<s>
Circumscription	B-Application
is	O
a	O
non-monotonic	O
logic	O
created	O
by	O
John	O
McCarthy	O
to	O
formalize	O
the	O
common	O
sense	O
assumption	O
that	O
things	O
are	O
as	O
expected	O
unless	O
otherwise	O
specified	O
.	O
</s>
<s>
Circumscription	B-Application
was	O
later	O
used	O
by	O
McCarthy	O
in	O
an	O
attempt	O
to	O
solve	O
the	O
frame	B-Application
problem	I-Application
.	O
</s>
<s>
To	O
implement	O
circumscription	B-Application
in	O
its	O
initial	O
formulation	O
,	O
McCarthy	O
augmented	O
first-order	O
logic	O
to	O
allow	O
the	O
minimization	O
of	O
the	O
extension	O
of	O
some	O
predicates	B-Algorithm
,	O
where	O
the	O
extension	O
of	O
a	O
predicate	B-Algorithm
is	O
the	O
set	O
of	O
tuples	O
of	O
values	O
the	O
predicate	B-Algorithm
is	O
true	O
on	O
.	O
</s>
<s>
This	O
minimization	O
is	O
similar	O
to	O
the	O
closed-world	B-Application
assumption	I-Application
that	O
what	O
is	O
not	O
known	O
to	O
be	O
true	O
is	O
false	O
.	O
</s>
<s>
Circumscription	B-Application
was	O
later	O
used	O
by	O
McCarthy	O
to	O
formalize	O
the	O
implicit	O
assumption	O
of	O
inertia	O
:	O
things	O
do	O
not	O
change	O
unless	O
otherwise	O
specified	O
.	O
</s>
<s>
Circumscription	B-Application
seemed	O
to	O
be	O
useful	O
to	O
avoid	O
specifying	O
that	O
conditions	O
are	O
not	O
changed	O
by	O
all	O
actions	O
except	O
those	O
explicitly	O
known	O
to	O
change	O
them	O
;	O
this	O
is	O
known	O
as	O
the	O
frame	B-Application
problem	I-Application
.	O
</s>
<s>
Other	O
solutions	O
to	O
the	O
frame	B-Application
problem	I-Application
that	O
correctly	O
formalize	O
the	O
Yale	O
shooting	O
problem	O
exist	O
;	O
some	O
use	O
circumscription	B-Application
but	O
in	O
a	O
different	O
way	O
.	O
</s>
<s>
While	O
circumscription	B-Application
was	O
initially	O
defined	O
in	O
the	O
first-order	O
logic	O
case	O
,	O
the	O
particularization	O
to	O
the	O
propositional	O
case	O
is	O
easier	O
to	O
define	O
.	O
</s>
<s>
Given	O
a	O
propositional	O
formula	O
,	O
its	O
circumscription	B-Application
is	O
the	O
formula	O
having	O
only	O
the	O
models	O
of	O
that	O
do	O
not	O
assign	O
a	O
variable	O
to	O
true	O
unless	O
necessary	O
.	O
</s>
<s>
Circumscription	B-Application
is	O
expressed	O
by	O
selecting	O
only	O
the	O
minimal	O
models	O
.	O
</s>
<s>
Intuitively	O
,	O
in	O
circumscription	B-Application
a	O
variable	O
is	O
assigned	O
to	O
true	O
only	O
if	O
this	O
is	O
necessary	O
.	O
</s>
<s>
For	O
example	O
,	O
at	O
least	O
one	O
of	O
and	O
must	O
be	O
assigned	O
to	O
true	O
according	O
to	O
;	O
in	O
the	O
circumscription	B-Application
exactly	O
one	O
of	O
the	O
two	O
variables	O
must	O
be	O
true	O
.	O
</s>
<s>
The	O
variable	O
cannot	O
be	O
false	O
in	O
any	O
model	O
of	O
and	O
neither	O
of	O
the	O
circumscription	B-Application
.	O
</s>
<s>
The	O
extension	O
of	O
circumscription	B-Application
with	O
fixed	O
and	O
varying	O
predicates	B-Algorithm
is	O
due	O
to	O
Vladimir	O
Lifschitz	O
.	O
</s>
<s>
Formally	O
,	O
the	O
extension	O
of	O
circumscription	B-Application
that	O
incorporate	O
varying	O
and	O
fixed	O
variables	O
is	O
