_(b) If $i$ and $j$, let $i\ll j$._  _(c) Also, $=1$... m for some $1$. For each $\leq$, $i=i,1$... i,m, where $i,1$. Thus if $i,j$ i,k $1,1$... m,m and $i,j\neq i,k$, let $i,j\not\leq i,k$._  _(d) The following transitivity relations hold for $\ll$ and $\ll$._
$(a)\;\;If\;\;\ll\;\;\;\;and\;\;\;\ll\;\;,\;then\;\;\;\ll\;\;.$  $(b)\;\;If\;\;\ll\;\;\;\;\;and\;\;\;\ll\;\;\;,\;then\;\;\;\ll\;\;.$  $(c)\;\;If\;\;\ll\;\;\;\;\;and\;\;\;\ll\;\;\;or\;\;\;\ll\;\;\;and\;\;\;\ll\;\;,\;then\;\;\;\ll\;\;.$  $The\;default\;theory\;\Delta\;is\;\text{ordered}\;if\;and\;only\;if\;\;\ll\;\;\;for\;no\;\;\;.$  . 
According to Theorem 1 in (Etherington, 1987), ordered default theories, like normal default theories and unlike seminormal default theories in general, have at least one extension.

## 4. Prioritized Default Logics By Brewka And By Baader And Hollunder

Priorities in default logic have been investigated by Baader and Hollunder (1995) and Brewka (1994). They view priorities as information that selects which defaults are applied next when constructing an extension. Priorities are strict partial orders, that is, transitive and asymmetric relations P on the defaults. If  P   then the application of is more desirable than the application of  ', and the default   is more significant or has a higher priority. Brewka, as well as Baader and Hollunder, give a definition of preferred extensions by modifying the semiconstructive definition of extensions (Theorem 3.2.) The construction of an extension starts from the set     of objective facts, and proceeds in stages by repeatedly applying the highest priority applicable defaults. In this section we analyze the complexity of the associated decision problems.

  **Definition 4.1**: _A default : is active in is active in if = and $\neq$ and $\neq$._
D 
be a default theory and P
be a set of formulae.  Define for all a strict partial order on   . Let

0 no  P —— is active in l i + 1 is a P-preferredBH extension of if and only if    = U i>o Cn(  i)
Then Brewka's definition is syntactically similar. We have expressed it in a way that highlights the differences to Baader and Hollunder's definition. The main difference is that the construction of a preferred extension proceeds by applying defaults in an order specified by some strict total order that extends the priorities. Furthermore, Brewka gives his definition for normal default theories only, and the consistency of justifications is tested against the sets instead of the set