background acg example: cfg semantics for tag cg cvg ... · background acg example: cfg semantics...

Background ACG Example: CFG Semantics for TAG CG CVG Conclusion 1 / 82

Course Material

At http://www.loria.fr/equipes/calligramme/acg/

Slides and handout (1st week and 2nd week)Lecture Notes (2nd week, moving version)Software and examples (2nd week)

http://www.loria.fr/equipes/calligramme/acg/


Advances in Abstract Categorial Grammars:Language Theory and Linguistic ModelingFormal-Language-Theoretic Properties of 2nd Order ACG1

Syntax-Semantics Interface: an ACG Perspective2

Makoto Kanazawa1 Sylvain Pogodalla2

[email protected] Tokyo

[email protected]/INRIA Nancy–Grand Est

France

21st ESSLLI – Bordeaux, FranceJuly 27–31, 2009

[email protected]

[email protected]

http://esslli2009.labri.fr/


OutlineFirst Lecture

1 Motivations and AffiliationsArchitecture of Grammatical Formalismsλ-terms in the Syntax... and Everywhere

2 Abstract Categorial GrammarPrinciples and DefinitionACG Composition: The PictureAbout Word Order

3 Example: Context Free GrammarsProviding a Syntax-Semantics Interface to Context-Free GrammarsModularity of the Components

4 Example: A Semantic Component for Tree Adjoining GrammarA Functional View on TAGTAG as ACG


Outline (cont’d)Second Lecture

4 Example: A Semantic Component for Tree Adjoining GrammarA Functional View on TAGTAG as ACG

5 On the Relation Between ACG and Categorial GrammarsThe CG Approach to Scope AmbiguityRemoving Ambiguity From Syntax

6 Example: Convergent GrammarsArchitectureImplementationCVG into ACG Encoding

7 Conclusion


The Tectogrammatical and Phenogrammatical Distinction

On the Grammar Architecture [Curry(1961)]

Tectogrammatical: abstract combinatorial structure of the grammarPhenogrammatical: concrete operations on syntactic data structures (strings,trees, descriptions)Contrary to the view that:

Syntactic objects are the main objectsSemantics (and phonology, and . . . ) are by-products

Related Works

[Montague(1974)], [Dowty(1982)],[Ranta(1994)],[Oehrle(1994), Oehrle(1995)],[Muskens(2001)], [Muskens(2003)], [Kracht(2003)], [Pollard(2004)],[Pollard(2008)]. . .


Some Observations on Various Grammatical Formalisms

Syntactic Objects (trees, proofs, f-structures) are somehow prior andsemantics must be parasitic on those syntactic objects

[Muskens(2001)]

Changing the syntactic analysis to simplify one mapping makes the othermapping more complex. A third possibility is to keep both correspondencessimple by localizing the complexity in the syntactic component itself.(...)[T]here is a mismatch between phonology and meaning, which has to beencoded somewhere in the mapping among the levels of structure. If thismismatch is eliminated at one point in the system, it pops up elsewhere.

[Jackendoff(2002), p.15]


Mainstream ArchitecturesOn the Place of the Syntactic Component

Three Components

Phonology

Syntaxsyntacticstructures

Semantics

Generative theory: “Free combinatoriality of language is due to a single source,localized in syntactic structure”Syntactocentric formalisms = function from the syntactic component to theother ones


A Tripartite Parallel Architecture

Three Components

Phonology

Syntax,Syntacticstructures

SemanticsInterfaceInterface

InterfaceInterface

Weakly Syntactocentric formalisms = relation between the syntactic component andthe other ones

Language comprises a number of independent combinatorial systems whichare aligned with each other by means of a collection of interface systems.Syntax is among the combinatorial systems, but far from the only one.[Jackendoff(2002)]


Grammatical FormalismsA Classification by Architecture

General Classification

Cascaded Parallel (Curryesque)Syntactocentric GB, MG CFG, TAG (?), ACG (?)(Weakly) syntactocentric HPSG, CVG

The Syntax-Semantics Interface

The combinatorial principles of syntax and semantics are independent;there is no “rue-to-rule” homomorphism. (...) [T]he mapping betweensyntactic and semantic combinatoriality is many-to-many.