as	O
follows	O
,	O
where	O
is	O
the	O
set	O
of	O
variables	O
to	O
minimize	O
,	O
the	O
fixed	O
variables	O
,	O
and	O
the	O
varying	O
variables	O
are	O
those	O
not	O
in	O
:	O
</s>
<s>
The	O
solution	O
to	O
the	O
frame	B-Application
problem	I-Application
proposed	O
by	O
McCarthy	O
is	O
based	O
on	O
circumscription	B-Application
with	O
no	O
fixed	O
conditions	O
.	O
</s>
<s>
For	O
example	O
,	O
of	O
the	O
domain	B-Algorithm
in	O
which	O
there	O
is	O
a	O
door	O
that	O
is	O
closed	O
at	O
time	O
0	O
and	O
in	O
which	O
the	O
action	O
of	O
opening	O
the	O
door	O
is	O
executed	O
at	O
time	O
2	O
,	O
what	O
is	O
explicitly	O
known	O
is	O
represented	O
by	O
the	O
two	O
formulae	O
:	O
</s>
<s>
The	O
frame	B-Application
problem	I-Application
shows	O
in	O
this	O
example	O
as	O
the	O
problem	O
that	O
is	O
not	O
a	O
consequence	O
of	O
the	O
above	O
formulae	O
,	O
while	O
the	O
door	O
is	O
supposed	O
to	O
stay	O
closed	O
until	O
the	O
action	O
of	O
opening	O
it	O
is	O
performed	O
.	O
</s>
<s>
Circumscription	B-Application
can	O
be	O
used	O
to	O
this	O
aim	O
by	O
defining	O
new	O
variables	O
to	O
model	O
changes	O
and	O
then	O
minimizing	O
them	O
:	O
</s>
<s>
For	O
example	O
,	O
is	O
not	O
yet	O
entailed	O
by	O
the	O
circumscription	B-Application
of	O
the	O
formulae	O
above	O
:	O
the	O
model	O
in	O
which	O
is	O
true	O
and	O
is	O
false	O
is	O
incomparable	O
with	O
the	O
model	O
with	O
the	O
opposite	O
values	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
situation	O
in	O
which	O
the	O
door	O
becomes	O
open	O
at	O
time	O
1	O
and	O
then	O
remains	O
open	O
as	O
a	O
consequence	O
of	O
the	O
action	O
is	O
not	O
excluded	O
by	O
circumscription	B-Application
.	O
</s>
<s>
Several	O
other	O
formalizations	O
of	O
dynamical	O
domains	O
not	O
suffering	O
from	O
such	O
problems	O
have	O
been	O
developed	O
(	O
see	O
frame	B-Application
problem	I-Application
for	O
an	O
overview	O
)	O
.	O
</s>
<s>
Many	O
use	O
circumscription	B-Application
but	O
in	O
a	O
different	O
way	O
.	O
</s>
<s>
The	O
original	O
definition	O
of	O
circumscription	B-Application
proposed	O
by	O
McCarthy	O
is	O
about	O
first-order	O
logic	O
.	O
</s>
<s>
The	O
role	O
of	O
variables	O
in	O
propositional	O
logic	O
(	O
something	O
that	O
can	O
be	O
true	O
or	O
false	O
)	O
is	O
played	O
in	O
first-order	O
logic	O
by	O
predicates	B-Algorithm
.	O
</s>
<s>
Namely	O
,	O
a	O
propositional	O
formula	O
can	O
be	O
expressed	O
in	O
first-order	O
logic	O
by	O
replacing	O
each	O
propositional	O
variable	O
with	O
a	O
predicate	B-Algorithm
of	O
zero	O
arity	O
(	O
i.e.	O
,	O
a	O
predicate	B-Algorithm
with	O
no	O
arguments	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
minimization	O
is	O
done	O
on	O
predicates	B-Algorithm
in	O
the	O
first-order	O
logic	O
version	O
of	O
circumscription	B-Application
:	O
the	O
circumscription	B-Application
of	O
a	O
formula	O
is	O
obtained	O
forcing	O
predicates	B-Algorithm
to	O
be	O
false	O
whenever	O
possible	O
.	O
</s>
<s>
Given	O
a	O
first-order	O
logic	O
formula	O
containing	O
a	O
predicate	B-Algorithm
,	O
circumscribing	O
