Head-driven Phrase Structure Grammar I

Head-driven Phrase Structure Grammar I 

Weiwei Sun 

Institute of Computer Science and Technology 

Peking University 

May 10, 2013

Basic Idea Typed feature structure HPSG: First glance 

Lexicalized grammar formalism 

◮ Linguistic rules to derive syntactic structures are mainly 

introduced by lexical entries, which are structures assigned to 

words. 

◮ A number of modern syntactic theories fall into lexicalized 

grammars: 

◮ Lexicalized tree-adjoining grammar (last lecture) 

◮ Head-driven phrase structure grammar (this lecture) 

◮ Lexical functional grammar 

◮ Combinatory categorial grammar 

◮ Lexicalized grammars provides a mathematical framework for 

describing linguistic structures. 

Weiwei Sun Head-driven Phrase Structure Grammar I 2/61


Outline 

Basic Idea 

Typed feature structure 

HPSG: First glance 



Motivation 

CFG treats each grammatical category symbol as atomic without 

internal structure. 

⇒ Two categories are either identical or different. 

⇒ There is no mechanism for saying that two categories are alike 

in some ways, but different in others. 

Words and phrases in NLs typically behave alike in certain 

respects, but not others. 

Cross-cutting grammatical properties 

3rd singular subject plural subject 

direct object NP denies deny 

No direct object NP disappears disappear 



Features and values 

◮ Linguistic feature: a property-like element that indicates 

the grammatical behavior of syntactic constituents. 

◮ The elements associated to linguistic expressions, such as 

words, can be broken down. 

◮ Features are usually modeled as pairs of feature names and 

feature values. 

Feature Example Value 

person I go, you go, he goes 1st, 2nd, 3rd 

number he dances, they dance singular, plural 

case he brings Bob, Bob brings him nominative accusative 

tense go, went, gone past, present, future 

modality may, can, conditional, subjunctive 

A nice summary of linguistic features 

http://www.grammaticalfeatures.net 



Features and values 

◮ Whereas NPs, VPs, NNs, etc. are conceived of as 

categories (first-class “citizens”) in the grammar, features 

are more property-like. 

◮ Features are conceived of as the atomic units that 

compose more complex categories. 

◮ The VP has the feature value past tense. 

◮ The verb is a past tense verb. 

◮ The noun has a case feature accusative. 

A noun is a feature set of semantic, morphological and phonological 

features: 

⎡ 

form 

⎢ 

⎣number 

animacy 

⎤ 

dog 

⎥ 

singular ⎦ 

animate 



Feature structure 

◮ Use a feature structure to specify of grammatical information. 

◮ Formally, a feature structure is a specification of a set of 

features, each of which is paired with a particular value. 

Descriptions 

Feature structures can be 

represented by a type of 

attribute-value matrix (AVM). 

⎡ 

⎤ 

FEATURE 1 VALUE 1 

FEATURE 2 VALUE 2 

⎢ 

⎣... 

⎥ 

⎦ 

FEATURE n VALUE n 

Example 

〈 [ ]〉 

pos noun 

bird, 

num singular 



More on feature values 

Atomic value 

An unstructured value, one with only one part 

[ 

tense 

] 

past 

person 2 

Complex value 

A structured value, itself a feature structure 

⎡ 

tense 

⎢ 

⎣agreement 

⎤ 

past 

[ ] 

person 2 ⎥ 

⎦ 

number singular 



Handling valence with a feature 

IV, TV and DTV 

IV: TV: DTV: 

[ ] 

pos verb 

[ ] 

pos verb 

[ ] 

pos verb 

word 

val 

itr 

word 

val 

tr 

word 

val 

dtr 

V 

] 

V= 

[pos verb 

word 

N, NP 

N: NP: 

[ 

pos 

word 

] 

noun 

[ 

pos 

phrase 

] 

noun 



Handling valence with a feature 

VP→V NP ∗ 

⎡ ⎤ 

] word 

⎢ ⎥ 

→ ⎣pos 1 ⎦ 

pos 1 

val itr 

b. [ ⎡ ⎤ 

] word 

phrase 

[ 

⎢ ⎥ 

→ ⎣pos 1 ⎦ pos 

pos 1 

val tv 

c. [ ⎡ ⎤ 

] word 

phrase 

[ 

⎢ ⎥ 

→ ⎣pos 1 ⎦ pos 

pos 1 

val dtv 

(1) a. [ phrase 

] 

noun 

][ 

noun pos 

] 

noun 



Handling agreement with a feature 

S→NP VP 

(2) S → 

[ 

pos 

] 

noun 

num 1 

phrase 

[ 

pos 

] 

verb 

num 1 

phrase 

With valence 

⎡ 

⎤ 

◮ [ ] pos noun 

pos noun ⎢ 

⎥ 

→ ⎣num 1 ⎦ 

num 1 

phrase 

val itr 

word 

⎡ 

⎤ 


pos noun ⎢ 

⎥ 

→ ⎣num 1 ⎦NP 

num 1 

phrase 

val tr 

phrase 

⎡ 

⎤ 


pos noun ⎢ 

⎥ 

→ ⎣num 1 ⎦NP NP 

num 1 

phrase 

val tr 

phrase 



HPSG: Assumptions 

Two assumptions 

1. Languages are systems of sorts of linguistic objects at 

avariety of levels of abstraction. 