[Culicover and Jackendoff(2005)]

In-situ Operators

Quantified noun phrases, interrogative wh-expressions, topicalization...Their semantic scope is under-determined by their syntactic position


λ-terms in the Syntax... and Everywhere

What are λ-terms useful for?Montague-like semanticsGeneralization of trees and stringsAny kind of signatures (atomic types and typed constants): FOL propositions,descriptions (LFG f-structures, URL), other logicsVery well studied generative systemVariable binding system

Not that New in Syntax

[Ranta(1994)],[Oehrle(1994), Oehrle(1995)], [Muskens(2001)],[Muskens(2003)], [Kracht(2003)], [Pollard(2004)], [Pollard(2008)]. . .Movements in GB/MG to get S-structures.Index Transfer syntactic rule in Binding TheoryTAG → MCTAG


ACG: a Grammatical Framework

Main Features

ACG is a (grammatical) frameworkAn ACG G generates two languages:

The abstract language A(G )The object language O(G )

Abstract language: Admissible structures (as in syntactic structures)Object language: Realizations of the admissible structures

Both languages are the same objects: sets of (linear) λ-terms


ACG Definition G = 〈Σa, Σo, L, s〉

Σanp, s : typeChris : npmet : np ( np ( s

Λ(Σa)

λox .met Chris x : np ( s

λoP.P met : ((np ( np ( s) ( s) ( s

Σoσ : type/Chris/ : σ/met/ : σ+ : σ ( σ ( σ

Λ(Σo)

L(α ( β) = L(α) ( L(β)L(t u) = L(t) L(u)L(λox .t) = λox .L(t)

L(np) = σL(s) = σL(Chris) = /Chris/L(met) = λos.s + /met/ + o

λox .x + /met/ + /Chris/ : σ ( σ

λoP.P (λoos.s + /met/ + o) : ((σ ( σ ( σ)( σ) ( σ

A(G )= {t| ` t : s}

O(G )


ACG ArchitectureComposition Ability

Functional compositionAbstract language sharing(bimorphism)Parsing and generation(syntactic realization) in theusual sense: functioninversion

(TAG derivation trees)

(TAG derived trees)

(TAG string languages)

G1

G1 ◦ G2

G2

G3


About Word OrderMy First “Chris met Sandy” ACG Program

Σa :np, s : typeChris, Sandy : npmet : np ( np ( s

Σo :σ : type+ : σ ( σ ( σ/Chris/, /Sandy/ : σ/met/ : σ

np := σs := σ

Chris := /Chris/Sandy := /Sandy/met := λoos.s + /met/ + o

met Sandy Chris : s

L(met Sandy Chris)

skip reduction

= L(met) L(Sandy) L(Chris)= (λoos.s + /met/ + o)(/Sandy/)(/Chris/)= (λos.s + /met/ + /Sandy/)(/Chris/)= /Chris/ + /met/ + /Sandy/

/Chris/np

/met/(np\s)/np

/Sandy/np

np\ss


About Word OrderGetting Higher-Order Exercise

Σa : np, s : typeChris, Sandy : npmet : np ( np ( sthat : (np ( s) ( np ( np

Σo :/Chris/ : σ/Sandy/ : σ/met/ : σ/that/, ε : σ

Chris := /Chris/Sandy := /Sandy/met := λoos.s + /met/ + othat := λoPn.n + /that/ + (P ε)

that (λot.met t Chris) Sandy : np

See example-01.acg

L(that (λot.met t Chris) Sandy)

skip reduction

= L(that) (λot.L(met) t L(Chris))L(Sandy)= L(that) (λot.(λoos.s + /met/ + o) t /Chris/)) /Sandy/= L(that) (λot.(λos.s + /met/ + t) /Chris/)) /Sandy/= L(that) (λot./Chris/ + /met/ + t) /Sandy/= (λoPn.n + /that/ + (P ε)) (λot./Chris/ + /met/ + t) /Sandy/= (λon.n + /that/ + ((λot./Chris/ + /met/ + t) ε)) /Sandy/= (λon.n + /that/ + (/Chris/ + /met/ + ε)) /Sandy/= /Sandy/ + /that/ + (/Chris/ + /met/ + ε)= /Sandy/ + /that/ + /Chris/ + /met/

