By Frank Richter
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Additional info for A Mathematical Formalism for Linguistic Theories with an Application in Head-Driven Phrase Structure Grammar
Every piece of partial information that is true of the language is said to model or to represent the language. Feature structures are located in a Heyting algebra according to their degree of informativeness. The operations of the algebra function as a recipe for constructing the total information that speakers have about their language from the set of feature structures specified by the grammar. 6 Occasionally, I adopt a reading of Pollard and Sag 1987 that is slightly closer to HPSG 94 than the more literal reading of King, and I change some minor technical details of the sketch of a formalism of Pollard and Sag where subsequent research showed that the technical choices of the authors had been erroneous or misled.
The basic idea is that 26 CHAPTER 2. FORMAL FOUNDATIONS I feature structures are recursively structured, complex entities. Informally, they consist of a set of nodes that are connected by attributes. There is a distinguished root node from which all other nodes of a feature structure can be reached via the attributes. In HPSG, feature structures are sorted: The root node of every feature structure has a sort label, as do all other nodes of a feature structure. The sort of the root node indicates about which kind of entity the feature structure provides information.
For each nonempty path, π, ∆∗ is the iteration of the application of ∆ on the attributes in π and the node that has been reached by the (iterative) application of ∆ to the preceding attributes in π. Extending my previous simplifying terminology, I say that a path, π, is defined on a node, q, if ∆∗ q, π is defined. Alternatively, I sometimes say that a concrete feature structure, = Q, qˆ, ∆, Λ , has a path π, meaning that ∆∗ qˆ, π is defined. The concrete feature structures of HPSG 87 are more restricted than the concrete feature structures of Definition 2.