Translating default theories to normal default theories

Normal default theories have many properties exclusively such as semi-monotonicity and existence of extensions, etc, and they have precisely denned proof systems and model theories. Hence, a translation of default theories to normal default theories is necessary in order to make use of the proof systems and model theories of normal default theories to study non-normal default theories. A default is translated to a normal default such that its prerequisite and justification are the prerequisites of the normal default; its consequence is the justification of the normal default. To draw the consequence of a default, the prerequisite of the default is provable and the negation of the justification of the default is not provable, however, to draw the consequence of its normal default, the prerequisite and justification of the default are provable and the negation of the consequence of the default is not provable. Thus a consequence which is derivable from a default may not derivable from its normal default. Hence, in the translation of a default theory to a normal default theory, there is information loss. Therefore, closed world assumption and a priority order on defaults are introduced to study the problem of information loss in the translation. It is proved that under closed world assumption and the priority order on defaults, the translation of default theories with atomic negative defaults to its normal default theories is complete, if they have extensions.