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From syntactic structures to logical semantics

Alexandre Dikovsky, LINA-FRE CNRS 2729, Université de Nantes Christian Retoré LABRI-CNRS, INRIA-Futurs & Université Bordeaux 1

ESSLLI 2006 MALAGA

July 31st August 4th 2006

These lecture notes (slides) contain a presentation of the syntax semantics

interface according to various syntactic and semantic models. It contains five parts:

1. An introductory case for Syntax and Logical Semantics 2. Classical notions and results for TAGs and for dependency grammars 3. Categorial and generating dependency grammars 4. Underspecified semantics and compositional grammar interface 5. Categorial minimalist grammars

This does not mean that no other subject will be addressed, in particular if

time allows, we will probably present more generative semantics, for instance a recent computational approach to binding theory for pronoun interpretation (Bonato 2006).

An introductory case for Syntax and Logical Semantics:categorial grammars

Christian Retoré, Université Bordeaux 1

Équipe Signes linguistiques, grammaire et sens :algorithmique logique de la langue

INRIA-Futurs, LaBRI -C.N.R.S. et Département des Sciences du Langage Université Bordeaux 3

Linguistic domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Mathematical models in linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5A syntax classic:formal language theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8Grammar and Logic:a natural and a traditional link . . . . . . . . . . . . . . . . . 14Syntax and semantics in type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Università di VeronaMaggio 2006

Plan Début Fin Préc. Suiv. J

Linguistic domainsLanguage is wide subject, hence one of the main outcomes of linguistics has

been to divide its study into domains:

phonetics studying the sounds of languageAccoustics - Phonatory/auditory system

phonology Abstract sounds, a discrete system

Bali / Paris indistinguishable for Japanese speakers

prosody pauses, intonation, stress

"Je serai très heureux de venir parler au LaBRI, laboratoire auquel jedois ma formation initiale en informatique, par exemple sur lalambda-DRT.""Je serai très heureux de venir parler au LaBRI — laboratoire auquel jedois ma formation initiale en informatique — par exemple sur lalambda-DRT."



morphologie word structure

derivational morphology [word creation]prefix, suffix, compoundscategory may change

noble→noblesse petit→petitessemaison→maisonnette camion→camionnette carpe→carpette?

flexional morphology [conjugation, agreement, case marking]no category change

arriver → arriv[er][ons]cheval → chevaux

syntax sentence structure

*Je fais la réparerJe la fais réparer* [[Peter [eats an]] apple]Pierre [eats [an apple]]



semantics word meaning, sentence meaning

lexical semantics word meanings and relation between them

livre, print (physical object), read (information)

logical semantics two independant aspectstruthconditional semantics under which conditions is a statement true(meaning = logical formula)compositional semantics computing the meaning of a compound expres-sionout of the meaning of its parts (λ-calculus handles meaning compositionand substitution)

pragmatics using language to communicate in a given situation1st and 2nd persons, now, here, there , . . .

Let’s rather go in this restaurant.



Mathematical models in linguistics

probabilities, statistics

example: part of speech tagging using the previous words : if theprevious words are article, noun, adjective the next word is probably notan article



formal grammars

• morphology: automata, transducers

numbers, dateschanterons → chanter 1pers. pl. , futur

• generative syntax, formal language theory

[La [petite brise]] [la glace][La petite] [[brise [la glace]]Il [regarde [une passante]] [avec des lunettes noires].Il [regarde [une [passante [avec des lunettes noires]]]Elle [[a trouvé] [son [parapluie bizarre]]]Elle [[[a trouvé] [son parapluie]] [bizarre]]Syntaxe étendue vers la sémantique:* Hei liked two books that Chomskyi wrote.How many books that Chomskyi wrote did hei like?



logic

• logic for semantics

all doctors are drivers(therefore) all French doctors are French drivers*(therefore) all good doctors are good drivers

I had three coins. I lost one of them. I look for it.I had three coins. I lost two of them. *I put it into my pocket.

• a particularity with linear logic and resource logic:logic also describing syntaxparse structure = proof, deduction (= graph)



A syntax classic: formal language theory

In linguistics, computer science, mathematics, biology:compilation, concurrence, group theory,

formal grammar: a linguistic notion du to Noam Chomsky (with mathematicalcontributions by Marcel-Paul Schützenberger)

first formal grammar (context free) Pan. ini, 5e s. av. J.C.

A human language is not the set of the utterances of its speakers :new sentences identified as correct by its speakers can always be producedfor instance Ei+1 = He believes that Ei

→ Hypothesis: A human language is a set of (unconscious) rules :children over generalize when they have acquired a rule: "holded"



hence, formal grammars and Chomsky’s hierarchy

• Distinction competence /performance : grammar / our actual use

The wolf ate the goat.

The goat that the wolf ate ate the cabbage.

? The the cabbage that the goat that the wolf ate atebelonged to theferryman.

?? The ferryman to whom the cabbage that the goat that the wolf ate atebelonged owns boats.

??? The boats that the ferryman to whom the cabbage that the goat thatthe wolf ate ate belonged owns are green.

(nevertheless correct, with a pencil and time)



what do rules (expressing speakers’s competence) look like,c.-à-d.where are human languages in the hierarchy?

two principles

sentences can be analysed (understandable) in a sensible (=polyno-mial?) time

grammar should be learnable from positive examples only (criterion of-ten left out)



sentences can be analysed in polynomial time

I regular languages are not enough:(previous example) Subject1 Subject2 Subject3 ... Verb3 Verb2 Verb1

I context free language are not enough(completives in dutch) Subject1 Subject2 Subject3 ... Verb1 Verb2 Verb3

I more than hors-context, but can be parsed in polynomial timeTAG or context free grammars with movements, with unification



Some properties of universal grammars (assumed because of learnability):

• A noun phrase receives a caseand only finite verbs give a case

It seems that Mary is arriving.Mary seems to arrive.*It seems (that) Mary (to) arrive.



• A pronoun must be bound (syntactic position) by its antecedent.(configuration in the parse tree)

Carlotta’s dog thinks that he hates him.he6= him other equalities are possibleCarlotta’s dog thinks that he hates himself.he=himself all other equalities are possible



Grammar and Logic: a natural and a traditional link

Ancient times (Aristotle, Denis from Thrax)middle age (scholastics), eighteenth century (Port-Royal)...The sentence has a logical structure.

My children will have a pizza.∀x (child(x) ⇒ (∃y pizza(y) ∧ eat−future(x,y) ) )

∃y (pizza(y) ∧ ∀x (enfant(x) ⇒ eat−future(x,y) ) )

A pizza will be served to my children.∃y (pizza(y) ∧ ∀x (enfant(x) ⇒ eat−future(x,y) ) )

∀x (child(x) ⇒ (∃y pizza(y) ∧ eat−future(x,y) ) )

Subtle differences: every, each, all, ... a ,some, ...



Generalised quantifiersMany quantifiers in natural language:most of, many, a few

Most politicians read a book.

numbers also have scope properties.

Put eight drops into three spoons of water.3x8=24 drops?8 drops?



Interpretation, possible worlds, intentionalityTruth conditional semantics:the meaning of a statement is the set of the possible words in which it is true.

This student thinks that Chomsky is a computer scientist. .

In any world in which the belliefs of this studnet are true, Chomsky is a com-puter scientist.

Lectures de re et de dicto

James Bond believes that there is a spy in the lab.James Bond thinks that Professor Busquets is a spy.James Bond found a microphone in the drawer.



CompositionalityFrege: the meaning of compound expression is a function from the meaning of

its parts.

Paul, that I know, has not yet arrived.

Limits:If a farmer owns a donkey, then hei beats itj.If (∃f∃d Donkey(d) ∧ Farmer(f) ∧Own(f,d)) then B(f,d).Second d and second f FREE??

Limits of purely logical approach

I had three coins. I lost one of them. I look for it.I had three coins. I lost two of them. *I put it into my pocket.



Semantical aspects of syntactic categoriesCategories and part of speech have a semantical or logical counterpart.Parallel between main categories Verbs et Nounsand Predicates et les Individuals in logic.

I Noun phrases: individuals (real individual or quantified individual variables)

I Verbs, verbal groups: predicates

I Adjectives: nouns (agreement) or verbs (express a property)

I Prepositional phrase: neither nouns or verbs.



Syntax and semantics in type theory

We focus on the red domains

phonetics

phonologyprosody

derivational morphology

flexional morphology

syntax

semantics

logical semanticslexical semantics

pragmatique

formal grammardeductive systemlambda calculushigher order logic



Objectives

• Parsing producing semantic representationsNatural language data-base querytranslation

• Generating sentences from semantic representationsText synthesis or generationnatrual language answers to data base queriestranslation

Key points: syntax and compositional semanticsStrengths of categorial grammars



Composition

Semantically oriented syntactic formalism: categorial grammars

Natural implementation of compositional semantics :higher order logic in simply typed lambda calculus.

