<s>
Default	B-Application
logic	I-Application
is	O
a	O
non-monotonic	O
logic	O
proposed	O
by	O
Raymond	O
Reiter	O
to	O
formalize	O
reasoning	O
with	O
default	O
assumptions	O
.	O
</s>
<s>
Default	B-Application
logic	I-Application
can	O
express	O
facts	O
like	O
“	O
by	O
default	O
,	O
something	O
is	O
true	O
”	O
;	O
by	O
contrast	O
,	O
standard	O
logic	O
can	O
only	O
express	O
that	O
something	O
is	O
true	O
or	O
that	O
something	O
is	O
false	O
.	O
</s>
<s>
Default	B-Application
logic	I-Application
aims	O
at	O
formalizing	O
inference	O
rules	O
like	O
this	O
one	O
without	O
explicitly	O
mentioning	O
all	O
their	O
exceptions	O
.	O
</s>
<s>
A	O
default	B-Application
theory	I-Application
is	O
a	O
pair	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
the	O
Closed-World	B-Application
Assumption	I-Application
,	O
and	O
is	O
formalized	O
in	O
default	B-Application
logic	I-Application
using	O
a	O
default	O
like	O
the	O
following	O
one	O
for	O
every	O
fact	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
computer	O
language	O
Prolog	B-Language
uses	O
a	O
sort	O
of	O
default	O
assumption	O
when	O
dealing	O
with	O
negation	O
:	O
if	O
a	O
negative	O
atom	O
cannot	O
be	O
proved	O
to	O
be	O
true	O
,	O
then	O
it	O
is	O
assumed	O
to	O
be	O
false	O
.	O
</s>
<s>
Note	O
,	O
however	O
,	O
that	O
Prolog	B-Language
uses	O
the	O
so-called	O
negation	B-Application
as	I-Application
failure	I-Application
:	O
when	O
the	O
interpreter	O
has	O
to	O
evaluate	O
the	O
atom	O
,	O
it	O
tries	O
to	O
prove	O
that	O
is	O
true	O
,	O
and	O
conclude	O
that	O
is	O
true	O
if	O
it	O
fails	O
.	O
</s>
<s>
In	O
default	B-Application
logic	I-Application
,	O
instead	O
,	O
a	O
default	O
having	O
as	O
a	O
justification	O
can	O
only	O
be	O
applied	O
if	O
is	O
consistent	O
with	O
the	O
current	O
knowledge	O
.	O
</s>
<s>
A	O
default	B-Application
theory	I-Application
is	O
called	O
categorical	O
,	O
normal	O
,	O
supernormal	O
,	O
or	O
seminormal	O
if	O
all	O
defaults	O
it	O
contains	O
are	O
categorical	O
,	O
normal	O
,	O
supernormal	O
,	O
or	O
seminormal	O
,	O
respectively	O
.	O
</s>
<s>
When	O
the	O
theory	O
is	O
such	O
that	O
no	O
other	O
default	O
can	O
be	O
applied	O
,	O
the	O
theory	O
is	O
called	O
an	O
extension	O
of	O
the	O
default	B-Application
theory	I-Application
.	O
</s>
<s>
The	O
Nixon	O
diamond	O
example	O
is	O
a	O
default	B-Application
theory	I-Application
with	O
two	O
extensions	O
:	O
</s>
<s>
This	O
particular	O
default	B-Application
theory	I-Application
has	O
therefore	O
two	O
extensions	O
,	O
one	O
in	O
which	O
is	O
true	O
,	O
and	O
one	O
in	O
which	O
is	O
false	O
.	O
</s>
<s>
The	O
original	O
semantics	O
of	O
default	B-Application
logic	I-Application
was	O
based	O
on	O
the	O
fixed	O
point	O
of	O
a	O
function	O
.	O
</s>
<s>
The	O
following	O
default	B-Application
theory	I-Application
has	O
no	O
extension	O
:	O
</s>
<s>
In	O
a	O
normal	O
default	B-Application
theory	I-Application
,	O
all	O
defaults	O
are	O
normal	O
:	O
each	O
default	O
has	O
the	O
form	O
.	O
</s>
<s>
A	O
