Default Logic - Variants of Default Logic

Variants of Default Logic

The following variants of default logic differ from the original one on both syntax and semantics.

Assertional variants

An assertion is a pair composed of a formula and a set of formulae. Such a pair indicates that is true while the formulae have been assumed consistent to prove that is true. An assertional default theory is composed of an assertional theory (a set of assertional formulae) called the background theory and a set of defaults defined as in the original syntax. Whenever a default is applied to an assertional theory, the pair composed of its consequence and its set of justifications is added to the theory. The following semantics use assertional theories:

• Cumulative default logic
• Commitment to assumptions default logic
• Quasi-default logic

Weak extensions

rather than checking whether the preconditions are valid in the theory composed of the background theory and the consequences of the applied defaults, the preconditions are checked for validity in the extension that will be generated; in other words, the algorithm for generating extensions starts by guessing a theory and using it in place of the background theory; what results from the process of extension generation is actually an extension only if it is equivalent to the theory guessed at the beginning. This variant of default logic is related in principle to autoepistemic logic, where a theory has the model in which is true just because, assuming true, the formula supports the initial assumption.

Disjunctive default logic

the consequence of a default is a set of formulae instead of a single formula. Whenever the default is applied, at least one of its consequences is nondeterministically chosen and made true.

Priorities on defaults

the relative priority of defaults can be explicitly specified; among the defaults that are applicable to a theory, only one of the most preferred ones can be applied. Some semantics of default logic do not require priorities to be explicitly specified; rather, more specific defaults (those that are applicable in fewer cases) are preferred over less specific ones.

Statistical variant

a statistical default is a default with an attached upper bound on its frequency of error; in other words, the default is assumed to be an incorrect inference rule in at most that fraction of times it is applied.