Morphologyagreement categoriesor feature unification?or preprocessing and already modified categoriesor module with transducers

On the syntactic side, we could compile the categorial grammar into a moreefficient one like (RCG)



General pictureOn the syntactic side:

I Words are mapped to formulae (syntactic categories) describing their syn-tactic behavior.

I A logical calculus which allows to map compound expression to syntacticcategories.(Rules are common to all languages.)

I if the category of a sequence of words is S, then it is a sentence.

On the semantics side:

I Each syntactic category a is mapped to a semantic type a∗

I Each word of syntactic category u is associated with a λ-term of type u∗

I Every syntactic composition operation corresponds to an operation on thesemantic λ terms.

I Each expression of syntactic category u is mapped into a λ-term of type u

I Sentences are mapped into logical formulae.Università di VeronaMaggio 2006


Lambek grammars (1958)A Lambek grammar= a lexicon LexLex(word) = a finite set of categories (depicting the syntactic behavior of word)

Syntactic categories L ::= P = S,sn,n,... | L− L | L − L

m1 · · ·mn ∈ Language(Lex) ssi ∀i ∃ti ∈ Lex(mi) t1, . . . ,tn ` S

Lexicalised grammar = rules are universal(only the lexicon may vary to describe different languages)



Regarding semantics: we use two types to describe logical formulae

I e individuals

I t truth values

Sleep: e → t function from individuals which maps individuals to truth valuesLike: e → (e → t) function from pairs of individuals to truth values

Notation λx.uthe function which maps the variable x to the term uTo compute (λx.u) t we replace x with t in u.

Example (λxλy.((like y)x))MariaPeter= λy.((like y)Maria)= likePeterMarialike: e → (e → t)Peter, Maria: e



Some examples of syntactic categories

• noun phrases, proper names: np

• common nouns: apple, car : n

• a, the, : np − n

• sleeps : np− S

• eats: (np− S) − np, np− S

• red : n− n

• who, that (subject relative pronouns),: (n− n) − (np− S)

Some examples of expected syntactic categories

• car: n

• the car that just passed: np

• eats an apple: np− S



Some examples of semantic types

• noun phrases, proper names: e

• common nouns: apple, car : e → t

• a, the, : (e → t) → e (or quantifier raised form)

• sleeps : e → t

• eats: e → (e → t), e → t

• red : (e → t) → (e → t)

• who, that (subject relative pronouns),: (e → t) → ((e → t) → (e → t))



Rules of the syntactic systemA =leftmostfreehypothesis

. . . [A] . . . . . .

B −i bounds AA−B

∆A

ΓA−B −eB

A =rightmostfreehypohtesis

. . . . . . [A] . . .

B −i bounds AB − A

ΓB − A

∆A −e

B



Meaning of the connectives − et ‖to and of the rulesu : B − A means that if u is followed by v : A then uv : B

u : B − A means that if u is before v : A then vu : B

example: the : np − n car : n permet d’obtenir the car : np

(modus ponens or elimination)

introduction rules allows to introduce and erase virtual constituents :if u : A −B and v : B − C we obtain with introduction rules uv : A − C

indeed taking x : Cwe have uvx : C with two introduction rulesand uv : A − C with an introduction rule

example: very : (n− n) − (n− n) yields veryvery : (n− n) − (n− n)



relation CFG ↔ Lambek grammarsWord: terminalsCategories and subcategories: non terminals.

• CFG → Lambek grammars

I CFG → CFG in Greibach normal formI if X → aTUV add to the lexicon a : ((X − V ) − U) − T

I The Lambek grammar is weakly equivalent to the CFG (rather easy)

• Lambek grammar → CFG

I Lambek grammars, maximum size of a category in the lexicon: k

I for every provable sequent A,B ` C where A,B,C smaller than k add arule C → AB

I for each a : A in the lexicon add a rule A → a

I The obtained CFG (in Chomsky normal form) is weakly equivalent to theoriginal Lambek grammars (conjectured in Chomsky 63, proof by Pentusin 93)



Syntactic categories and semantic types (Montague 1970)Logical formulae in typed λ with 2 types:

individuals e, truth values t.

n-ary predicate : e → (e → (e → (· · · → t)))n argument function : e → (e → (e → (· · · → e)))

Logical constants:

∧, ∨ , ⇒ : t → (t → t)∃,∀ : (e → t) → t

Morphism from syntactic types to semantic types

S∗ = t sentences: truth valuesnp∗ = e individualsn∗ = e → t unary predicates

(A−B)∗ = (B − A)∗ = A∗ → B∗ inductive extension of the morphisms to all syn-tactic categories



An example of a tiny lexicon We provide a lexicon analysing a single (!) sen-tence: « Some statements speak about themselves. »

word Syntactic category uSemantic type u∗

Semantic representation : λ-term of type u∗

xv variable or constant x of type vstatements n = St"statements" is a common noun

e → t = St∗

semantically statements is a unary predicate

λxe(statemente→t x)this predicate maps x onto the truth value of "x is a statement"





xv variable or constant x of type vspeak_about (np− S) − np = SpAspeak_about wiats for a noun phrase on its right, and then for a noun phrase onits left.

e → (e → t) = SpA∗

semantically "speak_about" is a binary predicate

λye λxe ((speak_aboute→(e→t) x)y)a function of two individuals which is true whenever the second one (the subject)speak about the first one (the object)





xv variable or constant x of type vthemselves ((np− S) − np)− (np− S) = X"themselves" (object) waits for a transitive verb on its en left and then produces asentence missing a subject

(e → (e → t)) → (e → t) = X∗

semantically, "themselves" maps a binary predicate P (x,y) (the transitive verb)into aunary predicate

λP e→(e→t) λxe ((P x)x)which is obtained by asking the two arguments to be equals P (x,y) est P (x,x)





xv variable or constant x of type vsome (S − (np− S)) − n = E"some" (subject) waits for a noun on its right and then for a sentence missing itssubject on its right and yields a sentence.

(e → t) → ((e → t) → t) = E∗

given a unary predicate P (noun) and a unary predicate Q (verb phrase) "some"builds a closed formulae

λP e→t λQe→t (∃(e→t)→t (λxe(∧t→(t→t)(P x)(Q x))))the formula built by "some" is ∃xP (x) ∧Q(x)





xv variable or constant x of type vsome (S − (np− S)) − n = S

(e → t) → ((e → t) → t) = S∗

λP e→t λQe→t (∃(e→t)→t (λxe(∧t→(t→t)(P x)(Q x))))statements n = St

e → t = St∗

λxe(statemente→t x)speak_about (np− S) − np = SpA

e → (e → t) = SpA∗

λye λxe ((speak_aboute→(e→t) x)y)themselves ((np− S) − np)− (np− S) = X

(e → (e → t)) → (e → t) = X∗

λP e→(e→t) λxe ((P x)x)



Syntactic analysis

(S − (np− S)) − n , n , (np− S) − np , ((np− S) − np)− (np− S) ` S ?

E ` (S−(np−S))−n St ` n −eE,St ` (S−(np−S))

SpA ` (np−S)−np X ` ((np−S)−np)−(np−S) −eSpA,X ` (np−S) −e

E,St,SpA,X ` S



Semantic skeleton of the sentence

E∗ ` (e → t) → (e → t) → t St∗ ` e → t →eE∗,St∗ ` (e → t) → t

SpA∗ ` e → e → t X∗ ` (e → e → t) → e → t →eSpA∗,X∗ ` e → t →e

E∗,S∗,SpA∗,X∗ ` t

corresponding λ-term :

((eE∗sSt∗)(xX∗

pSpA∗))

t



Computing the semantic representationvariable := semantic λ-terms (having the same type)

((λP e→t λQe→t (∃(e→t)→t (λxe(∧(P x)(Q x)))))(λxe(statemente→t x))

)((λP e→(e→t) λxe ((P x)x))(λye λxe ((speak_aboute→(e→t) x)y))

)↓ β

(λQe→t (∃(e→t)→t (λxe(∧t→(t→t)(statemente→t x)(Q x)))))(λxe ((speak_aboute→(e→t) x)x))

↓ β

(∃(e→t)→t (λxe(∧(statemente→t x)((speak_aboute→(e→t) x)x))))

in other words :

∃x : e (statement(x) ∧ speak_about(x,x))



Summary

• Grammar = lexicon providing words with (complex) syntactic categories

• Syntactic analysis = typing the sentence with S

= proof of S in a resource sensitive logic

• Morphism from syntactic categories to semantic typessyntactic analysis = non commutative resource sensitive proof→intuitionistic proof= λ-term expressing the compositional structure (actually still linear)

• Replace variables with λ-terms from the lexicon (non linear)perform β-reduction→formula representing the meaning of the analysed sentence

• Here syntactically poor system. Natural extension: more sophisticated logiclike Moorgat’multimodal extension with modalities and postulates.