normal	O
default	B-Application
theory	I-Application
is	O
guaranteed	O
to	O
have	O
at	O
least	O
one	O
extension	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
extensions	O
of	O
a	O
normal	O
default	B-Application
theory	I-Application
are	O
mutually	O
inconsistent	O
,	O
i.e.	O
,	O
inconsistent	O
with	O
each	O
other	O
.	O
</s>
<s>
A	O
default	B-Application
theory	I-Application
can	O
have	O
zero	O
,	O
one	O
,	O
or	O
more	O
extensions	O
.	O
</s>
<s>
Entailment	O
of	O
a	O
formula	O
from	O
a	O
default	B-Application
theory	I-Application
can	O
be	O
defined	O
in	O
two	O
ways	O
:	O
</s>
<s>
Skeptical	O
a	O
formula	O
is	O
entailed	O
by	O
a	O
default	B-Application
theory	I-Application
if	O
it	O
is	O
entailed	O
by	O
all	O
its	O
extensions	O
;	O
</s>
<s>
Credulous	O
a	O
formula	O
is	O
entailed	O
by	O
a	O
default	B-Application
theory	I-Application
if	O
it	O
is	O
entailed	O
by	O
at	O
least	O
one	O
of	O
its	O
extensions	O
.	O
</s>
<s>
As	O
this	O
example	O
shows	O
,	O
the	O
credulous	O
consequences	O
of	O
a	O
default	B-Application
theory	I-Application
may	O
be	O
inconsistent	O
with	O
each	O
other	O
.	O
</s>
<s>
The	O
following	O
alternative	O
inference	O
rules	O
for	O
default	B-Application
logic	I-Application
are	O
all	O
based	O
on	O
the	O
same	O
syntax	O
as	O
the	O
original	O
system	O
.	O
</s>
<s>
Rational	O
similar	O
to	O
constrained	O
default	B-Application
logic	I-Application
,	O
but	O
the	O
consequence	O
of	O
the	O
default	O
to	O
add	O
is	O
not	O
considered	O
in	O
the	O
consistency	O
check	O
;	O
</s>
<s>
The	O
justified	O
and	O
constrained	O
versions	O
of	O
the	O
inference	O
rule	O
assign	O
at	O
least	O
an	O
extension	O
to	O
every	O
default	B-Application
theory	I-Application
.	O
</s>
<s>
The	O
following	O
variants	O
of	O
default	B-Application
logic	I-Application
differ	O
from	O
the	O
original	O
one	O
on	O
both	O
syntax	O
and	O
semantics	O
.	O
</s>
<s>
An	O
assertional	O
default	B-Application
theory	I-Application
is	O
composed	O
of	O
an	O
assertional	O
theory	O
(	O
a	O
set	O
of	O
assertional	O
formulae	O
)	O
called	O
the	O
background	O
theory	O
and	O
a	O
set	O
of	O
defaults	O
defined	O
as	O
in	O
the	O
original	O
syntax	O
.	O
</s>
<s>
This	O
variant	O
of	O
default	B-Application
logic	I-Application
is	O
related	O
in	O
principle	O
to	O
autoepistemic	B-Application
logic	I-Application
,	O
where	O
a	O
theory	O
has	O
the	O
model	O
in	O
which	O
is	O
true	O
just	O
because	O
,	O
assuming	O
true	O
,	O
the	O
formula	O
supports	O
the	O
initial	O
assumption	O
.	O
</s>
<s>
Disjunctive	O
default	B-Application
logic	I-Application
the	O
consequence	O
of	O
a	O
default	O
is	O
a	O
set	O
of	O
formulae	O
instead	O
of	O
a	O
single	O
formula	O
.	O
</s>
<s>
Some	O
semantics	O
of	O
default	B-Application
logic	I-Application
do	O
not	O
require	O
priorities	O
to	O
be	O
explicitly	O
specified	O
;	O
rather	O
,	O
more	O
specific	O
defaults	O
(	O
those	O
that	O
are	O
applicable	O
in	O
fewer	O
cases	O
)	O
are	O
preferred	O
over	O
less	O
specific	O
ones	O
.	O
</s>
<s>
Default	B-Application
