Appropriateness

Actual systems using type feature structures, such as ALE [Carpenter and Penn2001], require a strict specification of the possible types a feature value can have and of the possible types a feature can be defined on. This strict specification is known as an appropriateness specification. Appropriateness was first proposed by Pollard and Sag pollard-sag-syntax in order to differentiate between cases where there is a lack of information about the value of a feature, and cases where that feature is irrelevant. The formal definition of appropriateness is given in [Carpenter1992]:

Definition 2.15   An appropriateness specification over the type hierarchy $\langle Type,\sqsubseteq\rangle$ and features Feat is a partial function $Approp:Feat\times Type\rightarrow Type$ such that:

• $\forall F\in Feat$, $\exists$ a most general type $Intro(F)$ for which $Approp(F,Intro(F))$ is defined (feature introduction),
• if $Approp(F,\sigma)$ is defined and $\sigma\sqsubseteq\tau$, then $Approp(F,\tau)$ is also defined and $Approp(F,\sigma)\sqsubseteq Approp(F,\tau)$ (upward closure).

The appropriateness specification cannot be treated independently of the type hierarchy, and their specification together is known as the type signature [Penn2000]:

Definition 2.16   A type signature is a structure $\langle Type,\sqsubseteq,Feat,Approp\rangle$, where $\langle Type,\sqsubseteq\rangle$ is a type hierarchy, Feat is a finite set of features, and Approp is an appropriateness specification.