Conclusion

I For a small fragment, we have a computable relation between some aspectsof meaning (roughly speaking who does what and quantifier scopes) and thesyntactic structure of a statement.

I From a syntactic viewpoint the fragment is much too restricted:

• discontinuous constituents

Je ne sais pas .

• medial extraction

The woman whoi called ti yesterday will call again.

• clitic pronouns in romance languages

Je la fais réparer.Je sais la réparer.



I Semantically restricted as well:

• Times, space are not well taken into account.

• To keep compositionality one must modify the semantic logic.

• Coreference between pronouns and antecedent is not handled.

• L’ajout de paramètres complique le modèle.

• Possible worlds interpretation is not very satisfactory.

• Incorporating lexical semantics, (relation between higher order predi-cates) is not easy to introduce.

I Practically, there are not many logical-categorial resources(except Grail for Dutch)

• Large lexicons? (automated acquisition from annotated corpora, Grail)

• Efficient parsing algorithms? (supertagging, distance minimization, Grail)

• Difficulties to cope with incorrect sentences.

Many open research directions.



Classical notions and results forTAGs and for dependency grammars

Alexander Dikovsky

LINA, University of Nantes

ESSLLI 2006 – p.1

PLAN

1. SYNSEM interface

2. Constituent structures

3. TAG, MCTAG

4. Dependency Graphs

ESSLLI 2006 – p.2

1. SYNSEM interface

sentence

Axioms (words)

Lambda-term

Compositional meaning

S

Rules

(universal

/ words?)

Questions:

- Rule to rule?

- Filter for syntax?

-Generation?

-Semantic analysis?

Syntax and semantics in a lexicalized view

Lexical

rules?

ESSLLI 2006 – p.3

1.1 Replies"Rule-to-rule?" - unrealistic

"Filter for syntax?"

inefficient (so inadequate)

"GenerationSyntAnalysis?" - Interfaces are asymmetric

Generation is simpler:lexical ambiguity: resolved (if not intentional)syntactic ambiguity: loweredanaphora: resolvedcommunicative structure: prescribed

Parsing must resolve these problems

ESSLLI 2006 – p.4

1.2 Compositional SYNSEM interfaceMorphisms between semantic and syntactic composition oprrators:

Generation: Γ : φ (semantic) f φ (syntactic)

Γφπ

1

sem, . . . , π

k

sem f φΓπ

1

sem, . . . ,Γπ

k

semParsing: Π : f (syntactic) φf (semantic)

ΠfS

1

syn, . . . , S

k

syn φf ΠS

1

syn, . . . ,ΠSk

syn

f φ

Γ :

f φ

Π :

...

φf

SkS1

f

φ

π1

πk

...

φf

...

ΠS1 ΠSk

ΓπkΓπ1 ...

ESSLLI 2006 – p.5

1.3 Compositional interface definitionalgebra Asyn of syntactic structures under syntacticcompositions

extension of Asyn to grammars G with efficient parsing

algebra Asem of semantic structures under semanticcompositions

compositional denotational semantics for Asem

extension of this semantics to Gefficient algorithms for Γ and Π

Syntactic structures: constituents and dependenciesTwo cases of grammars: TAGs/McTAGs for constituents,

D-grammars for dependencies

ESSLLI 2006 – p.6

1.4 Formal semantic structures

Logical expressions

Formulas valid in models

Proofs in formal systems

Static (intra-sentence) / Dynamic (of discourse)

Challenges:CompositionalityRealistic complexity

Underspecified semantic structures

ESSLLI 2006 – p.7

1.5 Underspecified semantics ideaEX [Hob83] (In most democratic countries) (most politicians) can fool

(most of the people) (on almost every issue) (most of the time)5 scopal NPs, 5! 120 combinations of scopes not entailedone by another, a single pertinent readingProposed solution:

to interpret the scopal NPs in situ (in their argumentpositions)

to put off the scope constraints to an extra-semantic(reasoning) level

Two cases of underspecified semantics: hole semantics for TAGs

and discourse plan semantics for D-grammars

ESSLLI 2006 – p.8

2. C-structuresCS over L ε ΣN (S N : sentence category) and F :features ( F : undefined)CS T : a labelled featured tree

U : nodes (u0: root),

(immediate dominance). C C1: C1: immediate

constituent of C

(tree PO of dominance),

(PO of precedence),

µ : UL (labelling),

ν : L FA (feature assignment)

NT: Xuf :v or Xu

v , when f is implied: u U, µu X, νX, f

v (node u labelled with X L, with f -value v)ESSLLI 2006 – p.9

2.2 Some axioms(complete axiomatization in [RVS94] )u u1 u u2 u1 u2 u2 u1

u1 u2 u1 u1 u2 u2 u1 u2

u1 u2 u1 u2

u1 u2 µu1 N

PR: defines a linear order (WO) on the leaves of TwT yield of T : string of labels of the leaves in WOEX: CS with head selection feature h:

V Ph

S

NP

John Adv V Ph

toh Mary

PPto

NPPtohwritesh

periodically Vh

ESSLLI 2006 – p.10

2.2 Substitution and adjunction

T

uB

T

uB1

SUBSTIUTION: T1 T uB T

T

uB

T

u u1

ADJUNCTION: T1 T uB a T Γ

uB2

uB1 uB

T

0

uB2

T

uB1

TuB

T0

FOOT

UT

0

UT0 u u2, T0 T

0

At the condition Γ on features in u, u1, u2

Important constraints: u: u is foot; uNA: no adjunction

REM: sub is monotone. adj can be made so

ESSLLI 2006 – p.11

3. TAG, MCTAGFor a finite set G of CS and a set of operations F,F G: the set of terminal CS in the closure of G under F ;LF Gdf wT T F

G

∆F df F

G finite G ; LF df LF G finite G

PR: Lsub LCF

TAG languages [JLT75] :

LTAG df Lsub, adj

PR: ∆sub, adj ∆adj , hence Lsub, adj Ladj

CS matrix: a finite list of CS T1, . . . , Tk

mc-adjunction: mcadjT ;u1, . . . , uk; T1, . . . , Tkdf

T u1 T1, . . . , uk Tk Γ simultaneouslymulti-component TAG languages [Wei88] :

LMCTAG df Lmcadj

ESSLLI 2006 – p.12

3.1 Expressiveness

PR: anbncndn n 0 LTAG LCF

PR: anbncndnanbncndn n 0 LMCTAG LTAGPR:

LMCTAG are NP-complete and non-semi-linear[Ram94]

They are polynomial and semi-linear under someconstraints to mcadj [VSW94]

ESSLLI 2006 – p.13

3.2 Copy language is TAGPR: COPY ww w a, b LTAG:G : α : β1 : β2 :

S

ε a S

SNA

S

NA a

b S

SNA

S

NA b

a

a a

S

SNA

b S

SNA

b

SNA

a

b

SNA

SNA

a

a

SNA

S

SNA a

bSNA

SNA a

SNA

SNA a

a

SNA

S

ε

ε

ε

ε

a

ESSLLI 2006 – p.14

3.3 TAGs for ALPlexicalized

each tree T G has a unique terminal head hT subcategorized

the head h corresponds to a predicate phT

there is a bijection between the arguments of phT andnonterminal leaves (slots) of T (EX: complements ofverbs). Hence, sub is needed .

modifiers and circumstantials are implemented byadjunction trees

semantically founded

each T has a semantics defined through that of phT

semantics is compositional

ESSLLI 2006 – p.15

3.4 Lexicalized TAG

?!?!