theories	I-Application
can	O
be	O
translated	O
into	O
theories	O
in	O
other	O
logics	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Faithful	O
this	O
condition	O
only	O
makes	O
sense	O
when	O
translating	O
between	O
two	O
variants	O
of	O
default	B-Application
logic	I-Application
or	O
between	O
default	B-Application
logic	I-Application
and	O
a	O
logic	O
in	O
which	O
a	O
concept	O
similar	O
to	O
extension	O
exists	O
,	O
e.g.	O
,	O
models	O
in	O
modal	O
logic	O
;	O
a	O
translation	O
is	O
faithful	O
if	O
there	O
exists	O
a	O
mapping	O
(	O
typically	O
,	O
a	O
bijection	O
)	O
between	O
the	O
extensions	O
(	O
or	O
models	O
)	O
of	O
the	O
original	O
and	O
translated	O
theories	O
;	O
</s>
<s>
Modular	O
a	O
translation	O
from	O
default	B-Application
logic	I-Application
to	O
another	O
logic	O
is	O
modular	O
if	O
the	O
defaults	O
and	O
the	O
background	O
theory	O
can	O
be	O
translated	O
separately	O
;	O
moreover	O
,	O
the	O
addition	O
of	O
formulae	O
to	O
the	O
background	O
theory	O
only	O
leads	O
to	O
adding	O
the	O
new	O
formulae	O
to	O
the	O
result	O
of	O
the	O
translation	O
;	O
</s>
<s>
autoepistemic	B-Application
logic	I-Application
;	O
</s>
<s>
propositional	O
default	B-Application
logic	I-Application
restricted	O
to	O
seminormal	O
theories	O
;	O
</s>
<s>
alternative	O
semantics	O
of	O
default	B-Application
logic	I-Application
;	O
</s>
<s>
Translations	O
from	O
propositional	O
default	B-Application
logic	I-Application
to	O
classical	O
propositional	O
logic	O
cannot	O
always	O
generate	O
a	O
polynomially	O
sized	O
propositional	O
theory	O
,	O
unless	O
the	O
polynomial	O
hierarchy	O
collapses	O
.	O
</s>
<s>
Translations	O
to	O
autoepistemic	B-Application
logic	I-Application
exist	O
or	O
not	O
depending	O
on	O
whether	O
modularity	O
or	O
the	O
use	O
of	O
the	O
same	O
alphabet	O
is	O
required	O
.	O
</s>
<s>
The	O
computational	B-General_Concept
complexity	I-General_Concept
of	O
the	O
following	O
problems	O
about	O
default	B-Application
logic	I-Application
is	O
known	O
:	O
</s>
<s>
Existence	O
of	O
extensions	O
deciding	O
whether	O
a	O
propositional	O
default	B-Application
theory	I-Application
has	O
at	O
least	O
one	O
extension	O
is	O
-complete	O
;	O
</s>
<s>
Skeptical	O
entailment	O
deciding	O
whether	O
a	O
propositional	O
default	B-Application
theory	I-Application
skeptically	O
entails	O
a	O
propositional	O
formula	O
is	O
-complete	O
;	O
</s>
<s>
Credulous	O
entailment	O
deciding	O
whether	O
a	O
propositional	O
default	B-Application
theory	I-Application
credulously	O
entails	O
a	O
propositional	O
formula	O
is	O
-complete	O
;	O
</s>
<s>
Extension	O
checking	O
deciding	O
whether	O
a	O
propositional	O
formula	O
is	O
equivalent	O
to	O
an	O
extension	O
of	O
a	O
propositional	O
default	B-Application
theory	I-Application
is	O
-complete	O
;	O
</s>
<s>
Model	O
checking	O
deciding	O
whether	O
a	O
propositional	O
interpretation	O
is	O
a	O
model	O
of	O
an	O
extension	O
of	O
a	O
propositional	O
default	B-Application
theory	I-Application
is	O
-complete	O
.	O
</s>