/Sandy/np

/that/(np\np)/(s/np)

/Chris/np

/met/(np\s)/np [np]1

np\ss /1

is/npnp\np

np


About Word OrderMedial Extraction

Σa :Chris, Sandy : npmet : np ( np ( sthat : (np ( s) ( np ( npyesterday : s ( s

Σo :/Chris/ : σ/Sandy/ : σ/met/ : σ/that/, ε : σ/yesterday/ : σ

Chris := /Chris/Sandy := /Sandy/met := λoos.s + /met/ + othat := λoPn.n + /that/ + (P ε)yesterday := λos.s + /yesterday/

that (λot.yesterday (met t Chris)) Sandy : np

L(that (λot.yesterday (met t Chris)) Sandy)

skip reduction

= L(that) (λot.(λs.s + /yesterday/) (/Chris/ + /met/ + t)) /Sandy/= L(that) (λot./Chris/ + /met/ + t + /yesterday/) /Sandy/= (λon.n + /that/ + (/Chris/ + /met/ + ε + /yesterday/)) /Sandy/= /Sandy/ + /that/ + (/Chris/ + /met/ + ε + /yesterday/)= /Sandy/ + /that/ + /Chris/ + /met/ + /yesterday/

/Sandy/np

/that/(np\np)/(s/np)

/Chris/np

/met/(np\s)/np [np]1

np\ss

/yesterday/s\s

s /1is/np

np\npnp


Intermediate Conclusion

So far...Discussion on possible architectures of grammatical formalismsDiscussion on function-compositional properties of ACG and modularityWord order in ACG:

Not in the logic of combinatorics, rather at the object level⇒ (Possibly straightforward) Encoding of various formalisms

Next to come: Examples!

Modularity:CFG: at the syntax-semantics interfaceTAG: at the syntactic levelTAG: at the syntax-semantics interface

Parallel architecture:CG: What do we call syntax?CVG: What do we call interface?


CFG into ACG Encoding

Example (CFG)

ρ0 : s → np vpρ1 : vp → vt npρ2 : np→ /John/ρ3 : np→ /Mary/ρ4 : vp → /left/ρ5 : vt → /saw/

ρ0 : s

ρ2 : np

/John/

ρ1 : vp

ρ5 : vt

/saw/

ρ3 : np

/Mary/

CFG as ACG

ΣRules LCFG ΣStringsρ0 : np ( vp ( s := λxy .x + y : σ ( σ ( σρ1 : vt ( np ( vp := λxy .x + y : σ ( σ ( σρ2 : np := /John/ : σρ3 : np := /Mary/ : σρ4 : vp := /left/ : σρ5 : vt := /saw/ : σ


CFG into ACG Encoding (cont’d)

CFG as ACG


skip reduction

LCFG(ρ0 ρ2 (ρ1 ρ5ρ3) : s) = (λxy .x + y)/John/((λxy .x + y)/saw/ /Mary/)→β (λy ./John/ + y)((λy ./saw/ + y)/Mary/)→β (λy ./John/ + y)(/saw/ + /Mary/)→β /John/ + (/saw/ + /Mary/)


CFG Encoding

CFG derivation trees

CFG string languages CFG direct semantics

GCFG

Gsem


A Direct SemanticsSharing Abstract Languages

CFG syntax as ACG


CFG (direct) semantics as ACG

ΣRules Lsem ΣLogρ0 : np ( vp ( s := λsP.P s : e ( (e ( t) ( tρ1 : vt ( np ( vp := λPos.P s o : (e ( e ( t) ( e ( e ( tρ2 : np := John : eρ3 : np := Mary : eρ4 : vp := left : e ( tρ5 : vt := saw : e ( e ( t