T1 : T2 : T3 : T4 : T5 :

G :

T :

NP

PPtowritesh

V Ph

S

S

Advh V P

V Ph

PPto

NP

John

NP

PPto

Ptoh

toh periodicallyh

Advh V P

V P NP

John

NP

Mary

NP

toh Mary

writeshperiodicallyh

T sub, adjT1, T2, T3, T4, T5John periodically writes to Mary

Ptoh

Problems:

to express discontinuous dependencies

of unnatural modifier-circonstantial dependencies

of clauses in complement slots (defined through adj andnot sub) ESSLLI 2006 – p.16

4. Dependency Graphs4.1 Projections, projectivityR: syntactic relations, Σ: words, N : nonterminals.

Dependency graph: DS of w x1...xn ΣN : a graphD x1, . . . , xn,,, ρ, where:

xixj : xj depends on xi (xi : governor, xj : subordinate)

a LO on x1, . . . , xn

ρ : R. For xixj, ρxi, xj d denoted xid

xj

wdf wD: carrier of DDomination : reflexive-transitive closure of

Projection of x : projxdf x

xx

Dependency tree (DT): x1, . . . , xn, a tree

Projective DT: all node projections are continuous intervalsESSLLI 2006 – p.17

4.2 Dependencies vs. constituentsProjective DTs can be seen as a by-product of selection ofa head constituent in non-unit constituents ( [Ior63] ,

[Gla66] , [Rob70] , [Jac77] ):

head selection: in every non-unit constituent C there isexactly one selected immediate constituent: C selC

for a unit constituent C x, headCdf x

for a non-unit constituent C, headCdf headselC

if C1 is a non-selected immediate constituent of C(C C1, C1 selC), then headC governs headC1:

headC headC1

PR: The DG induced by head selection is a projective DT

(see [DM00] for more facts).ESSLLI 2006 – p.18

4.3 DT induced by head selection

EX: projective DT derived from CS with selected heads:

V Ph

S

NP

John Adv V Ph

toh Mary

PPto

NPPtohwritesh

periodically Vh

John periodically writes to Mary

ESSLLI 2006 – p.19

4.4 Non-projective DT

EX: Many adequate DTs are not projective:

ESSLLI 2006 – p.20

4.5 Generalized Dependency Structures

gDS: k-component DG D D0, ..., Dk1, k 1, where:

one maximal connected component D0 is selected ashead component

one node in D0 is selected as head of D0 (and of D)

...

D0 D1 Dk

EX: a two-component gDS:

MORE

pred preposcompar

NGPp

preposrestrdobj

prepiobj

THANNGPpNG5Pnpers Vtr

ESSLLI 2006 – p.21

4.6 Monotone gDS compositionFor gDS δ and δ1, the composition of δ1 for a node α in δδαδ1 (or simultaneous composition δα1, ..., αnδ1, ..., δn )

unites the dependencies in δ and δ1; the head of δ1inherits dependencies of α;

preserves order: wδαδ1 xw1y, ifwδ x labelα y and wδ1 w1.

...

......

... ... ...

α

D20 D2n

DR1i D1mDL

1iD10

δ

δ1

DL1iD10 D2nD20 DR

1i D1m

δαδ1

ESSLLI 2006 – p.22

4.7 Example of gDS composition

Bδ A

aδ1

bδ2

a b b B c ca AδA,Bδ1Aδ1, δ2Bδ2

A

B c

REM: wδ uxv and wδ1 z implies wδxδ1 uzv.

So Dependency grammars defined as closure of finite gDSsets under this composition generate the CF-languages.

A more realistic solution in the next Lesson

ESSLLI 2006 – p.23

Conclusion

Compositionality of syntax is related with monotonicityof syntactic structure composition

Challenge: compositional definition of syntacticstructures expressing discontinuous dependencies

Completely compositional definition of syntacticstructures leads to grammars with efficient parsing

ESSLLI 2006 – p.24

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24-2

Categorial and generatingdependency grammars

Alexander Dikovsky


ESSLLI 2006 – p.1

PLAN

1. Categorial dependency grammars

2. Dependency structure grammars

3. CDG vs. DSG and complexity

ESSLLI 2006 – p.2

1. Categorial Dependency Grammars (CDG)( [Dik04] )1.1 Categories CatC and Valencies V C :

primitive categories C : det, pred, copul etc.

elementary categories:

neutral valencies for local dependencies: primitive C,iterative C, repetitive C and optional C? categories,

polarized valencies for non-local discontinuousdependencies:

positive for the beginnings of left C and right C,negative for the ends of left C and right C,

host valencies C, C and anchored valencies#C, #C for the (adjacent) ends of left / rightdiscontinuous dependencies.

ESSLLI 2006 – p.3

1.2 Categories and dependenciesComplex categories (1st order):

L1 LiCRj R1,

where L1, , Li, C,Rj , , R1 are elementary.

EX 1. theory modifdetdirobjattrrel

theory

modif

det attrrel

dirobj

2. etre clitdobjpredSaux

auxiliary host verb for a clitic in pre-position, expecting a left-

subordinate subject and a right-subordinate participle.

ESSLLI 2006 – p.4

1.2 Categories and dependencies

he took Tom Maryfromthanfromcandiesmore

restr compconj [#compconjcompar

compconj

la commission ne la

neg composneg

refusee

#clitdobj

pas

#clitiobj

lui aclitiobj clitdobjaux

clitiobj

clitdobj

composneg

#composneg

ESSLLI 2006 – p.5

1.3 Monoid of potentialsDual valencies v and v, v V C : together, C,C ˘Cand C,C ˘C define discontinuous dependency C.

Potential: a string of polarized valencies in V.EX: π CBACBANeutralization through reduction: ΓΓ:Γ Γ1vΓ2vΓ3, Γ

Γ1Γ2Γ3 and the condition of FirstAvailable (FA) holds: Γ2 has no occurrences of v and of v.

Monoid of potentials: being terminal and confluent, thereis the unique FA-normal form ΓFA. Henceforth, the correctproduct:

Γ1 Γ2df Γ1Γ2FA

EX: π FA BB

ESSLLI 2006 – p.6

1.4 Tree constraintsIn complex categories L1 LiCRj R1 CatC,

Li, Ri cannot be negative end valencies C,C, thoughthey may be negative hosts C, C,

C is primitive or negative C,C, or anchored negative#C,#C.

TC are waved for the generalized polarized categories(gCatC) in which all polarized valencies are factored out:

CP or L1 LiCRj R1

P , where:

C,Li, Ri are neutral and P is a potential,

C and P may be empty. Cε C.

EX: predSauxclitdobj for clitdobjpredSaux,

auxclitdobjclitiobj for clitdobj clitiobjaux.ESSLLI 2006 – p.7

1.5 CDG definition frameCDG G W,C, S, δ : W : words, C: primitive categories,S C, δ : W 2CatC a finite substitution (lexicon).Generalized CDG (gCDG): gCatC in the place of CatC.

D-form D,Γ of a sentence w a1...an:

D: a dependency structure on w,

Γ: a string of positioned categories,

initial D-form: a1 . . . an,, C11 . . . Cnn,

Ci δai, 1 i n,

terminal D-form: D, Sj.

GD,w Γ0jΓ0 δw w,,Γ0 D, Sj

: derivability in a dependency calculus.

DGdf D w GD,w, LGdf w D GD,w.ESSLLI 2006 – p.8

1.6 Generalized dependency calculusLl. V,E,Γ1C

P1iCβP2jΓ2

V,E aiC

aj,Γ1β

P1P2jΓ2, C C.

Il. V,E,Γ1CP1iC

βP2jΓ2

V,E aiC

aj,Γ1C

βP1P2jΓ2, C C.

Ωl. V,E,Γ1C

βP iΓ2 V,E,Γ1βP iΓ2.

Dl. V,E,Γ1αP1CiP CjP2Γ2

V,E aiC

aj,Γ1αP1PP2Γ2,

if CP C satisfies the FA-condition.

Exr: Write the corresponding rules for C and C?.

ESSLLI 2006 – p.9

1.7 A gCDG for a non-CF language

G1 :

a AA, AAA

b BCA, ASCA

c C, BC

TH: LG1 anbncn n 0.

A proof of a3b3c3 LG1 :

Types assignment: a3b3c3

AA1AAA2AAA3ASCA4BCA5BCA6C7BC8BC9

AAAAA

Ll

AAA AAA

Ll

AAAAASCA

BCA

BCAC

Lr

BA BCLl

CA

Lr

BAA BCLl

CAA

Ll

ASAAA

Ll

SAAAAAA

Dl 3

S ESSLLI 2006 – p.10

1.7 Corresponding dependency structure

is not a dependency tree

ESSLLI 2006 – p.11

1.8 Dependency calculus for DT [DD04]

Ll. V,E,Γ1CiCβjΓ2

V,E aiC

aj,Γ1βjΓ2, C C.

Vl. V,E,Γ1βαjΓ2 V,E,Γ1βjαjΓ2,

β C, lC, lC

Al. V,E,Γ1#αi

lαjΓ2 V,E,Γ1αiΓ2, α C,C.

Cl. V,E,Γ1 αiβj Γ2 V,E,Γ1 βjαi Γ2, where α pC

C,C,C C C C and either β X X pC

or β CAT C has no subexpressions α, α,#α, α.

