Editing Link grammar (section)

=== Type theory ===
The link grammar link types can be understood to be the types in the sense of [[type theory]].<ref name="intro"/><ref>{{cite conference |author1=Daniel Sleator |author2=Davey Temperley |title=Parsing English with a Link Grammar |conference=Third International Workshop on Parsing Technologies |year=1993 |url=https://www.cs.cmu.edu/afs/cs.cmu.edu/project/link/pub/www/papers/ps/LG-IWPT93.pdf}} (See section 6 on categorial grammar).</ref>  In effect, the link grammar can be used to model the [[internal language]] of certain (non-symmetric) [[compact closed category|compact closed categories]], such as [[pregroup grammar]]s.  In this sense, link grammar appears to be isomorphic or homomorphic to some [[categorial grammar]]s.  Thus, for example, in a categorial grammar the noun phrase "''the bad boy''" may be written as

:<math>
{\text{the} \atop \text{NP/N,}}
{\text{bad} \atop \text{N/N,}}
{\text{boy} \atop \text{N}}
</math>

whereas the corresponding disjuncts in link grammar would be

 the: D+;
 bad: A+;
 boy: D- & A-;

The contraction rules (inference rules) of the [[Categorial_grammar#Lambek_calculus|Lambek calculus]] can be mapped to the connecting of connectors in link grammar. The '''+''' and '''-''' directional indicators correspond the forward and backward-slashes of the categorical grammar.  Finally, the single-letter names '''A''' and '''D''' can be understood as labels or "easy-to-read" mnemonic names for the rather more verbose types ''NP/N'', etc.

The primary distinction here  is then that the categorical grammars have two [[type constructor]]s, the forward and backward slashes, that can be used to create new types (such as ''NP/N'') from base types (such as ''NP'' and ''N''). Link-grammar omits the use of type constructors, opting instead to define a much larger set of base types having compact, easy-to-remember mnemonics.