A Direct Semantics (cont’d)

CFG (direct) semantics as ACG

ΣRules Lsem ΣLogρ0 : np ( vp ( s := λsP.P s : e ( (e ( t) ( tρ1 : vt ( np ( vp := λPos.P s o : (e ( e ( t) ( e ( e ( tρ2 : np := John : eρ3 : np := Mary : eρ4 : vp := left : e ( tρ5 : vt := saw : e ( e ( t

skip reduction

Lsem(ρ0 ρ2 (ρ1 ρ5ρ3) : s) = (λsP.P s)John((λPos.P s o)saw Mary)→β (λP.P /John/)((λos.saw s o)Mary)→β (λP.P John)(λos.saw s Mary)→β (λos.saw s /Mary/)John→β saw John Mary

See example-02.acg.


A Continuized (Higher–Order) Semantics

CFG continuized semantics as ACG [Barker(2002)]

ΣRules Lsem ΣLogs := (t ( t) ( tnp := (e ( t) ( tvp := ((e ( t) ( t) ( tvt := ((e ( e ( t) ( t) ( tn := ((e ( t) ( t) ( tdet := (((e ( t) ( t) ( t) ( (e ( t) ( tρ0 : np ( vp ( s := λosvp.v(λoP.s(λox .p(Px)))ρ1 : vt ( np ( vp := λovoP.v(λoR.o(λoy .P(Ry)))ρ2 : np := λoP.P Johnρ5 : vt := λoP.P sawρevery : det := λoKP.K (λoQ.∀x .(Q x) ⇒ (P x))

See example-03.acg:Scope ambiguityScope displacementNP as a scope island


CFG Encoding

CFG derivation trees

CFG string languages

CFG direct semantics

CFG continuized semantics

GCFG

Gsem

Gcont. sem


Exercise

Let ` : N → N be: `(n) =

{k if n = 2k

1 otherwiseIf w = x1 . . . xn, w t = xn . . . x1.|w |a is the number of occurrences of a in w|w |b is the number of occurrences of b in w

ExerciseCan you find G1 and G2 such that:

O(G1) = {wcw t |w ∈ (a|b)∗};for all t ∈ A(G1) such that t :=G1 w , t :=G2 `|w |b(|w |a)


Intermediate ConclusionExemplified on CFG

Abstract structures are mapped to object structuresObject structures:

StringsSimple semantic objectsComplex semantic objects:

Continuized semantics: s := (t ( t) ( tResults of ACG composition: s := N× N := (N ( N ( N) ( NDynamic semantics: s := γ ( (γ ( t) ( t [de Groote(2006)]Underspecified representations

Algorithms for parsing and generation (in the usual sense) are essentially thesame: ACG parsing : finding the abstract antecedent of an object

Abstract structures?


Trees as λ-Terms

Trees Build on a Ranked Alphabet

s

np

John

vp

likes np

Mary

s

np

John

vp

sleeps

s2(np1John)(vp2likes (np1Mary)) s2(np1John)(vp1sleeps)

s of arity 2 (non-terminal)np of arity 1 (non-terminal)vp? vp1 of arity 1 and vp2 of arity 2(non-terminals)John of arity 0 (terminal)

s2 : τ ( τ ( τ

np1 : τ ( τ

vp1 : τ ( τ , vp2 : τ ( τ ( τ

John : τ


A Functional View on TAG

Tree Adjunction:

XXX

X ∗

−→

XX

X


Auxiliary Trees as Functions

Example

vp

apparently vp∗

s

np vp

likes np

−→

s

np vp

apparently vp

likes np

λox .vp

apparently x

vp

likes np→β

vp

apparently vp

likes npλa.

s

np a

vp

likes np

λox .

vp

apparently x


Adjunction as Functional Application

γ′likes γ′

apparently =λoa.