Dl. V,E,Γ1CiCjΓ2 V,E aiC

aj,Γ1Γ2.

ESSLLI 2006 – p.12

1.8 Iteration rules

Il. V,E,Γ1CiC

βjΓ2

V,E aiC

aj,Γ1C

βjΓ2, C C.

Ωl. V,E,Γ1C

βiΓ2 V,E,Γ1βiΓ2.

Generalized dependency calculus - gCDG - LgCDG

Dependency calculus - CDG - LCDG

ESSLLI 2006 – p.13

1.9 CDG examples

w the red wines of Bordeaux are excellent

the det,

red modif,

Bordeaux prepos,

excellent copulattr

wines modifdetpredpostattrof postattrprepos

are predScopulattr

ESSLLI 2006 – p.14

1.9 CDG examples

EX: w les vins rouges bordelais sont excellents

les det

bordelais modif,

rouges modif,

excellents copulattr

vins detpredmodif

sont predScopulattr

ESSLLI 2006 – p.15

1.10 A DT through a proof in CDG

EX: δ : Les vins rouges bordelais sont excellents

detdetpredmodifmodif modif predScopulattrcopulattr

detdetpredmodif Ll

predmodif modif

Ir

predmodif modif

Ir

predmodif Ωr

pred

predScopul attrcopul attrLr

predSLl

S

ESSLLI 2006 – p.16

1.11 CDG for anbncn n 0

a #A, A#A

b ABC, A ASCc C, B C

Types assignment: a3b3c3

#A1A#A2A#A3A ASC4ABC5ABC6C7B C8B C9

#A, A#A, A#AA ASC

Vr,Cl

A,A, SC

ABC

Vl

A, BC

ABCC

Lr

AB

Vl

A,B BCLl

A,C

Cl,Lr

A,A,B BCLl

A,A,C

Cl, Lr

A,A,A,A,S

Vl, Ar

S

ESSLLI 2006 – p.17

1.12 CDG languagesProposition 1. For every CDG G, dependency structures inDG are trees.

Projective CGD: those not using the polarized valencies.

Proposition 2. For every projective CDG G, DS in DG areprojective DTs.

Proposition 3. Projective CDG are weakly equivalent to

CF-grammars: LCDGproj LCF .

CDG are more expressive than CF-grammars:Proposition 4. anbncn n 0 LCDG LCF .

They are weakly equivalent to gCDG.

Proposition 5. LCDG LgCDG. DCDG ! DgCDG.

ESSLLI 2006 – p.18

2. Dependency Structure Grammars( [BDF05] )

2.1 Generalized DSG (gDSG) G T,N, S,R

T : terminals

N : nonterminals

S N : axiom

R: rewriting rules A δ , where δ : a gDS over T N

with assignment of left and right potentials to nodes.

EX:restr

NG5NG6 MOREcompconjCompRestrG compconjTHAN

compar

PG

ESSLLI 2006 – p.19

2.2 Derivation treesDerivation trees generate gDS together with their potentials:

α

α1αk...

A

r

π1 δ1 πk δk

π δ

r : A δ0

wr : A X1...Xk

ΓL1 X1Γ

R1 ΓL

k XkΓRk πk ΓR

kπ π1df ΓL1

, ...,δ1 δk δneutrdf δ0α1, ..., αkδ

ΓR1 ... ΓL

k

Complete derivation tree T : πT ε

Languages: δT ∆G and wδT LG

iff T is a complete derivation tree of G.ESSLLI 2006 – p.20

2.3 Non-CF gDSG-languageS aa

G1 :

S A c

A ab c abA

LG1 anbncnn 0aa aa c c cabaa ab

S

A

ab

S

S

S

A

A

a

a

a

ε

a

a

a a

a a

a a a

a a a

ESSLLI 2006 – p.21

2.4 Derivation of discontinuous dependencies

than Maryfromhe took more candies from Tom

predpred restr

dobj

prepos compar

compconj

V G1

V G3

THANCompar

SV G

PGNGPp

NG V G4

V G7

FROMPG

NGV

NG1MOREcompconj

NG

FROMiobj

V G6

Ppprepos

compconjTHAN

ESSLLI 2006 – p.22

2.5 Dependency discontinuity measureValency deficit σG: max size of potentials in completederivation trees of G.

σG1 ". It is unlikely the case of natural languages,where σG is bounded by 2 or 3 .

!! Dutch cross-serial dependencies ( [BKPZ82] ) and thesimilar:

Wim Jan Marie omdat de kinderen zag helpen leren zwemen

infdobjinfdobjdet

Wim Jan Marie the children saw help teachbecause swim

iobj

predofinf

dobj

conj

pred

infiobj

Wimpred

zag and Janpred

helpen wouldt conflict with FA rule.

But Wimpred

zag and Janpredofinf

helpen do not!!

ESSLLI 2006 – p.23

3. DSG vs. CDG and Complexity

Strong generative power:TH DCDGproj ! DCDG ! DgCDG ! DgDSG

Weak generative power:TH LCFG LCDGproj LgDSGσω !

LCDG LgCDG LgDSG

(gDSGσω : gDSG with bounded valency deficit).

TH [DD04]

1. Worst case complexity of CDG is On54p, where p is thenumber of polarized valencies.

2. If σG is bounded by a constant , then the complexity is

On3.

ESSLLI 2006 – p.24

3.1 DSG,CDG vs. TAG,MCTAG (1)

For Lmdf d0an0d1a

n1 . . . dma

nmdm1n # 0 and for all m 4,

Lm $ LTAG.

TH [Dik04] Lm is generated by CDG Gm :

a0 D0D0Am . . . A1,

d0 SD0 and dm1 Dm,

ai #rAi

rAi, #rAiDi,

di Di1

rAi0 % i m.

Lcopydf wcw w a, b LTAG.Hypothesis: Lcopy $ LDSG.

Unlikely: LDSG closed under L and L1 L2.

ESSLLI 2006 – p.25

3.2 DSG,CDG vs. TAG,MCTAG (2)MIXdf w a, b, cwa wb wc

Hypothesis[E.Bach]: MIX $ LMCTAG.

TH [BDF05] MIX LCDG .

TABLE OF CATEGORY ASSIGNMENTS

a BCS a SCB a BSC, CSB

a BCSS a SSCB a BSSC, CSSB

b B b B

c C c C

Lcopy $? DSGTAG w LinIG w CombCG

L5

DSG w CDG

MIX $? MCTAG

CF w DSGσc w CDG

σcMCTAG w MinG w MCFG

ESSLLI 2006 – p.26

Conclusion

First-available-dual-valency rule underliescompositional definitions of discontinous dependencies

FA-rule is linguistically founded

FA-rule valency based grammars are efficientlyparsable

ESSLLI 2006 – p.27

Referen

ces

[BD

F05]

D.B

echet,A

.D

ikovsky,and

A.Foret.

Dependency

Structure

Gram

mars.

InP.

Blache

andE

.S

ta-

bler,editors,

Proc.

ofthe

5thInt.

Conf.

“Logical

Aspects

ofCom

putationalLinguistics”(LA

CL’2005),

LNA

I3492,pages18–34,2005.

[BK

PZ82]

J.Bresnan,R

.M.K

aplan,S.P

eters,andA

.Zaenen.

Cross-serial

dependenciesin

Dutch.

LinguisticIn-

quiry,13(4):613–635,1982.

[DD

04]M

ichaelD

ekhtyarand

Alexander

Dikovsky.

Cate-

gorialDependency

Gram

mars.

InM

.M

oortgatand

V.P

rince,editors,

Proc.

ofIntern.C

onf.on

Catego-

rialGram

mars,pages

76–91,Montpellier,2004.

[Dik04]

Alexander

Dikovsky.

Dependencies

asC

ategories.

In“R

ecentA

dvancesin

Dependency

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CO

LING

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orkshop,pages90–97,2004.

