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HPSG : featureless or
    2 papers by
  Andrew Moshier
    Comparative Introduction to lexicalist
      syntactic theories WS 2005/2006

       Antske Fokkens
      Today’s presentation
• Mostly based on the article Featureless HPSG.
• Why all this trouble?
  – Two main questions the model of HPSG leads to..
  – Three ways to approach HPSG
  – The engineering approach: importance of modularity
  – The interaction of feature geometry and principles: a
    modularity problem
  – Towards a solution
• Categorial model for HPSG
• Definitions for a model of HPSG
      Two main questions
• The HPSG model leads to two interesting
  (1) What should the model look like?
  (2) What should the role of a model be?
• While the second question logically comes
  first, only the first has been addressed so
     The second question
• ….so what is the answer to the
  second question?
               Role       of the   Model
• A model
  – Firstly is there to ensure coherence in the theory (“or so
    logicians tell us”)
  – But more importantly a model:
  – Allows for independency,
  – Non-entailment and
  – Relative coherence
• These two properties are also mentioned by
  Pollard and Sag (1987)

  “…HPSG shares a concern about matters of
  modularity of design and the deduction of
  particular facts from the complex interaction of
  general principles.”
   The problem with HPSG
• Lack of modularity this can be seen
• Comparing the different definitions of
  the Head Feature Principle
• Organizing agreement features
• From a theoretical point of view, it is
  difficult to proof that certain
  definitions are equal, but there is
  also another side of the story….
 Three approaches to HPSG
• HPSG is often looked at from a
  linguistics point of view, or
• A theoretic point of view, but
• Can also be looked at from an
  engineering point of view
• Looking at HPSG from the
  engineering point of view, there are
  a couple of things one needs to take
  into account….
Implementing a grammar…
• Is a task that is too complex for one
  person to complete by him/herself.
• Involves many revisions/refinements
  of rules at different places in the
• Therefore, the notion of modularity is
       The importance of
• Several engineers need to be able to work
  on the grammar at the same time
• without changes in the area of one
  engineer causing problems in a domain
  another engineer is working on…
• Modularity allows the engineer to make
  changes in one place, without it causing
  many problematic changes in another part
  of the grammar.
How to improve modularity
        in HPSG?
• Moshier’s proposal : the way HPSG is
  designed, it reminds of a system of
  category theoretic
  constraints…therefore it can be
  reanalyzed using only category-
  theoretic properties.
            Category theory
• A category is a model of the first order theory of
• Every arrow has a specific domain and codomain
• If f : A → B , then there is an arrow f with domain A and
  codomain B
• Arrows compose iff their codomain and domain match
• If f : A → B and g : B → C then there is an arrow g○f
  : A → C and
• If g○f : A → C , then there is a domain B for which
  f : A → B and g : B → C
• Entities that can serve as domains are called objects.
   Category theory (cont)
• Composition is associative
   f○ (g ○ h) = (f ○ g) ○ h
• Every object has an identity arrow,
  so if B is an object,
  -then there exists an arrow
  idB : B → B for which
• h ○ idB = h for all h : B → C’ and
• j = idB ○ j for all j : A’ → B
 Category theory –denoting
    objects and arrows
• For any category A, let |A| denote
  the collection of objects of the
• And for A,B є |A|, let A(A,B) denote
  the collection of arrows from A to B.
• A functor from category A to B is a
  structure preserving map from A to B.
  The structure of a category corresponds to
  the composition of the (identity) arrows.
• A functor F from A to B maps arrows in A
  to arrows in B, in a way that
  – (i) there exists an identity arrow F(idA)
  – (ii) F(f ○ g) = F(f) ○ F(g) wherever f ○ g is defined
• For any category A an A-diagram is a triple
  D = (G,A,f) where
• G is a directed graph
• A labels vertices and f edges, so that each
  vertex v is labeled by an object Av є |A|
  and each edge v to v’ is labeled by an
  arrow fe є A (Av, Av’).
•       .v      e               .v‘
 label                                label
         . Av   fe            . Av’
            Diagram (cont)
            .v          e           .v‘     e    . v’’
   label                                         label
            . Av        fe          . Av’        . Av’’

