Assume that       2 and   = : ' ' for some and appl( :  '   '  ). Because  ' is consistent,  ' /  ' and hence appl( :  '  '  '  '  '  By definition  :  '   'P: '  '. Assume that 2 and 2 and 2 for 'for some 'for some 'for some 'for appl(: 'for Because not appl( :  ). Hence By definition :  Assume that   =   :      . Clearly   /   , appl(:  for   =  ':      is similar. Assume that = : ' '. Hence appl( : ' ' '). By definition : ' ' 'T: ' '. Proof for =   :    '     ' is similar. Assume that = ': false false. Hence appl(: ' ' ' ) and appl(: ' ' ' ' '). By definition :  '   'T  : false. This exhausts all 
preferred L extension of
(f) Assume that    is a P-preferred 2 extension of A such that false /   . We show that there is a model    such that    =   . Define the model as     =    iff
, for
. Let   = 
Because false /   , not appl( ': false false   ). Because   false is not in any conclusion of a default in
( 
Hence 
' where OT
If = 1, then ' / because of the following. Assume that ' Let 12 =
,        )\   '       ' ). It is easy to show that Cn( )
12 is an extension of 1 2 2 2 2 . By Theorem 3.2 in (Reiter, 1980) there is an extension
' of           such that GD(   12     1      2    )     GD(   '         ). Now appl(  :    ' 
C 
and there is no such that appl( T such that P T . This contradicts the P -preferredness L of T , and hence it must be the case that  ' /   . Hence by definition    =   . If   =  ', then clearly  '     . Hence by definition    =    . Therefore    =  . Because this holds for any clause
=

## 6. Related Work On Prioritized Default Reasoning

Marek and Truszczynski (1993) introduce a prioritized default logic that is similar to Brewka's (1994) logic. The existence of preferred extensions is not guaranteed in general, but for normal defaults it is. We believe that for normal default theories, the complexity of the Marek and Truszczyński logic coincides with the complexity of Brewka's logic.

Delgrande and Schaub (1997) present a translation from prioritized default theories to unprioritized default theories, so that the extensions of the resulting theories obey the priorities.

Their translation can be performed in polynomial time. Buccafurri et al. (1998) present a knowledge representation language that extends logic programs with priorities, classical negation, and disjunction. They claim that brave reasoning for their language is in general
..

Brewka and Eiter (1998) present the notion of preferred answer sets for extended logic programs. Their definition diverges from earlier work in that even with a total ordering on the rules there may be more than one preferred answer set, and preferred answer sets