27-1

Underspecified semantics andcompositional grammar interface

Alexander Dikovsky


ESSLLI 2006 – p.1

PLAN

1. Underspecified semantics

2. TAG - hole semantics interface

3. Discourse Plans

4. DP - CDG interface

ESSLLI 2006 – p.2

1. Underspecified semanticsWhy underspecification: e.g. for generation:

generate from valid logical form leads to excessivecomplexity

in machine translation, resolution of quantifier scope hasminor effect on the translation. So to couple thecorresponding syntactic structures, a partially definedsemantic structure may suffice

on the other hand, flattening the recursive semanticstructure can lead to spurious ambiguity; so anintermediate abstraction level is needed

Ideally: underspecified semantic structure should preserve

enough information to construct all and only possible readings

ESSLLI 2006 – p.3

1.1 Underspecified MRS [CFSP99]

Language: conventional predicate calculus extended with flatconjunction

and disjunction

and generalized quantifiers(GQ) GenQx,R,Bdf λB λR. Q xφQRx, BxEX: Genx, dogx, Geny,

whitey, caty, chasex, y

for every dog chases some white cat

Partial description of scopes of GQ using handle variables H:

h0 handle, h1 : Genx, h2, h3, h2 : dogx, h4 : chasex, y, h5 : Geny, h6, h7,

h6 : whitey, h7 : caty body, constraints subsumes

h0, h1 : Genx, h2, h3, h2 : dogx, h4 : chasex, y, h5 : Geny, h6, h7,

h6 : whitey, h7 : caty, h3 h4 Constraint h3 h4 implies: h1 overscopes h4

REM: Subsumption is monotone and includes composition of MRS

ESSLLI 2006 – p.4

1.2 Hole semantics [Bos95]

Language: another extension of predicate calculus with GQ:H (hole constants), Lc (label constants), Lv (label variables),K (basic language variables and constants)Formulas F :

l : Rni1, . . . , in F (including GQ), l Lc Lv;

ij K H Lc Lv

h l F (h overscopes l), h H, l Lc

φ, ψ F for φ F and ψ F

Saturated formulas φ: without label variables

Not:Sφ: all scope (i.e. label and hole) constants in φ

ESSLLI 2006 – p.5

Hole semantics. ContinuedScoping relation of a saturated formula φ is the minimal PO

φ on Sφ such that:

k φ k for all k Sφ

k φ k

if k k in φ

k φ k

, k φ k if l : Rn. . . , k, . . . , k, . . . is in φ

Plugging is an injection P : HφLφ

Plugging P possible for φ: k P φ k k k k φ k

Possible pluggings P of φ define its scoping models, i.e. the

basic language formulas resulting from φ by specializations

of scopes consistent with φ

ESSLLI 2006 – p.6

Hole semantics. FinishedEX: Every dog chases a cat

φ : l0 : x, h1, h2, h1 l1, l1 : Dx, h2 l2, l2 : Chx, y, l3 :

y, h3, h4, h3 l4, l4 : Cy, h4 l2

the only two possible pluggings for φ:P1 h1 l1, h2 l3, h3 l4, h4 l2P2 h1 l1, h2 l2, h3 l4, h4 l0correspond to two different readings:l0 : x, l1, l3, l1 : Dx, l2 : Chx, y, l3 : y, l4, l2, l4 : Cy

l0 : x, l1, l2, l1 : Dx, l2 : Chx, y, l3 : y, l4, l0, l4 : Cy.

This technique is applied to other basic languages, in particu-

lar to DRT [KvGR]

ESSLLI 2006 – p.7

Analytic SYNSEM interface. TAG case

Π : f (syntactic) φf (semantic)

ESSLLI 2006 – p.8

2. TAG - hole semantics interface [GK03]

Interface: Predicate arguments (linked with NP-groups byidentical variables) are added through substitutionAdverbials and quantifiers are added through adjunctionEX: John loves Mary

S

NP x1V P

NP x2 NPm

NP j

loves Mary

John

namej, john

namem,maryl0 : lovex1, x2

V

resulting in:

l0 : lovej,m, namej, john, namem,mary

ESSLLI 2006 – p.9

TAG - hole semantics interface finished

EX: every dog barks

Not: N s vs. Ns: variable s unifies with superscript / underscriptterm

dog

Nx1,l1

every

Nx,s2

Det

S

Nx2,l2 V P

V

barks

N

x,s1

l0 : x, h1, h2

h1 s1, h2 s2

l1 : dogx1l2 : barkx2

resulting in:

l0 : x, h1, h2, h1 l1, h2 l2, l1 : dogx, l2 : barkx

ESSLLI 2006 – p.10

3. Discourse Plans[Dik03, DS05] Underspecified semantic structure languageDPL extending the standard logical syntax with cognitivefeatures sufficient to represent the various linguistic featuresmarked for by surface language means (prosodical,morphological, syntactic or lexical)

uses abstract situations (different from referentialsituations of [BP83] ) as elementary predications

accounts for communicative structure

is closely related with dependency structures

has a dynamic compositional semantics close to that ofDRT [Kam81, vEK97]

ESSLLI 2006 – p.11

3.1 Abstract situationsInvariants of communicative views (= semantic diatheses)

gives

VIEWS: salience, typeof actants

dthinf

canonical

given . . by

. . .

to give

giving

dthgerund

sit give

surface form

dthpassive

. . .

EX(G. Frege, "Begriffsschrift" [Fre79] ):Bei Platae siegten die Griechen uber die Perser andBei Platae wurden die Perser von den Griechen besiegtuse the same situation: siegenSBJ,OBJ

Arguments of situations vary from diatheses to diathesesaccording to their intended communicative ranks

Their types constrain values: s, n, q, c and instances(e.g., na n, scauseff s, qquant q, cintens c)

ESSLLI 2006 – p.12

3.2 Situation definition (profiles)Canonical profile EX: open:

give

SBJna ,OBJn, RCPna

scausmov

Diatheses (specified by intended commranks: T, O, , , )dthfpassive PAGT SBJ, SBJ OBJT

spsv

one of passive profiles in English:

dthfpassivegive

SBJn, PAGTna , RCPna

spsv

. . .

(The appleSBJ:T was given to AdamRCP: by EvePAGT:)Actants: arguments identified by thematic roles obligatory inat least one profileCircumstantials: all other arguments (identified by attributes)Lexical test: Circumstantials are not used in the gloss:giveSBJ : X, OBJ : Y, RCP : Z :: “X deliberately changes belonging of Y to X for itsbelonging to Z at will of Z”

Circumstantials obligatory in DP: ILLOCUTIVE STATUS (e.g. DCL,

INTERROG, CONDIT), ASP (NEUTR, PERF, PROGR), TIME (PRES, PAST)ESSLLI 2006 – p.13

3.3 DP examples (in place of DPL definition)Situation open canonical profile:

open

SBJna ,OBJn, INSTRn

seff

openseff :

ILL=DCL

ASP=PERF

TNS=PRES

DS

g Johnna

SBJna : T

x doorn

OBJn :

y keyn

INSTRn :

newqprop

Qualqprop

easilycint

Intenscint

ESSLLI 2006 – p.14

3.4 A decausation diathesis

open: dthdecausins , SBJ, SBJ INSTRT,OBJ OBJ

seff

dthdecausinsseff

openseff :

SBJna : OBJn : INSTRn : T

ILL=DCL

ASP=PERF

TNS=PRES

DS

x keyn

SBJn

newqprop

Qualqprop

y doorn

OBJn

easilycint

Intenscint

ESSLLI 2006 – p.15

3.5 Another decausation diathesis

open

dthdecausO , SBJ, SBJ OBJT, INSTR sfpd

dthdecausO

sfpd

openseff :

SBJna : OBJn : T INSTRn :

ILL=DCL

ASP=PERF

TNS=PRES

DS

x doorn

SBJn

easilycint

Intenscint

ESSLLI 2006 – p.16

3.6 Ultimate decausation diathesis

open: dthdecaus , SBJ, OBJ, INSTR

qs

leave2 canonical profile:

leave2

SBJna,OBJn, DFSq

seff

leave2

seff :

ILL=DCL

ASP=PERF

TNS=PRES

DS

g Johnna

SBJna : T

x doorn

OBJn :

dthdecaus

qs

openseff :

SBJna : OBJn : INSTRn : DFSq :

ESSLLI 2006 – p.17

3.7 Actant’s scopeScoping rule: follows the actants’ topicality order :

SBJ > OBJ > oblique roles (cf. [Cro03] )(SBJ actant outscopes the OBJ which outscopes the oblique)

take2

SBJna ,OBJn

snef

take2snef :

ILL=DCL

ASP=PERF

TNS=PRES

DS

all newsmanna :ncoll

SBJna : T

some trainn

OBJn :

g Geneva site

DESTsite

Genx,

newsmanx, Geny,

trainy, desty,Geneva, take2x, y

ESSLLI 2006 – p.18

3.8 Actant’s scope continued

take2snef :

ILL=DCL

ASP=PERF

TNS=PRES

DS

all newsmanna :ncoll

SBJna : T

x one trainn

OBJn :

g Geneva site

DESTsite

besatt :

ILL=DCL

ASP=PERF

TNS=PRES

DS

x

SBJn : T

latec

DFSc :

(represents the reading:Geny,

trainy, desty,Geneva, latey, Genx,

newsmanx, take2x, y )

