ssensitive to the lengths of sequences of defaults involved in deriving certain facts: among two conflicting defaults the one with the lower priority may become applied solely because the sequence of defaults needed for deriving its prerequisite is shorter (Brewka & Eiter, 1998). Approaches to prioritizing defaults that are based on lexicographic comparison (Lifschitz, 1985; Brewka, 1989; Geffner & Pearl, 1992; Ryan, 1992) do not exhibit that kind of behavior.

Lexicographic comparison has properties that are favorable from the point of view of knowledge representation. Extensions of a default theory are possible interpretations of the default theory, representing different ways of resolving the conflicts between default rules. Priorities express the plausibility of different ways of resolving the conflicts, and consequently act as an implicit representation of preferences between the extensions. One useful property of lexicographic comparison is that every finite default theory has at least one preferred extension whenever it has at least one Reiter extension. It does not seem plausible that the priorities could contain information that indicates that none of the extensions is a plausible meaning of the default theory. Another useful property is that every extension of a default theory is a P-preferred 1 extension for a suitably chosen P. In other words, the way priorities are used in ranking the extensions should not per se rule out the possibility that a certain extension is preferred or that it is the unique preferred extension.

A distinguishing difference between the lexicographic prioritized default logic and other work on priorities in default logic is that the highest priority default   if there is one applied in all preferred 1 extensions if there is an extension where it is applied. Preferred B
and preferred BH extensions do not always apply the applicable highest priority defaults.

Example 5.1 (Brewka, 1994) Consider the default theory ∆ =
and l

$\frac{1}{2}$  . 
. Define the relation P
$\mathrm{W}$ II. 
where According to Definitions 4.2 and 4.3 the sets ¡ are as follows, and the unique P -preferred B
and P -preferred BH extension of ∆ is    = U i > 1 Cn (  i ).

$$\begin{array}{r l}{0}&{{}={}}\end{array}$$
$$\begin{array}{r l}{1}&{{}={}}\end{array}$$
2 for all   
1. In other words, initially the highest priority default :
 is not applicable, and hence the second default   :
is applied, and - is obtained. The highest priority default is still not applicable, and hence the third default  :
is applied, and   is obtained. Now  :
were applicable if the contradicting default   :
would not have been applied first. 

□
The application of the highest priority default whenever possible would seem a useful declarative property for nonmonotonic reasoning with priorities. The satisfaction of this property leads to lexicographic prioritization. Also, the behavior of lexicographic prioritized default logic is more consistent for default theories with normal defaults