s

np a

vp

likes np

λox .

vp

apparently x

→β

s

np

λox .vp

apparently x

vp

likes np

→β

s

np vp

apparently vp

likes np


Substitution Operation

X

X−→

X


Substitution as Functional Application

Example

np

Chris

s

np vp

likes np np

Sandy

−→

s

np

Chris

vp

likes np

Sandy

λoos.

s

s vp

likes o

np

Chris

np

Sandy

→β

s

np

Chris

vp

likes np

Sandy


Putting Everything Together

Σtrees :

τ : type

γapparently = λoax .a

vp

apparently x

: (τ ( τ) ( τ ( τ

I = λox .x : τ ( τ

γJohn =np

John: τ

γlikes = λoSaso.S

s

s a

vp

likes o

:

(τ ( τ)( (τ ( τ)( τ ( τ ( τ


TAG Derivation as Term Application

Example

γlikes I I γJohn γMary =λoSaso.S

s

s a

vp

likes o

(λox .x) (λox .x)

np

John

np

Mary

→β

λoso.

s

s

vp

likes o

np

John

np

Mary

→β

s

np

John

vp

likes np

Mary

See example-04.acg


Yield as an ACG

Derived Tree Signature

s2 : τ ( τ ( τ

np1 : τ ( τ

vp1 : τ ( τ , vp2 : τ ( τ ( τ

John : τ

s

np

John

vp

likes np

Johns2(np1John)(vp2likes (np1Mary))

String signature (as before):

σ : type/John/, /likes/ . . . : σ

GYield

τ := σ John := /John/X1 := λx .x X2 := λxy .x + y· · ·

s2(np1John)(vp2likes (np1Mary)) := /John/ + (/likes/ + /Mary/)


TAG as ACGThe Current Picture

Λ(Σtrees)

Λ(Σstring)

Λ(Σderivations)

TAG derivation trees

TAG derived trees

TAG string languages

Gyield

Gtyped trees


A(Gyield) = TAG Derived Trees?

Σtrees :

τ : type

γapparently = λoax .a

vp

apparently x

: (τ ( τ) ( τ ( τ

I = λox .x : τ ( τ

γJohn =np

John: τ

γapparently IγJohn =

vp

apparently np

John


TAG as ACG

Category Induced Constraints

The site of an adjunction has the same category as the root (and foot) node ofthe auxiliary treeThe site of a substitution has the same category as the root node of thesubstituted tree

ΣderivationsLtyped trees−→ Σtrees

cJohn : np := γJohn : τcapparently : (vp ( vp) ( vp ( vp := γapparently : (τ ( τ) ( τnp, vp, s . . . : types := σ

Example

There is no t : vp ∈ Λ(Σderivations) such that t :=

vp

apparently np

John


Control on the Derived Trees

Gtyped trees = 〈Σderivations ,Σtrees ,Ltyped trees, s〉

np, vp, s . . . := σ

cJohn : np :=np

John

capparently : (vp ( vp) ( vp ( vp := λoax .a

(vp

apparently x

)clikes : (s ( s) ( (vp ( vp)

( np ( np ( s := λoSaso.S

ss a

(vp

likes o

)

clikes(capparently Ivp)cJohncMary : s =

snp

Johnvp

apparently vplikes np

Mary


TAG Derivation Trees as Abstract Terms

clikes Is (capparently Ivp)cJohncMary : s =

clikes

Is

0

capparently

Ivp

1

cJohn

2

cMary

3

clikes : (s ( s)︸︷︷︸0

( (vp ( vp)︸︷︷︸1

( np︸︷︷︸2

( np︸︷︷︸3

( s


TAG as ACGIntermediate Picture

Λ(Σtrees)

Λ(Σstring)

Λ(Σderivations)


TAG derived trees


Gyield

Gtyped trees

See example-05.acg.


