Strong Generative Capacity in HPSG

1
Strong Generative Capacity in HPSG

• SRL A language for the description of formal
  grammars

2
Definitions

• Formal Grammar A mathematical idealization of
  the mental grammar. A scientific theory which
  predicts tat an utterance token will be judged
  grammatical if and only if it is a model of the
  grammar.
• SGC The set of canonical representatives of the
  isomorphism classes of singly generated models of
  the grammar.
• SRL A particular logic designed by King(1989) in
  order to talk about species and their features
  with precision, both in describing linguistic
  entities and in writing grammars that predict
  what linguistic entities are well-formed.

3
HPSG 87, 94

• 87 The formalization was based on the
  intuitionistic feature structures to model
  partial information about linguistic entities.
• 94 Sharp distinction between feature structures
  and feature structure descriptions
• Feature structures Total mathematical
  idealizations of linguistic entities
• Descriptions Pick out that subset of FSs that
  will be regarded as possible well-formed
  linguistic entities.
• Gap Relationship between HPSG and its SGC.

4
SRL

• Fundamental ontological assumptions of HPSG 1.
  closed world assumption (fixed set of disjoined
  ontological categories-species), 2. total
  well-typedness (different kinds of species have
  different kinds of identifiable parts (features)
  which in turn must be of certain kinds (species).

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SRL

• An SRL language is determined by 2 disjoined sets
  of nonlogical symbols (species and features) and
  a notion of appropriateness that relates them to
  each other.
• A class of interpretations each of which provides
  meanings for the nonlogical symbols and for
  descriptions.

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SRL

• Signature SltS,F,Agt, A is a total function from
  SxF to Pow(S) called the appropriateness
  function.
• Interpretation of S IltU,S,Fgt
• Each species denotes a set of objects, each
  object is in the denotation of exactly one
  species and S assigns to each object the unique
  species to whose denotation the object belongs
• F assigns to each feature the partial function it
  denotes

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SRL

• The appropriateness function encodes a strict
  relationship between the denotations of species
  and features if A(s,f)empty set then the domain
  of F(f) does not intersect with the denotation of
  s.
• E.g. A(republic,MONARCH) empty set

8
Formal vs Informal Grammars

• HPSG 94
• Sort hierarchy and set of principles. Sorts that
  label terminal nodes are called maximal) in the
  sense of maximally informative-specific).
• A sort inherits the feature declarations of its
  supersorts.
• The finite set of sort symbols is partially
  ordered, with sort symbols corresponding to more
  inclusive types lower in the ordering.

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Formal vs Informal Grammars

• Maximal HPSG sorts SRL species in that each
  linguistic object is in the denotation of one and
  only one maximal sort.
• A Nonmaximal HPSG sort is the disjunction of its
  maximal subsorts (e.g. sign word phrase
• HPSG feature SRL feature
• An SRL A can express the HPSG feature
  declarations for the maximal sorts
• The S of an SRL grammar can express the sort
  hierarchy component of an HPSG grammar.

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Types vs isomorphism classes

• Replacement of types with isomorphism classes
• A unique (abstr.) feature structure is
  associated with each distinct isomorphism class
  of feature structures.
• How? We associate with each object ? in an
  interpretation I a unique abstract feature
  structure Abst1(?), namely the abstract feature
  structure associated with the isomorphism class
  of the FS determined by the ?.

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SGC

• In case the interpretation happens to be a model
  of the grammar -gt the SGC of a grammar could be
  identified with the objects in a model of a
  grammar.
• Linguistic objects in HPSG feature structure
  descriptions
• Each interpretation determines a unique set of
  abstract feature structures. If the
  interpretation is a model of a given grammar,
  then the set of abstract feature structures
  determined by the model is the SGC of the grammar.

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Kinds of SGCs for a grammar

• The set of abstract FSs which are ABST1(?) for
  some object ? in some interpretation I which is a
  model of the grammar.
• The set of abstract FSs determined by some
  exhaustive model (it contains for any object ? in
  any model, an object ? which determines the same
  abstract FS.
• The set of all singly generated models that each
  belongs to a one isomorphism class.
• Summary The SGC of an HPSG grammar is the set of
  its models which are abstract FSs.