◮ Grammars assign structural descriptions and semantic 

interpretations to linguistic objects. 

2. Grammars are best represented as process-neutral systems 

of declarative constraints. 

◮ TG defines contraints in terms of operations on objects 

(movements). 

◮ TAG defines contraints in terms of adjunctions. 



Developing a feature structure based syntactic theory 

Questions 

Define feature structures in a linguistic way 

◮ What kinds of features go together 

◮ What values are appropriate for each particular feature 

Manipulate feature structures in a systematic way 

◮ How to specify which feature structures are well-formed 

◮ How to combine small feature structures to larger ones 

Mathematically sound model is employed to guarantee that rich 

features can be widely defined for modeling various linguistic 

phenomena. 



Developing a feature structure based syntactic theory 

Type ⇒ Typed feature structure 

◮ One of the first things in developing a theory of grammars 

is to classify linguistic entities. 

◮ Paradigmatic relations: We cluster “good” feature 

structures into various types. 

◮ Syntagmatic relations: We combine “good” feature 

structures according to several general principles. 

◮ HPSG employs Typed feature structure (TFS), a 

mathematical device, to express symbolic and structured 

constraints of linguistic objects. 

1. Types are associated with feature-value constraints 

expressed as TFS’s. 

2. Linguistic objects “of the same type” share its 

feature-value constraints. 

3. A concrete linguistic object may define other feature-value 

constraints, which are compatible with its type. 



Reference 

HPSG is a syntactic theory that follows the lexicalist framework. 

1. (Sag and Wasow, 1999) ⇒ (Sag, Wasow and Bender 

2003): 

◮ Syntactic Theory: A Formal Introduction 

2. (Pollard and Sag, 1994): 

◮ Head-Driven Phrase Structure Grammar 

HPSG describes linguistic structures with typed feature structures 

◮ (Carpenter, 1992): 

◮ The Logic of Typed Feature Structures 

◮ (Copestake, 2000): 

◮ Appendix: Definitions of typed feature structures. 



Outline 

Basic Idea 





Type hierarchy: Informal idea 

A type hierarchy 

◮ states what kinds of objects we claim exist (the types) 

◮ organizes the objects hierarchically into classes with shared 

properties (the type hierarchy) 

◮ states what general properties each kind of object has (the 

feature and feature value declarations). 

◮ describes a classification of feature structures. 

⇒ It classifies the corresponding linguistic objects modeled by the 

feature structures. 

Types are occasionally referred to as sorts. 




Multiple inheritance allows classification on multiple dimensions 

◮ In the CFG way, strings are classified in only one dimension. 

What do these CFG categories have in common 

NP VP are both phrases 

N V are both words 

NP N are both ‘nouny’ 

VP V are both ‘verby’ 




Multiple inheritance allows classification on multiple dimensions 

◮ In the CFG way, strings are classified in only one dimension. 

What do these CFG categories have in common 

NP VP N V 

NP VP are both phrases 

N V are both words 

NP N are both ‘nouny’ 

VP V are both ‘verby’ 

phrase word nouny verby 

sign 

func 

⊥ 




Given two types σ, τ, we usually want to talk about: 

◮ σ is more general than τ 

◮ τ is more specific than σ 

◮ σ is a supertype of τ 

◮ τ is a subtype of σ 

◮ One unique type subsumes all other types 

◮ Types without any subtype are called maximal types. 



Type hierarchy: Subsumption 

◮ T : is a finite set of types 

◮ ⊑: a partial order defined on T 

◮ Reflexive: σ ⊑ σ 

◮ Antisymmetric: if σ ⊑ τ and τ ⊑ σ then σ = τ 

◮ Transitive: if σ ⊑ ω and ω ⊑ τ then σ ⊑ τ 

◮ Subsumption: For types σ, τ ∈ T , σ subsumes or is more 

general than or is a supertype of τ, when σ ⊑ τ. 

◮ τ is subsumed by or is more specific than or is a subtype of σ. 