Let’s Build Some Semantic Representation

Forgetting few seconds about TAG, we have:

A higher-order signature Σderivations :

vp, np, s : typescjohn, cmary : npcapparently : vp ( vpclikes (vp ( vp) ( np ( s

Some knowledge about Montague-like semantics?

A standard interpretation

s := t np := (e ( t) ( tvp := e ( tcjohn := λoP.P j capparently := λoaP.a(λx .apparently(P x))Ivp := λx .x clikes := λaos.s(a(λx .o(λy .like x y)))

See example-06.acg and esslli09-tag.acg.How to get the object wide scope reading?


TAG with Semantics

Λ(Σtrees)

Λ(Σstring)

Λ(Σderivations)


TAG derived trees


Gyield

Gtyped trees

Λ(ΣLog)

GLog


Intermediate Conclusion

So farTrees as λ-termsYield as an ACGTyping control: ACG from derivation trees to derived treesSome semantics added. Is it a function from syntax?

Questions?Any TAG feature missing?Order of Gtyped trees? (clikes : (vp ( vp) ( np ( s)


Features in TAG

Nouns as NP

np[+]

a np∗[−]

np[−]

big np∗[−]

np[−]

man

s

np[+] vp

leftca : np[−] ( np[+] cbig : cman : cleft : . . . (

(np[−] ( np[−]) (np[−] ( np[+]) ( np[+] ( s( np[−] ( np[−] (np[−] ( np[−])

( np[+]

No Inp[−+] : np[−] ( np[+]

Different categories for the root node and the foot node [Vijay-Shanker(1992)]


TAG Derivation Trees

Enough typing control?

clikes : (s ( s) ( (vp ( vp) ( np ( np

cis uncertain : s ( s :=

s

s vp

is uncertains node: substitution node (Whether he will actually survice the experience isuncertain)cis uncertain : s ( s: typed as auxiliary trees (with root node of type s)

Adding more control: yet another ACG

Not all the terms of type s are correct TAG derivationsDistinguish between s ( s (substition node of type s to tree of type s) ands ( s (auxiliary tree whose root and foot node are of type s)


TAG Derivation Trees (cont’d)

GTAG = 〈ΣTAG ,Σderivations ,LTAG, s〉

Clikes : sA ( vpA ( np ( np := clikesCis uncertain : s ( s := cis uncertainCapparently : vpA := capparentlyIvp : vpA := Ivp

Semantics unchanged! (GLog = 〈Σderivations ,ΣLog ,LLog, s〉)See example-07.acg.


The Actual Picture

Λ(ΣTAG )

Gyield

Gtyped trees

GLog

GTAG


Let’s Practice

Let ` : N → N be: `(n) =

{k if n = 2k

1 otherwise .

ExerciceCan you find GTAG , Gtyped trees , Gyield and GLog such that:

O(Gyield ◦ Gtyped trees ◦ GTAG ) = {an|n ∈ N};for all n ∈ N and for all t ∈ A(GTAG ) such thatLyield ◦Ltyped trees ◦LTAG(t) = an, LLog ◦Ltyped trees ◦LTAG(t) = `(n).


MCTAG

⟨s

does s∗,

vp

seem vp∗

⟩ ⟨s∗,

vp

likely vp∗

⟩ ⟨ s

np vp

to love np

⟩

Extending the current signatures and lexicons

Cdoes : sAcdoes : s ( sγdoes : τ ( τγdoes = λx .s2 does x

s

does s∗

Cs : sA ( sAcs : (s ( s) ( s ( sγs : (τ ( τ) ( τ ( τγs = λax .a x

s∗

mto love : ((sA ( vpA ( s) ( s) := λoPso.P(λoxy .Cto love x y s o)( np ( np ( s

mdoes seem : (sA ( vpA ( s) ( s := λo f .f Cdoes Cseemmlikely ((sA ( vpA ( s) ( s) := λoPf .P(λoxy .f (Cs x)