ESSLLI 2006 – p.19

3.9 Abstracted DPtake1

seff :

ILL=DCL

ASP=PERF

TNS=PRES

DS

(x)na (he)

SBJna : T

candynmass

OBJn :

ιmore

Quantqrestr

ιcandy nmass

1 dthrelseff

take1seff : O

SBJna : O OBJn : O ORIGn :

2

ILL=REF

ASP=PERF

TNS=PRES

DS

x

SBJnaOBJng Mary na

ORIGn

g Tom na

ORIGn :

ESSLLI 2006 – p.20

3.10 Aggregation / Coordinationaliassfpd :

ILL=DCL

ASP=PERF

TNS=PRES

DS

ι

SBJna : T

x (itna)

1 dthrelseff

inviteseff :

SBJna : T OBJn : O

2

ILL=REF

ASP=PERF

TNS=PRES

DS

g John na

SBJna OBJn

A,contr

DFSq :

no g Maryna

1

g Janena

2

ESSLLI 2006 – p.21

Generation SEMSYN interface. CDG case

Γ : φ (semantic) f φ (syntactic)

ESSLLI 2006 – p.22

4. DP - CDG interface4.1 Transduction Dependency Grammar (TDG) ∆ Γ, G:

G Σ, Cat, S, δ: a CDG,

Γ: a top-down finite transducer from DP π to typed stringsw1 : c1 . . . wn : cn (wi Σ, ci Cat)

∆π,w1 . . . wn, c1 . . . cn, D

π, q0

Γ w1 : c1 . . . wn : cn w,, c1 . . . cn

G D,S

Transitions of Γ:

ΓK, q w : c (K a leaf of π, w Σ, c δw)

ΓKs1 : π1, . . . , sm : πm, q

Γwj1 , qj1 . . .Γπi1 , qi1 . . .ΓK, qk . . .Γπiu, qiu . . .Γwjv , qjv

Language of ∆: L∆ x Σ π,C,D∆π, x, C,DESSLLI 2006 – p.23

4.2 Examples of transition rulesNotation: States DS , GS : DS : discourse status,GS HEAD, LEFT, RIGHT, CLS, TNS, NUM, PER: grammatical status.

E.g., HEAD , LEFT , RIGHT determine categories. Namely,

α1 ... αl γ!β1!...!βr δ GS if

LEFT α1 ... αl, RIGHT β1!...!βr, HEAD γ

EX: they have recently fired twenty people

firescauseff :

ILL=“dcl”ASP=“perf”TNS=“past”

DS

ytheyna : ncoll

SBJna : T

peoplencoll

OBJna :

twentyqqnt

NUMqrestr

recentlycrestr

TMPcrestr :

ESSLLI 2006 – p.24

4.3 Main situation ruleRule pattern:

Γ

π1 SBJnT, OBJ seff, DS ILL “Dcl”

β1sf1SBJ, GS NUM n, PER p, HEAD “pred”,vf1" GS ,of1OBJ, GS HEAD “dobj”

Rule instances (under mutually exclusive conditions)Past perfect instance:π1 SBJnT, OBJ ; TMPt

seff DS .ASP “Perf” DS .TIME “Past”

β1sf1

“have”CLS VAux, NUM n, PER p, TNS DS .TIME, HEAD “S”,

LEFT sf1 . HEAD , RIGHT “aux # ppart”!“circ # time”"

TMP, HEAD “circ # time”

vf1FORM “Partc”, LEFT ε, HEAD “aux # ppart”, RIGHT “dobj”

of1ESSLLI 2006 – p.25

4.4 Main situation rule application

This rule creates the categories:

pred S!aux# ppart!circ# time" for the auxiliary verb and

aux# ppart!dobj for the participle

So it defines the analysis:

ESSLLI 2006 – p.26

ConclusionSemantics underspecification serves for semanticscompositionality and efficiency

Ideally, underspecified semantics expresses all andonly meaning components marked for by surfacelanguage means

Challenge: how to capture the adequateunderspecification level

Underspecified semantics for compositional syntaxgrammars provide natural compositional

Semantics GrammarGrammar Semantics

interfaces

Both interfaces are to be coupled in a natural way with alogical reasoning formal system

ESSLLI 2006 – p.27

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27-2

Categorial minimalist grammars

Christian Retore, Universite Bordeaux 1

joint work withAlain Lecomte (U. Grenoble II) and Maxime Amblard (U. Bordeaux 1)

Universita degli studi di Verona – Maggio 2006

Equipe SignesINRIA-Futurs & LaBRI-C.N.R.S.

& dep. sciences du langage, Universite Bx3 Michel de Montaigne

Contents

1 General remarks 3

2 Reminder on syntax and semantics in categorial grammars 10

3 Stabler minimalist grammars 19

4 Categorial Minimalist Grammars ( a la Lambek) 25

5 Syntax/semantics 36

6 Results 43

7 In progress: CMG proofnets 44

8 Perspectives 45

1. General remarks

1.1. Syntax boundaries

• Inflectional morphology

– for: depending on the language syntactic construction with ex-plicit words OR inflection

– against: different techniques (finite state automata, transducers— although in Navajo....)

• Logical semantics (who does what)

– for: rather syntactic phenomena (logical syntax)– against: different techniques (e.g. in dependency approach

formal grammars 6= dependency graphs)

• prosody (which is related to syntactic structure)

• lexical semantics (→ restricted selection)

• encyclopaedic knowledge (→ getting rid off some ambiguities)

1.2. Linguistic theories and their mathematical models

Theories: generative grammar dependency grammar others?

Mathematical models:

• context-free grammars, tree grammars, composition different fromsubstitution and term rewriting (e.g. adjunction in TAGs)

• unification grammars

Algorithmic complexity of parsing

• Unification grammars DCG GPSG HPSG (undecidable parsing)

• Context sensitive unification grammars like LFG (decidable parsing)

• TAGs, Range Concatenation Grammars (polynomial)

Given a theory, is there a privileged model?(generative grammar −→ TAGs?)Given a model, is there and underlying theory(cf. exegeses of HPSG by Pollard & Sag)

1.3. Modelling, Parsing, Generation

Parsing or generation?

• As far as analyse is concerned:

– word order does not mind,sentences are grosso modo correct.

– transformations and empty elements are a challenge

– what do we do with parse structures?

• As far as generation is concerned

– word order if crucial.

– transformations and empty elements are welcome

– out of what kind of object do we build (parse structure of) sen-tences.

1.4. Cognitive realism, empirical coverage

What do we model?

• corpora? normed language from some norm?

• examples representative of language faculty(internal language of X)

• pathological examples (like magma study for physics)

Problems

• linguistic resources

(annotated corpora, grammars)

• are the solution to specific phenomena compatible

• surgeneration (never adressed in main stream NLP:

corpora are a priori assumed to be correct)

1.5. Convergence towards generalized categorial grammars

Structure MLLlinear λ-calculus

Phonological Form Logical Form

(word order,...) (logical structure)

• Pollard 2004: High-Order Categorical Grammar

• De Groote 2001: Abstract Categorial Grammars

• Muskens 2003: Lambdas, Language and Logic

• Lecomte Retore 2001 Minimalist Categorial grammars

• Perrier 2001: Interaction Grammars

1.6. Generative grammars

Usual criticisms:

• transformations are algorithmically untractable

(analyse / generation)

• derivation −→ representations levels −→ conditions on each

• what is Logical Form?

Awards:

• links between languages (principles and parameters)

• transformations: links between related sentences (questions / an-swers)

• syntax and semantics (coreference, (generalized) quantifier scopes)

1.7. Outcome of Stabler’s formalisation of the minimalist program

• Good computability: polynomial, like LCFRS simple positive RCG)

• Derivational formalisation (generative-enumerative syntax)

and representational formalisation (model-theoretic syntax)

(cf. Pullum et Scholz 2001)

Monnich, Morawietz et Michaelis (2001-2004):Set of an MG parse trees =image by a binary relation definable in monadic second order logicof a set of regular tree definable in monadic second order logic(hence can be analyzed with pushdown of pushdown automaton)

Very fine, but ... it’s quite difficult to write lexicalized grammar:One must follow some linguistic theory.

2. Syntax and semanticsin categorial grammars (reminder)]

2.1. Syntactic categories

B = S, sn, n, ...