Example 

dolphin elephant salmon 

aquatic-animal terrestrial-animal mammal fish 

animal 



Type hierarchy: Most general common subtype of two types 

Definition 

◮ For types τ 1 , τ 2 ∈ T , τ 1 and τ 2 are consistent if they have 

a common subtype or an upper bound σ ′ , such that 

τ 1 ⊑ σ ′ and τ 2 ⊑ σ ′ . 

◮ A least upper bound σ (lub(τ 1 , τ 2 )) of τ 1 and τ 2 is defined 

as follows: 

◮ σ is an upper bound of τ 1 and τ 2 . 

◮ For any upper bound σ ′ of τ 1 and τ 2 , σ ⊑ σ ′ . 

dolphin elephan salmon 


animal 



Type hierarchy: Formal definition 

A type system consists of a type hierarchy plus a set of constraints 

which determine which typed feature structures are well-formed. 

Definition 

Type hierarchy H is a pair 〈T , ⊑〉 that satisfies the following 

condition: For all τ 1 , τ 2 ∈ T , 

◮ lub(τ 1 , τ 2 ) = ∅ or 

◮ there exist one unique σ, such that lub(τ 1 , τ 2 ) = {σ}. 

We denote the most general type in a type hierarchy by ⊥. 



Type unification: Definition 

Definition 

Unification of types τ 1 , τ 2 ∈ T is an operation to compute the 

least upper bound of τ 1 and τ 2 . Type unification of τ 1 and τ 2 if 

denoted by τ 1 ⊔ τ 2 . That is, 

{ σ if lub(τ1 , τ 

τ 1 ⊔ τ 2 = 

2 ) = {σ} 

undefined otherwise 

dolphin elephan salmon 


animal 



Typed feature structure: Informal Idea 

 

 

 

 

 

 

◮ TFS’s can be represented as rooted directed labeled graphs: 

◮ Nodes are labeled with types 

◮ Edges are labeled with features. 

◮ A feature path is a list of features through a feature structure 

leading to a particular value. 

◮ Reentrancy/structure-sharing: two attributes denote identical 

information. 

Example 

predicate 

subj 

P RED 

noun 

verb 

AGR 

agreement 

P ERS 

AGR 

NUM 

3rd 

singular 



Attribute-Value Matrix (AVM) 

Attribute-value matrix (AVM) notation is a description language to 

describe sets of feature structures, with the following three building 

blocks: 

◮ Type descriptions selects all objects of a particular type. 

◮ Attribute-value pairs describe objects that have a particular 

property. 

◮ The attribute must be appropriate for the particular type, and 

the value can be any kind of description 

◮ Tags to specify token identity 



AVM: An example 

Type, attribute-value pairs and tags. 

Example 

⎡ 〈〉 

⎤ 

phon she 

⎡ 

⎡ ⎡ 

[ 

] ⎤ 

⎤ 

head 

case nom 

noun 

⎡ ⎤ 

cat 

subj 〈〉 

⎢ 

⎢ ⎥ 

valence 

⎣ 

⎣comps 〈〉 ⎦ ⎥ 

⎦ 

spr 〈〉 

cat 

val 

⎡ 

⎡ 

⎤⎤ 

ss 

per 3rd 

loc 

⎢ 

⎥ 

index 1 ⎣num 

sing⎦ 

cont 

⎢ 

gend fem ⎥ 

⎣ 

ref 

⎦ 

restr {} 

ppro 

⎡ 

⎧ [ ]⎫ ⎤ 

⎨ 

⎢ 

⎢ ⎢ 

reln female ⎬ ⎢ ⎣ 

⎣conx 

⎣bkgr 

⎩ inst 1 

⎭ 

⎦ 

⎦⎥ 

⎣ 

psoa ⎦ 

synsem 

local 

context 

word 



TFS: Graphs or AVMs 

 

 

 

 

 

 

Graph 

noun 

3rd 

predicate 

SUBJ 

AGR 

agreement 

P ERS 

P RED 

verb 

AGR 

NUM 

singular 

AVM 

⎡ 

⎡ 

[ ]⎤ ⎤ 

person 3rd 

subj ⎣agr 1 

⎦ 

num singular 

agreement 

⎢ noun 

⎣ 

] 

⎥ 

⎦ 

pred 

[agr 1 

verb 



TFS: Formal definition 

Definition 

A typed feature structure is defined on a finite set of features F 

and a type hierarchy 〈T , ⊑〉. It is a tuple 〈Q, r, δ, θ〉, where: 

◮ Q is a finite set of nodes 

◮ r ∈ Q (r is the root node) 

◮ δ : Q × F ↦→ Q is a partial feature value function 

◮ θ : Q ↦→ T is a partial typing function 

subject to the following conditions: 

1. r isn’t a δ-descendant. 

2. all members of Q except r are δ-descendants of r. 

(*) there is no node n or path π such that δ(n, π) = n. 