this	O
predicate	B-Algorithm
amounts	O
to	O
selecting	O
only	O
the	O
models	O
of	O
in	O
which	O
is	O
assigned	O
to	O
true	O
on	O
a	O
minimal	O
set	O
of	O
tuples	O
of	O
values	O
.	O
</s>
<s>
Formally	O
,	O
the	O
extension	O
of	O
a	O
predicate	B-Algorithm
in	O
a	O
first-order	O
model	O
is	O
the	O
set	O
of	O
tuples	O
of	O
values	O
this	O
predicate	B-Algorithm
assign	O
to	O
true	O
in	O
the	O
model	O
.	O
</s>
<s>
First-order	O
models	O
indeed	O
includes	O
the	O
evaluation	O
of	O
each	O
predicate	B-Algorithm
symbol	O
;	O
such	O
an	O
evaluation	O
tells	O
whether	O
the	O
predicate	B-Algorithm
is	O
true	O
or	O
false	O
for	O
any	O
possible	O
value	O
of	O
its	O
arguments	O
.	O
</s>
<s>
Since	O
each	O
argument	O
of	O
a	O
predicate	B-Algorithm
must	O
be	O
a	O
term	O
,	O
and	O
each	O
term	O
evaluates	O
to	O
a	O
value	O
,	O
the	O
models	O
tells	O
whether	O
is	O
true	O
for	O
any	O
possible	O
tuple	O
of	O
values	O
.	O
</s>
<s>
The	O
circumscription	B-Application
of	O
a	O
predicate	B-Algorithm
in	O
a	O
formula	O
is	O
obtained	O
by	O
selecting	O
only	O
the	O
models	O
of	O
with	O
a	O
minimal	O
extension	O
of	O
.	O
</s>
<s>
Given	O
a	O
formula	O
and	O
a	O
predicate	B-Algorithm
,	O
circumscribing	O
in	O
is	O
the	O
following	O
second-order	O
formula	O
:	O
</s>
<s>
In	O
this	O
formula	O
is	O
a	O
predicate	B-Algorithm
of	O
the	O
same	O
arity	O
as	O
.	O
</s>
<s>
This	O
is	O
a	O
second-order	O
formula	O
because	O
it	O
contains	O
a	O
quantification	O
over	O
a	O
predicate	B-Algorithm
.	O
</s>
<s>
This	O
formula	O
states	O
that	O
extension	O
minimization	O
has	O
to	O
be	O
done	O
:	O
in	O
order	O
for	O
a	O
truth	O
evaluation	O
on	O
of	O
a	O
model	O
being	O
considered	O
,	O
it	O
must	O
be	O
the	O
case	O
that	O
no	O
other	O
predicate	B-Algorithm
can	O
assign	O
to	O
false	O
every	O
tuple	O
that	O
assigns	O
to	O
false	O
and	O
yet	O
being	O
different	O
from	O
.	O
</s>
<s>
This	O
definition	O
only	O
allows	O
circumscribing	O
a	O
single	O
predicate	B-Algorithm
.	O
</s>
<s>
While	O
the	O
extension	O
to	O
more	O
than	O
one	O
predicate	B-Algorithm
is	O
trivial	O
,	O
minimizing	O
the	O
extension	O
of	O
a	O
single	O
predicate	B-Algorithm
has	O
an	O
important	O
application	O
:	O
capturing	O
the	O
idea	O
that	O
things	O
are	O
usually	O
as	O
expected	O
.	O
</s>
<s>
This	O
idea	O
can	O
be	O
formalized	O
by	O
minimizing	O
a	O
single	O
predicate	B-Algorithm
expressing	O
the	O
abnormality	O
of	O
situations	O
.	O
</s>
<s>
Minimizing	O
the	O
extension	O
of	O
this	O
predicate	B-Algorithm
allows	O
for	O
reasoning	O
under	O
the	O
implicit	O
assumption	O
that	O
things	O
are	O
as	O
expected	O
(	O
that	O
is	O
,	O
they	O
are	O
not	O
abnormal	O
)	O
,	O
and	O
that	O
this	O
assumption	O
is	O
made	O
only	O
if	O
possible	O
(	O
abnormality	O
can	O
be	O
assumed	O
false	O
only	O
if	O
this	O
is	O
consistent	O
with	O
the	O
facts	O
.	O
)	O
</s>
<s>
Pointwise	O
circumscription	B-Application
is	O
a	O
variant	O
of	O
first-order	O
circumscription	B-Application
that	O
has	O
been	O
introduced	O
by	O
Vladimir	O
Lifschitz	O
.	O
</s>
<s>
In	O
the	O
propositional	O
