Graph Properties of HPSG Feature Structures

A general principle for grammars for languages is that they must use finite means to produce the infinitely many structures of the languages. When the term production is understood algebraically, there are at least two notions of finiteness involved. Firstly in each set of structures each structure is generated, i.e., the result of finitely many applications of operators. Secondly, the set of operators as such is also finite. For algebraic theories, this is eo ipso the case. But there are linguistic frameworks that are not generating but rather licensing linguistic structures such as Head-Driven Phrase Structure Grammar or GB-Theory. Within these, structures are not generated. Rather only general properties of admissible structures are stated. But these structures must be provided somehow. Thus one may ask if there are algebraic approaches to licensing theories. We show here that HPSG is non-algebraic in the following sense. There is no finite family of operations such that the members of every finite set of feature structures can be constructed by this finite family of operations. We also show that HPSG feature structures are abstractly recognisable in the sense that for every set of finite HPSG feature structures there exists a (manysorted) algebra with finite sort sets of infinitely many operations such that every feature structure in that set is recognised by the algebra (seen as an abstract automaton). In other words, the first finiteness demand can be fulfilled, the second one cannot.