Reentrancy: Informal Idea 

Question 

Shouldn’t a person live together with his/her spouse 




 

 

 

 

 

 

◮ Reentrant structure: one in which the attribute has a value 

that is another feature structure. 

◮ Reentrancy is also called token identity or path equivalence 

predicate 

SUBJ 

P RED 

noun 

verb 

AGR 

agreement 

P ERS 

AGR 

NUM 

3rd 

singular 

Structure sharing 

The main explanatory mechanism in HPSG is that of structuresharing, 

equating two features as having the exact same value. 




Agreement and valence 

◮ S → 

[ ] 

pos noun 

[ ] 

pos verb 

num 1 

num 1 

phrase 

phrase 

⎡ ⎤ 

◮ [ ] pos verb 

pos verb ⎢ ⎥ 

→ ⎣num 1 ⎦ 

num 1 

phrase 

val itr 

word 

⎡ ⎤ 



→ ⎣num 1 ⎦NP 

num 1 

phrase 

val tr 

phrase 

⎡ ⎤ 



→ ⎣num 1 ⎦NP NP 

num 1 

phrase 

val tr 

phrase 



Reentrancy: Formal Definition 

◮ A path is understood as a sequence of features: π ∈ F + . 

◮ δ(n, π) is the value node starting from n following path π. 

Definition 

If δ(r, π) = δ(r, π ′ ) and π = π ′ , i.e. two paths start from the 

root of the feature structure and point to the same node, then 

it is said there is a reentrancy between path π and π ′ 

Feature path 

⎡ 

⎡ 

[ ]⎤ ⎤ 

person 3rd 

subj ⎣agr 1 

⎦ 

num singular 

agreement 

⎢ noun 

⎣ 

] 

⎥ 

⎦ 

pred 

[agr 1 

verb 

Path: 〈SUBJ, AGR, NUM 〉 = singular 



TFS subsumption: Informal idea 

Subsumption between feature structures: 

◮ A less specific (more abstract) feature structure subsumes 

an equally or more specific one. 

The subsumption symbol: ⊑. 

Example 

[ 

[ ] ⎡ 

⎤ 

◮ number PL] 

⊑ number PL ⊑ number PL 

⎢ 

⎥ 

person 3 ⎣person 3 ⎦ 

phon dog 

[ ] [ 

] 

◮ number PL ¬ ⊑ number PL 

person 3 

[ 

] [ 

] 

◮ number PL ¬ ⊑ number SG 

] 

◮ [] ⊑ 

[number SG 



TFS subsumption: Formal definition 

◮ π ≡ F π ′ : feature structure F contains path equivalence or 

reentrancy between the paths π and π ′ 

Definition 

◮ δ(r, π) = δ(r, π ′ ) where r is the root node of F. 

F subsumes F ′ , written F ⊑ F ′ , if and only if: 

◮ π ≡ F π ′ implies π ≡ F ′ π ′ 

◮ θ(δ(r, π)) = σ implies θ(δ(r ′ , π)) = σ ′ and σ ⊑ σ ′ . 

In other words, feature structure F subsumes feature structure F ′ 

(F ⊑ F ′ ) iff: 

1. if path π is defined in F then π is also defined in F ′ and the 

type of the value of π in F is a supertype or equal to the type 

of the value of π in F ′ , and 

2. all paths that are reentrant in F are also reentrant in F ′ . 



Unification: Informal idea 

◮ Unification is a binary operation over two features 

structures F and F ′ , used for comparing or combining 

information. 

◮ Unification of F and F ′ either returns 

◮ a merged feature structure with the information from both 

F and F ′ , 

◮ or false if F and F ′ are incompatible. 

◮ The unification symbol is ⊔. 

Example 

[ 

◮ number 

[ 

◮ number 

◮ [ number 

] 

PL ⊔ 

[ 

number 

] [ 

PL ⊔ number 

] [ 

] 

PL ⊔ person 2 = 

] [ 

PL = number 

] 

SG = False 

[ ] 

number PL 

person 2 

] 

PL 



Unification: Informal idea 

◮ Decide whether two typed feature structures are mutually 

compatible; 

◮ determine combination of two TFS’s to give the most general 

feature structure which retains all information which they 

individually contain; 

◮ if there is no such feature structure, unification fails; 

◮ unification monotonically combines information from both 

“input” TFS’s; 

◮ relation to subsumption: the unification of two structures F 

and F ′ is the most general TFS which is subsumed by both F 

and F ′ (if it exists). 