case	O
,	O
pointwise	O
and	O
predicate	B-Algorithm
circumscription	B-Application
coincide	O
.	O
</s>
<s>
The	O
rationale	O
of	O
pointwise	O
circumscription	B-Application
is	O
that	O
it	O
minimizes	O
the	O
value	O
of	O
a	O
predicate	B-Algorithm
for	O
each	O
tuple	O
of	O
values	O
separately	O
,	O
rather	O
than	O
minimizing	O
the	O
extension	O
of	O
the	O
predicate	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
there	O
are	O
two	O
models	O
of	O
with	O
domain	B-Algorithm
,	O
one	O
setting	O
and	O
the	O
other	O
setting	O
.	O
</s>
<s>
Since	O
the	O
extension	O
of	O
in	O
the	O
first	O
model	O
is	O
while	O
the	O
extension	O
for	O
the	O
second	O
is	O
,	O
circumscription	B-Application
only	O
selects	O
the	O
first	O
model	O
.	O
</s>
<s>
In	O
pointwise	O
circumscription	B-Application
,	O
each	O
tuple	O
of	O
values	O
is	O
considered	O
separately	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
model	O
in	O
which	O
is	O
selected	O
by	O
pointwise	O
circumscription	B-Application
because	O
turning	O
only	O
into	O
false	O
does	O
not	O
satisfy	O
the	O
formula	O
,	O
and	O
the	O
same	O
happens	O
for	O
.	O
</s>
<s>
An	O
earlier	O
formulation	O
of	O
circumscription	B-Application
by	O
McCarthy	O
is	O
based	O
on	O
minimizing	O
the	O
domain	B-Algorithm
of	O
first-order	O
models	O
,	O
rather	O
than	O
the	O
extension	O
of	O
predicates	B-Algorithm
.	O
</s>
<s>
Namely	O
,	O
a	O
model	O
is	O
considered	O
less	O
than	O
another	O
if	O
it	O
has	O
a	O
smaller	O
domain	B-Algorithm
and	O
the	O
two	O
models	O
coincide	O
on	O
the	O
evaluation	O
of	O
the	O
common	O
tuples	O
of	O
values	O
.	O
</s>
<s>
This	O
version	O
of	O
circumscription	B-Application
can	O
be	O
reduced	O
to	O
predicate	B-Algorithm
circumscription	B-Application
.	O
</s>
<s>
Formula	O
circumscription	B-Application
was	O
a	O
later	O
formalism	O
introduced	O
by	O
McCarthy	O
.	O
</s>
<s>
This	O
is	O
a	O
generalization	O
of	O
circumscription	B-Application
in	O
which	O
the	O
extension	O
of	O
a	O
formula	O
is	O
minimized	O
,	O
rather	O
than	O
the	O
extension	O
of	O
a	O
predicate	B-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
formula	O
can	O
be	O
specified	O
so	O
that	O
the	O
set	O
of	O
tuples	O
of	O
values	O
of	O
the	O
domain	B-Algorithm
that	O
satisfy	O
the	O
formula	O
is	O
made	O
as	O
small	O
as	O
possible	O
.	O
</s>
<s>
Circumscription	B-Application
does	O
not	O
always	O
correctly	O
handle	O
disjunctive	O
information	O
.	O
</s>
<s>
Circumscription	B-Application
can	O
therefore	O
be	O
used	O
to	O
minimize	O
the	O
extension	O
of	O
predicate	B-Algorithm
,	O
so	O
that	O
is	O
false	O
even	O
if	O
this	O
is	O
not	O
explicitly	O
stated	O
.	O
</s>
<s>
The	O
idea	O
is	O
that	O
the	O
model	O
that	O
circumscription	B-Application
fails	O
to	O
select	O
,	O
the	O
one	O
in	O
which	O
both	O
and	O
are	O
true	O
,	O
is	O
a	O
model	O
of	O
the	O
formula	O
that	O
is	O
greater	O
(	O
w.r.t.	O
</s>
<s>
Theory	O
curbing	O
selects	O
such	O
least	O
upper	O
bounds	O
models	O
in	O
addition	O
to	O
the	O
ones	O
selected	O
by	O
circumscription	B-Application
.	O
</s>