( (sA ( vpA ( s) (Clikely y))( s

See example-08.acg


Controlling the MCTAG Derivations

Higher-order types: allows one to mix tuples. Non-local MCTAG

m0 =

⟨S

A B

⟩m1 =

⟨A

C A∗

⟩m2 =

⟨B

D B∗

⟩m3 = 〈C∗,D∗〉

(see example-09.acg)Add an ACG with atomic types [sA, vpA] := (sA ( vpA ( s) ( s: set-localMCTAG2nd order ACG


The Final Picture

Λ(ΣTAG )

Λ(Σtuple)

Λ(ΣMCTAG )

Gyield

Gtyped trees

GLog

GTAG

GMCTAGGtuple


About Scope Ambiguity in TAG

Main approaches

In the lexical entriesUnderspecified representation language as the semanticstarget [Gardent and Kallmeyer(2003), Kallmeyer and Romero(2007)]MCTAG: higher-order term in syntax⟨

s∗ ,np

/everyone/

⟩ ⟨ s

np vp

likes np

⟩

Amounts to add lexical entries [Weir(1988)]See the next sections

ConclusionArchitecture: splitting the syntax-semantics interface from the control onadmissible structuresStatus of the derivation trees interpreted in various ways (t, e ( t, k, etc.)


Scope Ambiguities

every man loves some woman →{

∀x .man x → (∃y .woman y ∧ love x y)∃y .woman y ∧ (∀x .man x → love x y)

Underspecified framework

every man loves some woman → → ? →∀x .man x → (∃y .woman y ∧ love x y)∃y .woman y ∧ (∀x .man x → love x y)

Type Logical framework

every man loves some woman→ ∀x .man x → (∃y .woman y ∧ love x y)→ ∃y .woman y ∧ (∀x .man x → love x y)


Strengths and Weaknesses

Underspecified framework

Pros:One syntactic analysisExpressivity

Cons:Description languageAmbiguity in the semanticrecipe, not in the interface

TL frameworkPros:

No intermediate languageAmbiguity handled by theprocess

Cons:Syntactic ambiguity

QuestionIs there an ACG way provifing a proof-theoretic approach with only one syntacticstructure and no intermediate language?


Scope Ambiguity in Categorial grammars

The standard way

/everyone/

s/(np\s)

/loves/

np\s/np [np]

np\ss

s/np

/someone/

(np/s)\ss

/everyone/

s/(np\s)

[np]

/loves/

np\s/nps/np

/someone/

(np/s)\ss

np\ss

Csomeone (λoy .Ceveryone (λox .Cloves y x)) Ceveryone (λox .Csomeone (λoy .Cloves y x))

The ACG way

Replace \ and / by (

Ceveryone : (np ( s) ( s


Scope Ambiguity in ACG

ΣCGCloves : np ( np ( sCeveryone : (np ( s) ( sCsomeone : (np ( s) ( s

Σstring/loves/ σ/everyone/ : σ/someone/ : σ

Lamb-stringCloves := λoos.s + /loves/ + oCeveryone := λoP.P /everyone/Csomeone ::= λoP.P /someone/

Ceveryone(λox .Csomeone(λ

oy .Cloves x y)):=amb-string (λoP.P /everyone/)((λox .λoP.P /someone/)(λoy .(λoos.s + /loves/ + o) y x))→β (λoP.P /everyone/)((λox .λoP.P /someone/)(λoy .x + /loves/ + y))→β (λoP.P /everyone/)(λox .(λoy .x + /loves/ + y) /someone/)→β (λoP.P /everyone/)(λox .x + /loves/ + /someone/)→β (λox .x + /loves/ + /someone/) /everyone/→β /everyone/ + /loves/ + /someone/


Scope Ambiguity in ACG (cont’d)


Σstring/loves/ σ/everyone/ : σ/someone/ : σ

Lamb-stringCloves := λoos.s + /loves/ + oCeveryone := λoP.P /everyone/Csomeone ::= λoP.P /someone/