F ::= B | F \ F | F / F

if u : A and f : A \B then uf : B (AB and Lambek)if u : A et f : B / A then fu : B (AB and Lambek)if u : A et uf : B then f : A \B (Lambek only)if u : A et fu : B then f : B / A (Lambek only)

2.2. Semantic types

Church 1930, Curry 1940, Montague 1970

2.2.1. Logical formulae in simply typed λ-calculus with 2 basic types:

• individual e

• truth values t

• n-ary predicate : e → (e → (e → (· · · → t)))

• n-ary function : e → (e → (e → (· · · → e)))

• logical constants ∧,∨,⇒ : t → (t → t)∃,∀ : (e → t) → t

2.2.2. Syntactic categories and semantic types

S∗ = t sentence: truth values / propositionssn∗ = e individualn∗ = e → t unary predicate

(A \B)∗ = (B / A)∗ = A∗ → B∗ propagation to every formula

2.2.3. Lexicon: example

aimer (np\S)/np e → e → t λxλy.aimer(y, x)

tout ((S/np)\S)/n (e → t) → (e → t) → t λPλQ.∀xP (x) ⇒ Q(x)

enfant n e → t λx.enfant(x)

une (S/(np\S))/n (e → t) → (e → t) → t λPλQ.∃xP (x) ∧Q(x)

institutrice n e → t λx.student(x)

2.2.4. Parsing example:

Two syntactic analysesfor two possible readings.

1. (tout enfant)(λy (une institutrice) (λx aimer (x,y)))∀z enfant(z) ∧ (∃s instit(x) ⇒ aimer(z, s))

2. (une institutrice)(λx (tout enfant)(aimer x))∃s (instit(x) ⇒ ∀zenfant(z) ∧ aimer(z, s))

S

HHHHH

S/(np\S)

HHH

S/(np\S)/ntout

nenfant

(np\S)2

S

HHHHHH

HH

S/np1

S

HHHH

np2

(np/S)

HHH

(np/S)/npaimer

np1

(S/np)\S

HHHH

(S/np)\S/nune

ninstitutrice

S

HHHH

HHH

S/np1

S

HHHHHH

S/(np\S)

HHH

S/(np\S)/ntout

nenfant

(np/S)

HHH

(np/S)/npaimer

np1

(S/np)\S

HHHH

(S/np)\S/nune

ninstitutrice

2.2.5. Explanation

Why does it works?

• syntactic analyse = proof in the Lambek calculus

• forgetting directions ⊂ proof in MLL ⊂ intuitionnistic logic

• type morphism −→ intuitionistic proof, lambda-term

• variable := lexical lambda-terms (same type)

• beta redution −→ proof of S∗ = t i.e. a proposition

2.2.6. Critics

• Too restricted syntactic formalism:

discontinuous constituents: ne...pas

middle extraction: Le livre quei [tu lis (ti) ces jours-ci] est Samarcande

• some analyses do not have a semantic counter part

(type raising is mandatory)

e.g. Joan: (e → t) → t and not e because of

Joan et tous les invit:’es sont partis.

an analysis with Joan: sn has no semantic counterpart

• the syntactic cateogry of the quantifiers depends on their syntacticposition

”tout” has a syntactic type for subject position

another for object position, etc. pour celle objet, etc...

3. Stabler minimalist grammars

3.1. Overview

• based on the minimalist program

• lexicalised grammars

• generative capacity : MC-TAG / MCFG

• polynomial parsing

• principle and parameters approach to language variation

• relation bewteen related sentences

(by movement and transformations):

? questions

(1)Combien de livres que Tabucchi a ecrit aime-t-il?

(2) Il aime trois livres que Tabucchi a ecrit.

? passive

(3) Ce livre a ete ecrit par Pavese

(4) Pavese a ecrit ce livre.

• raises some important syntactic/semantic questions:

possible or impossible coreference

il=Tabucchi (1) possible (2) impossible

3.2. Analysis structures

• binary trees

• leaves: list of features

• internal node : ”<” or ”>” leading to the head

• maximal projection of h: largest tree whose head is h

<

HHH

· · · > HH

· · · >HH· · · h

3.3. Lexicon

Features

• base d, n, v

• select =d for d in base

• licensees -case, -wh

• licensors +case, +CASE, +wh, +WH

Lexicon : list of features /mot/ (mot)

3.4. Generative rules

MERGE

<HH

t1 t2

>HH

t2 t1

if t1 ∈ Lex otherwise

MOVE

<

HHH

tete+f

•

HHHH

projection maximale−f

>

HHH

• HH

<

HHH

· •

HHHH

si +f riensi +F PF seulement

3.5. Lexicon example

aimer =d +case =d v

une =n d −case

institutrice n

tout =n d −case

enfant n

infl =v +case tcomp =v c

Exemple a faire au tableau

4. Categorial Minimalist Grammars

Only elimination rule

AB or Lambek grammars commutative product

b HH

b/a a

bHHb a\b

Merge Move

[Partially commutative linear logic, de Groote, 1996]

4.1. Some differences:

• internal subject hypothesis

(like in Radford 97 ands some other minimalist papers)

• commutative product

set of features instead of list of features

4.2. Example of a categorial minimalist lexicon

aimer (d\k1\v)/d1 =d +case =d vune k × d/n =n d −caseinstitutrice n ntout k × d/n =n d −caseenfant n n

infl (k\v)/v =v +case tcomp v/c =v c

5. Syntax/semantics

5.1. Logical system for semantics

As usual:

?logical formulae as λ-terms.Base types e and t a la Montague.

BUT moreover?λ-terms with explicit contexts:list of free variables

t

HHHH

HH

t → t t

HHHHHH

(e → t) → t e → t

e ` t

HHHHH

t → t e ` t

HHHHH

(e → t) → t e ` e → t

e, e ` t

HHHH

e ` e e ` e → t

HHH

e ` e e → e → t

5.2. Semantic rules

• application : [→]

• abstraction in the tree hosting the move

Γ, z : Z ` u : U

Γ ` (λz. u) : Z → U [EXTRACT ]

• application, for type raising

∆ ` z : (T → U) → V Γ ∪ [x : T ] ` u : U[RAISE]

∆ ∪ Γ ` z(λx.u) : V

• l’application, sans montee de type

∆ ` z : T Γ, x : T ` u : U[NORAISE]

∆ ∪ Γ ` (λx. u)z : U

5.3. Syntax/semantics

SY N , syntactic calculus

• connectives: ×, /, \

• only elimination rules (encoding move and merge

SEM , semantic calculus

• connective →

• semantic rules (derived rules)

parallel SY N ‖ SEM :

syntaxe semantique

merge : [/][\] [→]move [Extract]projection [RAISE]ou[NORAISE]

• every leaf in SEM has a coindexed part in SY M

• each step and its counterpart are executed

in the same order in their respective derivations

5.4. Sample lexicon

aimer =d +case =d v k\d\v/d ` λxλy.aimer(y, x)

une =n d −case k × d/n ` λPλQ.∃xP (x) ∧Q(x)

institutrice n n ` λx.instit(x)

tout =n d −case k × d/n ` λPλQ.∀xP (x) ⇒ Q(x)

enfant n n ` λx.enf(x)

infl =v +case t (t/k)/v ` λP.pass comp(P )comp =v c v/c ` λP.P

6. Results

• syntax/semantics correspondence

extended to a richer syntactic system.

• a single syntactic category for a quantifier,

whatever might be its syntactic position.

• understanding movement

– in the structure which host the moved constituent: λ-abstraction

– for the moved constituent : type raising

7. In progress: CMG proofnets

Minimalist grammars without movement : bounded pushdown for partialstructures, insertion only when there will be no further movement.

Word order can be reconstructed without distinct \ and /

• first application of a lexical function : argument after fucntion (lexicalmerge)

• otherwise argument before the function (non lexical merge)

Proof-nets (graphs):

• equivalent formalism

• better for product(complicated normal forms)

• avoid co-indexation of hypothesis to be cancelled simultaneously

• better algorithms for constructing analysis(e.g. minimizing axioms length, Moot 2004

• formulae −→ trees taking into account the order of the operations

8. Perspectives

• possible or impossible coreference for anaphora resolution

incremental calculus of binding principles and small clauses

or of Reinhardt/Reuland semantic binding (Bonato)

(1) Carlotta’s dog thinks that he hates him.

(2) * Ili aime trois livre que Tabuchi i a ecrit.

• semantics of questions

(Maxime Amblard)

(3) Quel train Pierre prend?

(4) Quel train prend Pierre? (plus difficile)

• clitics, clitic climbing with correct control intepretation (in progress,Amblard)

(similar to Moot/Retore 2005 for multimodal categorial grammars orto Stabler 2001)

(5) Je repare ma voiture.

(6) Je la repare.

(7) Je sais la reparer.

(8) Je la fais reparer. (”la” is being repared)

(9) Je te permets de venir. (”te” viens)

(10) Je te promets de venir (”je” viens)

Extending Montague semantics to a richer syntax.

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