Unification: Formal definition (skip) 

Definition 

Suppose F, F ′ ∈ FS such that F = 〈Q, r, θ, δ〉, F ′ = 

〈Q ′ , r ′ , θ ′ , δ ′ 〉, and Q ∩ Q ′ = ∅. We first define an equivalence 

relation ⊲⊳ on Q ∪ Q ′ as the least equivalence relation such that: 

◮ r ⊲⊳ r ′ 

◮ δ(f, q) ⊲⊳ δ(f, q ′ ) if both are defined and r ⊲⊳ r ′ 

The unification of F and F ′ is defined as: 

F ⊔ F ′ = 〈(Q ∪ Q ′ )/ ⊲⊳ , [r] ⊲⊳ , θ ⊲⊳ , δ ⊲⊳ 〉 

where: θ ⊲⊳ ([q] ⊲⊳ ) = ∪{(θ ∪ θ ′ )(q ′ )|q ′ ⊲⊳ q} and 

⎧ 

⎨ [(δ ∪ δ ′ )(f, q)] ⊲⊳ if (δ ∪ δ ′ )(f, q) is a well 

δ ⊲⊳ (f, [q] ⊲⊳ ) = 

defined function 

⎩ 

undefined otherwise 

Similarly, [(δ ∪ δ ′ )(f, q)] ⊲⊳ denotes ∪{〈q ′ , (δ ∪ δ ′ )(q ′ )〉|q ′ ⊲⊳ q}. 



Unification: An example (1) 

q 1 AGR q 2 P ERS q 3 

q 0 

SUBJ 

q 6 

P RED 

P ERS 

q 4 AGR q 5 NUM 

q 7 

◮ Q = {q 0 , q 1 , q 2 , q 3 , q 4 , q 5 , q 6 , q 7 } 

◮ r = q 0 

◮ θ = {q 0 → predicate, q 1 → noun, q 2 → agreement, q 3 → 

3rd, q 4 → verb, q 5 → agreement, q 6 → 3rd, q 7 → 

singular} 

◮ δ = {(SUBJ, q 0 ) → q 1 , (P RED, q 0 ) → q 4 , (AGR, q 1 ) → 

q 2 , (P ERS, q 2 ) → q 3 , (AGR, q 4 ) → q 5 , (P ERS, q 5 ) → 

q 6 , (NUM, q 5 ) → q 7 } 




 

 

SUBJ 

q ′ 1 

AGR 

 

q ′ 0 

P RED 

 

AGR 

q ′ 2 

q ′ 3 

◮ Q ′ = {q ′ 0 , q′ 1 , q′ 2 , q′ 3 } 

◮ r ′ = q ′ 0 

◮ θ ′ = {q 0 → predicate, q 1 → noun, q 2 → agreement, q 3 → 

verb} 

◮ δ ′ = {(SUBJ, q ′ 0 ) → q′ 1 , (P RED, q′ 0 ) → q′ 3 , (AGR, q 1) → 

q 2 , (AGR, q 3 ) → q 2 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




 

 

 

 

 

 

 

 

 

q 0 

SUBJ 

P RED 


q 1 

′ 

SUBJ 

q 6 q 0 

′ 

P ERS 


P RED 

q 7 

q ′ 3 

AGR 

 

q 2 

′ 

AGR 

{q 1 , q 1 ′ } 

SUBJ 

AGR P ERS 

 

{q 0 , q 0 ′ } {q 2 , q 5 , q 2 ′ } 

P RED NUM 

 

{q 4 , q ′ 3 } AGR 

{q 3 , q ′ 6 } 

{q 7 } 




{q 3 , q ′ 6 } 

{q 0 , q ′ 0 } SUBJ 

P RED 

 

{q 1 , q ′ 1 } AGR 

 

{q 4 , q ′ 3 } AGR 

{q 2 , q 5 , q ′ 2 } P ERS 

NUM 

{q 7 } 

◮ (Q ∪ Q ′ )/ ⊲⊳ = {{q 0 , q ′ 0 }, {q 1, q ′ 1 }, {q 4, q ′ 3 }, {q 2, q 5 , q ′ 2 }, {q 3, q 6 }, {q 7 }} 

◮ [r] ⊲⊳ = {q 0 , q ′ 0 } 

◮ θ = {{q 0 , q ′ 0 } → predicate, {q 1, q ′ 1 } → noun, {q 4, q ′ 3 } → 

verb, {q 2 , q 5 , q ′ 2 } → agreement, {q 3, q 6 } → 3rd, {q 7 } → singular} 

◮ δ = {(SUBJ, {q 0 , q ′ 0 }) → {q 1, q ′ 1 }, (P RED, {q 0, q ′ 0 }) → 

{q 4 , q ′ 3 }, (AGR, {q 1, q ′ 1 }) → {q 2, q 5 , q ′ 2 }, (P ERS, {q 2, q 5 , q ′ 2 }) → 

{q 3 , q 6 }, (AGR, {q 4 , q ′ 3 }) → {q 2, q 5 , q ′ 2 }, (NUM, {q 2, q 5 , q ′ 2 }) → {q 7} 



Unification: More examples 

Example 

◮ 

[ 

agreement 

[ 

⎡ 

[ 

⎣ agreement number 

number PL 

number PL] ] ⊔ [ number PL] 

= 

] ⎤ 

PL 

⎦ 

Example 

t 3 t 4 

t 1 t 2 

⊥ 

1. 

