If , as in any of the cases illustrated above, then the subsumed is redundant if all its AV pairs are completely specified. Consequently, the information in subsumer may be subtracted from , leaving a non-redundant set of AV specifications, and a redundancy rule can be formulated which will allow the `missing' features to be inferred or `added in'. This is standard procedure in the rule notation of generative phonology and morphology:
|The phonological redundancy rule:||/ #|
|expands conventionally to:||#||#|
|or, in terms of subsumption:||#||#|
The subtraction operation between a subsumed AVM and a subsumer AVM yields a non-redundant AVM in an inheritance relation with AVM. The inheritance relation whereby AVM inherits the features of AVM, and thereby reconstitutes AVM, s the inverse of the subtraction operation, and is expressed as a special case of unification: AVM = AVM AVM, where AVM AVM = 0. The generalisation (feature intersection) operator `' is defined as the set of features shared by AVM and AVM and and the specialisation (unification) operator `' is defined recursively for compatible AVMs: two attribute-value pairs unify either if the values are identical atoms, or if an attribute in one AVM is not specified in the other, or if the values of identical attributes in the AVMs unify. Under the type inheritance operation expressed by unification, the AVMs in Figure 2 and the AVMs in Figure 1 are equivalent. The elementary case of non-recursive unification has been familiar in linguistics since the introduction of the lexical insertion operation by Chomsky [Chomsky 1965]; Shieber [Shieber 1986] summarises the more general unification operation used in unification grammars.
In the DATR formalism, a form of default inheritance is defined, in which the subsumption relation and the unification operation do not hold. Instead, there is a default-override relation and a default unification operation. In the default-override relation, a value for a given attribute may be specified more than once in the same inheritance path, and the specification of the lower (more specific) class overrides the specification of the higher (more general) class. In a famous illustration, Tweety, qua penguin cannot fly, but Tweety, qua bird can fly. Clearly, the penguin specification is more specific than the bird specification, therefore the dispositional predicate `cannot fly' overrides the dispositional predicate `can fly'.
In the ILEX version of Inheritance Lexicon Theory, default inheritance is used in order to explain exceptions and subregularities of this kind.