• Let E(v, v’) denote the set of edges in G from v to v’
• Let G(v, v’) denote the set of paths in G from vertex
  v to vertex v’
• The labeling of edges in G determines, for each p є
  G(v, v’), an arrow fp : Av → Av’
• If <e0,…,en-1> є G(v, v’) and en є E(v’,v’’), then
  f<e ,…,en> = fen ○ f<e ,..,en-1>
     0              0
             Directed graph G
• Determines [G] with vertices as objects and
  paths as arrows.
• Composition is defined as concatenation of
        .v   p    .v‘     F   .v   p     . v‘
L:[G]                                            L:[G]
        . Av fp   . Av’   F   . B v gp   . Bv’
   – L : [G]       A
   –F:A            B
   – (G, B, g) is the diagram determined by F ○ L
• An A-diagram (G, A, f) commutes iff
• For each to vertices v and v’ and
• For every two paths p,q є G(v, v’)
  fp = fq
          A model of HPSG
• A functor Μ : H → A
• ‘H is a formally defined category of the form [H]
  for the directed graph H in which both vertices
  and edges are certain strings over a fixed
  alphabet…H takes the place of a formal
  language for HPSG’.
• If A is a Set (and we will suppose it is for
  reasons of simplification), then a model of HPSG
  is a Set-diagram
• M sends objects of H to sets and arrows of H to
• Objects of H correspond to sign and
• Arrows correspond to paths of
  features (f.i SYNSEM|LOC|CAT)
                 The Codomain
• Is the “semantic category”
• M picks out sets as the interpretations of sort
  names and functions as the interpretations of
  feature paths

H                       FEATURE-PATH

      M                 M

A     Interpretation
           of sign      Interpretation
                        of feature path
          To model HPSG…
• “For each sort σ there must be
  corresponding sorts that represent lists of
  σ and sets of σ.”
• This means that
  – H must be a rich enough vocabulary to say
    things about lists and sets.
  – A must have suitable interpretations of lists
    and sets
  – M must carry the vocabulary of H over to
    interpretations in A correctly interpreting lists
    and sets.

• Each of these points will be discussed in
Formal categories for HPSG
• H should be defined formally, as a
  formal language
• H should have a vocabulary which is
  rich enough
• A particular H should be determined
  by something analogous to HPSG
  appropriateness conditions
           Sets S and L
• S and L
• Let S and L be disjoint sets, a formal
  category over alphabets S and L is of
  the form [H], for a directed graph H
  satisfying that
  – the vertices of H are drawn from S*
  – the edges of H are drawn from L*
  Basic Objects and Arrows
     and their symbols
• Basic Objects (BO) and Basic Arrows
  (BA) are disjoint infinite sets.
  – BO includes objects like sign,
  – BA includes arrows like PHON.
• Symbols for Objects are
  {1, x, L, P, M, [, ]}
• Symbols for Arrows are
  {id, ◊, , , <,>, π,π’,nil, cons, Λ,[,]}
  H = [H] is a formal category
  over BO U SO and BA U SA
Then H has:
• Formal Identities (FI)
• Formal Terminals (FT)
• Formal Products (FPr)
• Formal Lists (FL)
• Formal Powers (FPo)
H is definite provided that canonical names
  are used only occur as required by the
  definition of formal HPSG categories (if FI,
  FT, FPr, FL and FPo are defined)
          Formal identities
• H has formal identities iff
  id : σ → σ is an edge in H for all
  σ є |H |

• Defined:
If id : σ → τ is an edge, then σ = τ
         Formal Terminals
• H has formal terminals iff
1. 1 є |H |; and
2. σ є |H | implies ◊ : σ → 1 is an edge

•   Defined:
If ◊ : σ → τ is an edge, then τ = 1
             Formal Products
•    H has formal products iff
1.   σ, τ є |H| implies that [σ x τ ] є |H|, and that
     π: [σ x τ] → σ and π’ [σ x τ] → τ
     are edges; and
2. If p : ρ → σ and q: ρ → τ are paths,
     then <p,q> : ρ → [σ x τ] is an edge.
•    Defined :
If [ω x ω’] є |H|, then ω, ω’ є |H| if π : ρ → σ is an edge then ρ
     = [σ x τ] for some τ; and if π’ : ρ → τ is an edge,
     then ρ = [σ x τ] for some σ.
Also if <p,q> : ρ → ρ’ is an edge, the ρ’ = [σ x τ],
     and p: ρ → σ and q : ρ → τ are paths for some σ and τ.
                  Formal Lists
•   H has formal lists, iff it has formal products and
1.  σ є |H| implies that L[σ] є |H| and that nil : 1 → L[σ]
    and cons : [σ x L[σ]] → L[σ] are edges; and
2. If p : 1 → τ and q : [σ x τ] → τ are paths, then fold[p,q] :
    L[σ] → τ is an edge.
•   defined
L[ω] є |H| , then ω є |H|. If nil : ρ → τ is an edge then ρ = 1
    and τ = L[σ] for some σ. If cons : ρ → τ, then ρ = [σ x
    L[σ]] and τ = L[σ] for some σ. If fold[p,q] : ρ → τ is an
    edge, then for some σ є |H|, ρ = L[σ], and p : 1 → τ
    and q : [σ x τ] → τ are paths.
                Formal Powers
•   H has formal powers iff
1.  σ є |H| implies that P[σ], M[σ] є |H| and that leg : M[σ]
    → P[σ] and leg’ M[σ] → σ are edges
2. If l : ρ →τ and r : ρ → σ are paths Λ[l,r] : τ → P[σ] is an
•   Defined
P[ω] є |H| iff M[ω] є |H| . If P[ω] є |H| then ω є |H|. If leg : ρ
    → τ is an edge, then ρ = M[σ] and τ = P[σ] for some σ.
    If leg’ : ρ → τ is an edge then ρ = M[τ]. If Λ[l,r] : τ → u
    is an edge, then there exists σ,ρ є |H| so that u = P[σ]
    and r : ρ → σ and l: ρ → τ are paths.
 Appropriateness conditions
• How to specify this formal definite H ?
• In HPSG specifications are introduced
  via appropriateness conditions
  specifying the existence of a feature f
  for a sort σ.
• Moshier looks for a formal counterpart
  for appropriateness conditions.
             Feature structure
• In HPSG:
      f0     τ0                    PHON        L(phon-string)
s:                        sign :
      fn-1   τn-1                  SYNSEM      synsem