2. 

3. 

[ ] 

[ 

F1 t 2 

F 1 t 1 

]⊔ 

= 

t 4 F 2 t 3 

t 4 

[ ] [ ] 

F1 t 1 F1 1 

⊔ 

= 

F 2 t 2 F 2 1 

t 4 ⊥ 

[ ] [ ] 

F1 1 t 1 F2 1 

⊔ 

F 2 1 F 3 1 t 4 

⊥ 

⊥ 

[ ] 

F1 t 3 

F 2 t 3 

t 4 

[ ] 

F1 1 

F 2 1 t 3 

t 4 

= False 



TFS for linguistic description 

How should I model linguistic objects by FEATURES 

In practice, 

◮ The grammar specifies partial descriptions of type constraints, 

which are automatically expanded to give constraint 

structures which meet the conditions given. 

◮ Once a type system is defined, specifications of rules and 

lexical entries are expanded so that they are well-formed 

according to the type system. 

Question 

Why so many Greek symbols 



A real-world ontology of linguistic objects 



TFS for linguistic description: An example 

Example 

⎡ 〈〉 

⎤ 

phon she 

⎡ 

⎡ ⎡ 

[ 

] ⎤ 

⎤ 

head 

case nom 

noun 

⎡ ⎤ 

cat 

subj 〈〉 

⎢ 

⎢ ⎥ 

valence 

⎣ 

⎣comps 〈〉 ⎦ ⎥ 

⎦ 

spr 〈〉 

cat 

val 

⎡ 

⎡ 

⎤⎤ 

ss 

per 3rd 

loc 

⎢ 

⎥ 


sing⎦ 

cont 

⎢ 

gend fem ⎥ 

⎣ 

ref 

⎦ 

restr {} 

ppro 

⎡ 

⎧ [ ]⎫ ⎤ 

⎨ 

⎢ 

⎢ ⎢ 

reln female ⎬ ⎢ ⎣ 

⎣conx 

⎣bkgr 

⎩ inst 1 

⎭ 

⎦ 

⎦⎥ 

⎣ 

psoa ⎦ 

synsem 

local 

context 

word 



Appropriateness of feature structures: Motivated idea 

Pollard and Sag (1994: 8) require the HPSG architecture to use 

TFS as models of linguistic objects. They assume the underlying 

mathematical models to be totally well-typed and sort-resolved 

feature structures. 

Totally well-typedness 

◮ What attribute labels can appear in a feature structure is 

determined by its sort. 

◮ Every feature that is appropriate for the sort assigned to 

that node is actually present. 

Sort-resolvedness 

◮ Every node is assigned a most specific type as value. 

◮ Types that do not subsume any other sorts (except for 

themselves) are called maximally specific types/sorts. 

◮ A type is a disjunction of its maximally specific subtypes. 



Sort-resolvedness 

◮ The idea: Linguistic objects are entities of maximally specific 

sorts whose components are again of maximally specific sorts. 

∗ Sort-resolvedness is not a necessary property of feature 

structures. 

An example drawn from (Pollard and Sag, 1994) 

◮ word and phrase 

◮ are immediate subsorts of sign and 

◮ have no proper subsorts. 

◮ We can talk about signs with AVMs, but in the real world 

◮ we only find feature structures of the maximally specific 

sorts word or phrase; 

◮ there is no such thing as a feature structure of type sign 



Type constraints and appropriate features: Informal idea 

◮ Well-formed TFS’s satisfy all type constraints from the 

type hierarchy. 

◮ Type constraints are typed feature structures associated 

with a type. 

◮ The top-level features of a type constraint are appropriate 

features. 

◮ Type constraints express generalizations over a “class” 

(set) of objects. 

Type Constraint Appropriate features 

*ne-list* 

] FIRST, REST 

[ 

first ⊥ 

*ne-list* 

rest 

*list* 



Type constraints and appropriate features: Formal stuff (skip) 

Types are associated with constraints expressed as TFS’s. 

◮ Constraint function C : T ↦→ FS 

◮ Define conditions on the constraint function. 

◮ For each type there is a set of features appropriate to that 

type 

AppFeat : T ↦→ F 

Definition 

◮ Feat(F, q) is defined to be the set of features labeling 

transitions from the node q in some feature structure F 

i.e. f ∈ Feat(F, q) such that δ(f, q) is defined. 

