O
value	O
3	O
for	O
c	O
.	O
</s>
<s>
Implemented	O
as	O
a	O
logical	B-Language
spreadsheet	I-Language
,	O
the	O
user	O
would	O
be	O
able	O
to	O
enter	O
values	O
into	O
any	O
of	O
the	O
cells	O
.	O
</s>
<s>
The	O
user	O
could	O
type	O
1	O
into	O
a	O
and	O
2	O
into	O
b	O
,	O
and	O
the	O
spreadsheet	B-Application
would	O
compute	O
the	O
value	O
3	O
for	O
c	O
.	O
Alternatively	O
,	O
the	O
user	O
could	O
type	O
2	O
into	O
b	O
and	O
3	O
into	O
c	O
,	O
and	O
the	O
spreadsheet	B-Application
would	O
compute	O
the	O
value	O
1	O
for	O
a	O
.	O
</s>
<s>
More	O
generally	O
,	O
we	O
could	O
build	O
spreadsheets	B-Application
with	O
symbolic	O
,	O
rather	O
than	O
numeric	O
data	O
,	O
and	O
write	O
arbitrary	O
logical	O
constraints	O
on	O
this	O
data	O
.	O
</s>