• S is a basic sort name, f are feature names and τ are
  intended to denote other sorts.
• Another specification in HPSG is that sorts are
  regarded as partially ordered.
• In a formal HPSG category, arrows can be used
  to define subsorts, f.i., if σ is a subsort of τ, then
  there is an arrow σ → τ
• This arrow (with certain additional properties
  (?!)) can be seen as an appropriate condition
  that is running backwards…
• In both cases there is a basic sort and a set of
  pairs (f, τ)
    Appropriateness condition
            for HPSG
•  ‘Define an appropriateness condition for
   a formal category H to be a triple α = (s,
   φ, d) where
1. s є BO
2. φ is a finite set of pairs in BA x |H|
3. And d є {f,b}, here d is intended to
   indicate the direction that arrows are
   supposed to run’
• If d = f, then α is called a forward
   appropriateness condition, else it is
   called a backward appropriateness
  Approp. condition (cont.)
• The extent of α = (s,φ, d) , denoted by
  E(α) is defined:
• E(s,φ,d) =   {(s,f,τ) | (f,τ) є φ} for d = f
               {(τ,f,s) | (f,τ) є φ} for d = b.

• The focus of α, denoted F(α) is defined as
• If H is a formal HPSG category and α is an
  appropriate condition for H, then
   – Let H + α denote the least formal HPSG
    category so that
  – F(α) |H + α| and
  – For all (σ, f,ρ) є E(α), f: σ → ρ is an edge in H + α
• H + α is guaranteed to exist (formal
  categories of HPSG are inductive, and
• When triples E(α) already correspond to
  edges in H, then H + α = H
  Interpretations of formal
      HPSG categories
• An interpretation of a formal HPSG
  category is a functor I : H → A, for
  which the following properties hold:
• Formal identity:
  I(id : σ → σ) is the identity for I(σ)

• Formal Terminal:
  I(1) is a terminal object.

• Formal Products :

  I((σ x τ)) is a product of I(σ) and I(τ). I(:→) and
  I(π’: [σ x τ] → τ) are corresponding projections for p : ρ → σ and
  q: ρ → τ,
       I(p : ρ → σ) = I(<p,q>)| π : ρ → σ)
       I(q: ρ → τ) = I(<p,q>)| π’ : ρ → τ)
       Interpretations (cont.)
• Formal lists

  I(nil : 1 → L[σ]) and
  I(cons : [σ x L[σ]] → L[σ]) define lists for I(σ ).
  For p : 1 → τ and q : (σ x τ) → τ,
  I(p : 1 → τ) = I(nil|fold[p,q] : 1 → τ)
  I([id x fold[p,q]]|q : (σ x L[σ]] → τ) =
  I(cons|fold[p,q] : (σ x L[σ]] → τ)
               Interpretations (cont)
• Formal Powers

        I(leg : M[σ] → P[σ]) and I(leg’ M[σ] → σ ) constitute a universal relation to
        I(σ). For p : ρ →τ and q : ρ → σ holds;
•       A               g                I(M[σ])

    f                                       I(leg)

        I(τ)           I(Λ[p,q])         I(P[σ])

    is a pullback of I(Λ[p,q] : τ → P[σ]) and I(M[σ] → P[σ]), then for every
    irreducible table
           B                   I(σ)
If there is an arrow h : I(ρ) → B so that I(p : ρ →τ ) = f’h and I(q : ρ → σ )
    = g’h, then there is also an arrow h’ : A → B, so that f = f’h’ and g = g’h’.
• Today we have seen
  – How a formal model of HPSG can be
  – A precisely defined vocabulary of sort
    names and arrow names has been
    defined, in which HPSG can be
  – How such a category can be interpreted
    in Set.
Other parts we will see next
• Constraints on Interpretations
• The principles
• Equivalent theories…
            Next week
• Please read the articles again ;-)…

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