◮ If C(t) = 〈Q, r, δ, θ〉, then the appropriate features of t are 

defined as Appfeat(t) = Feat(F, r). 




Definition: Well-formed feature structures 

F = 〈Q, r, δ, θ〉 is a well-formed feature structure if and only if 

for all q ∈ Q, we have that 

1. C(θ(q)) ⊑ F ′ = 〈Q ′ , q, δ, θ〉 

2. Feat(F, q) = Appfeat(θ(q)). 




Constraint function C : T ↦→ FS obeys the following conditions: 

1. Type For a given type t, if C(t) is the feature structure 

〈Q, r, δ, θ〉 then θ(r) = t. 

2. Monotonicity Given type t 1 and t 2 , if t 1 ⊑ t 2 then 

C(t 1 ) ⊑ C(t 2 ). 

3. Compatibility of constraints For all q ∈ Q the feature 

structure C(θ(q)) ⊑ F ′ = 〈Q ′ , q, δ, θ〉 and 

Feat(q) = Appfeat(θ(q)) 

4. Maximal introduction of features For every feature 

f ∈ Feat there is a unique type t such that f ∈ Appfeat(t) 

and there is no type s such that s ⊏ t and f ∈ Appfeat(s). 



Type inference: Making a TFS well-formed 

◮ Apply all type constraints to convert a TFS into a well-formed 

TFS. 

◮ Determine most general well-formed TFS subsumed by the 

input TFS; 

◮ Specialize all types so that all features are appropriate: 

[ ] [ ] 

head pos head pos 

→ 

args *list* 

args *list* 

⊥ 

phrase 

◮ Expand all nodes with the type constraint of the type of that 

node: 

[ ] 

head pos 

→ 

args *list* 

phrase 

⎡ 

head 

args 

⎢ 

⎣spr 

comps 

phrase 

⎤ 

pos 

*list* 

*list* 

⎥ 

⎦ 

*list* 



Outline 

Basic Idea 





Signature 

A signature consists of 

◮ A type inheritance hierarchy 〈T , ⊑〉. 

◮ A corresponding constraint function C : 〈T , ⊑〉 ↦→ FS. 

◮ FS: a collection of TFS’s. 

◮ Appropriate features and constraints on types 

◮ Linguistic theories are developed by describing the inheritance 

type hierarchy together with proper constraints 

⇒ A constraint-based grammar framework. 

Linguistic objects are modeled by feature structures, they are 

total with respect to the ontology declared in the signature. 



Signature 

These feature structures are 

◮ Totally well-formed: every node has all the features 

appropriate for its type and each attribute has an appropriate 

value 

◮ Type-resolved: every node is of a maximally specific type 

◮ Sometimes partially described: underspecifying information 

Linguistic description 

◮ Linguistic theories are described using AVMs. 

◮ A set of description statements comprises the constraints: 

◮ What are the admissible linguistic objects 

⇔ there is a corresponding well-formed TFS satisfying all 

the constraints 



Linguistic description 

Example 

(3) a. Ja spal. I masc.sg slept masc.sg 

b. On spal. He masc.sg slept masc.sg 

⎡ 

⎡ 

⎢ 

⎣synsem | loc 

word 

cat | head 

⎢ 

⎣cont | index 

⎤ 

noun 

[ ] 

num 3 

⎥ 

⎦ 

gen masc 

This AVM specifies the “partial” constraints on the complete 

(totally well-typed) feature structure of the subject. 



Feature structures in HPSG 

◮ HPSG is nonderivational, but in some sense, HPSG has 

several different levels (layers of features) 

◮ Each of these feature values is itself a complex object. 

Signs in HPSG 

Sign is the basic sort/type in HPSG used to describe lexical items 

(of type word) and phrases (of type phrase). 

◮ The type sign has the features PHON and SYNSEM. 

◮ The feature SYNSEM has a value of type synsem 

◮ This type itself has relevant features (LOCAL and 

NONLOCAL) 

◮ ... 

sign 

⎡ 

phon 

⎢ 

⎣synsem 

⎤ 

list(phone-string) 

[ ] 

local local ⎥ 

⎦ 

nonlocal nonlocal 

synsem 




◮ local introduces CAT(cateogry), CONT(content) and 

CONX(context) 

◮ non-local is connected with unbounded dependencies 

◮ category includes, 

◮ The sign’s syntactic category 

◮ Given via the feature [HEAD head], where head is the 

supertype for noun, verb, adjective, preposition, determiner, 

marker 

◮ each of these types selects a particular set of head features 

◮ The sign’s subcategorzation frame/valence. 

◮ 

⎡Three list-valued features 

⎤ 

⎡ 

⎤ 

subj list(synsem) 

⎢ 

⎢ 

⎥ 

⎣ 

synsem | loc | cat | valence ⎣spec 

list(synsem) ⎦⎥ 

⎦ 

comps list(synsem) 

valuence 

◮ If any of these lists are non-empty, the sign has the potential 

to combine with another sign 




Example 

⎡ 〈〉 

⎤ 

phon she 

⎡ 

⎡ ⎡ 

[ 

] ⎤ ⎤⎤ 

head 

case nom 

noun 

⎡ ⎤ 

cat 

subj 〈〉 

⎢ 

⎢ ⎥ 

⎣valence 

⎣comps 〈〉 ⎦ 

⎥ 

⎦ 

spr 〈〉 

cat 

val 

⎡ 

⎡ 

⎤ 

ss 

per 3rd 

loc 

⎢ 


sing⎦ 

cont 

⎢ 

gend fem ⎥ 

⎣ 

ref 

⎦ 

restr {} 

ppro 

⎡ 

⎧ [ 

]⎫ ⎤ 

⎨ 

⎢ ⎢ ⎢ 

reln female ⎬ ⎢ ⎣ ⎣conx 

⎥ 

⎣bkgr 

⎣ 

⎩ inst 1 

⎦ 

⎭ ⎦⎥ 

psoa ⎦ 

synsem local context 

word 



The building blocks of HPSG 

From a linguistic perspective, an HPSG consists of 

1. A lexicon: licensing basic words 

2. Lexical rules: licensing derived words 

3. Immediate dominance (ID) schemata: licensing 

constituent structure 

4. Linear precedence (LP) statements: constraining word 

order 

5. Grammatical principles: expressing generalizations about 

linguistic objects 



The lexicon 

◮ The basic lexicon defines the ontologically possible words that 

are grammatical: 

word → lexical entry1 ∨ lexical entry2 ∨ ... 

◮ Each lexical entry is described by an AVM. 

Example 

◮ Ja/On spal. I masc.sg /He masc.sg slept masc.sg 

spal v1 le 

⎡ 

phon 

ss | loc 

⎢ 

⎣ 

〈〉 

spal 

⎡ ⎡ 

] 

head 

[vform fin 

⎡ verb cat 

〈 

⎢ 

⎣val 

⎣ subj comps 〈〉 

⎢ 

⎣ 

[ 

] 

cont sleeper 1 

sleep 

′ 

NP[NOM] 1 [misc,sing] 

⎤ 

⎤⎤ 

〉 ⎤ 

⎥ 

⎦ 

⎥ 

⎦ 



Types of phrases 

◮ Each phrase has a DTRS attribute which has a 

constituent-structure value 

◮ This DTRS value corresponds to what we view in a tree as 

daughters (with additional grammatical role information, e.g. 

adjunct, complement, etc.) 

◮ By distinguishing different kinds of constituent-structures, we 

can define different kinds of constructions in a language 



Types of phrases 

Example 

⎡ 

〈 

〉 ⎤ 

phon she, drinks, wine 

⎡ 

⎤ 

head 3 

[ ] 

ss | loc | cat ⎢ 

⎣ subj 〈〉 ⎥ 

⎢ 

val 

⎦ 

⎣ 

comps 〈〉 ⎥ 

⎦ 

DTRS head-subj-struc 

S 

H 

⎡ 〈〉 ⎤ 

⎣ phon she 

⎦ 

ss 1 

⎡ 

phon 

ss | loc | cat 

⎢ 

⎣ 

⎡ 

〈 

〉 ⎤ 

phon drinks, wine 

⎡ 

⎤ 

head 3 

⎡ 〈〉 ⎤ 

ss | loc | cat ⎢ 

⎣val 

⎣ subj 1 ⎥ 

⎦ 

⎢ 

⎣ 

comps 〈〉 ⎥ 

⎦ 

DTRS 

H 

head-comps-struc 

〈〉 

⎤ 

drinks 

⎡ 

[ 

] 

⎤ 

head 3 vform fin 

verb 

⎡ 

〈〉 ⎤ 

subj 1 

⎢ ⎢ 

⎥ 

⎣val 

⎣ 〈〉 

⎦ 

⎥ 

⎦ 

comps 2 

C 

⎡ 〈〉 ⎤ 

⎣ phon wine 

⎦ 

ss 2 



LingGO grammar 

◮ LinGO = Linguistic Grammar Online 

◮ http://lingo.stanford.edu/ 

◮ Research focus: Development of 

◮ linguistically precise grammars based on the HPSG framework, 

◮ general-purpose tools for use in grammar engineering, profiling, 

parsing and generation. 

◮ English Resource Grammar: 

◮ The LinGO grammar consists of a specification of a type 

system and of various TFS’s which are well-formed according 

to the type system. 

◮ The TFS’s function as grammar rules, lexical rules and lexical 

entries.

Head-driven Phrase Structure Grammar I

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