Csomeone(λoy .Csomeone(λ

ox .Cloves x y)):=amb-string (λoP.P /someone/)((λoy .λoP.P /everyone/)(λox .(λoos.s + /loves/ + o) y x))→β (λoP.P /someone/)((λoy .λoP.P /everyone/)(λox .x + /loves/ + y))→β (λoP.P /someone/)(λoy .(λox .x + /loves/ + y) /everyone/)→β (λoP.P /someone/)(λoy ./everyone/ + /loves/ + y)→β (λoy ./everyone/ + /loves/ + x) /someone/→β /everyone/ + /loves/ + /someone/


Scope AmbiguityNon Injective Lexicon

Λ(ΣSimpleSyn)

Λ(Σstring )

Λ(ΣCG )

Σsurface

Gamb

Gamb-string

Λ(ΣLog)

GLog

Non-functional relation


Scope Ambiguity in ACG (cont’d)


ΣSimpleSyncloves : np ( np ( sceveryone : npcsomeone : np

LambCloves := λoos./loves/ o sλoos.cloves o sCeveryone := λoP.P /everyone/ceveryoneCsomeone ::= λoP.P /someone/csomeone

Ceveryone(λox .Csomeone(λ

oy .Cloves x y)) :=amb cloves csomeone ceveryoneCsomeone(λ

oy .Csomeone(λox .Cloves x y)) :=amb cloves csomeone ceveryone


Scope AmbiguityNon Injective Lexicon

Λ(ΣSimpleSyn)

Λ(Σstring )

Λ(ΣCG )

Σsurface

Gamb

Gamb-string

Λ(ΣLog)

GLog

Non-functional relation


ConjunctionJohn and every kid ran

crun : np ( sckid : ncJohn : npcand : np ( np ( npcrun(candcJohn(ceveryckid))

Crun : np ( sCkid : nCJohn : npCand : ((np ( s) ( s)

( ((np ( s) ( s)( (np ( s) ( s

Cand(λoP.PCJohn)(ceveryckid)Crun

Σstring(/John/ + /and/ + (/every/ + /kid/)) + /ran/

ΣLog(run j) ∧ (∀x .kid x ⇒ run x)

Cand := λoPQR.P(λox .Q(λoy .R(cand x y)))Crun := crunCkid := ckidCJohn := cJohn

Crun := runCkid := kidCJohn := jCand := λoPQ.

λR.(PR) ∧ (QR)

crun := λos.s + /ran/ckid := /kid/cJohn := /John/cand := λoxy .x + /and/ + y


De re and De dicto Readings

cseek : np ( np ( scbook : n

cseekcJohn(cacbook)

Cseek : np (((np ( s) ( s) ( s

Cbook : nCseekCJohn(CaCbook)(CaCbook)(λoy .CseekCJohn

(λoQ.Q y))

Σstring/John/ + /seeks/ + (/a/ + /book/)

ΣLog∃y .(book y) ∧ (try j (λox .find x y))try j (λox .∃y .(book y) ∧ (find x y))

Cseek := λoxP.P(λoy .cseek x y)Cbook := cbook

Cseek := λoxo.try x (λoz .o(λoy .find z y))

Cbook := book

cseek := λoxy .x + /seeks/ + ycbook := /book/


And More...

So farCoordination of quantified and non-quantified NPsVP ellipsis John saw a kid and so did Billde re and de dicto readingsQuantification and negation (every kid did’nt run)

Generalization: the Scoping Constructor

Γ `TL t : β ⇑ α ∆, x : β `TL u : α(E⇑)

Γ, ∆ `TL t(λx .u) : α

Γ `TL t : β(I⇑)

Γ `TL λx .(x t) : β ⇑ α

Syntactically behaves as a β and semantically as a (β ( α) ( α

Given a lexical entre w : a ∈ TL(A), we have cw : asyn and Cw : asem such that:if a ∈ A then asyn = a and asem = aif a = α ( β then asyn = αsyn ( βsyn and asem = αsem ( βsem.if a = α ⇑ β then asyn = αsyn and asem = (αsem ( βsem) ( βsem


ResultIn a commutative and associative TL system, given a suitable translation:

Theorem: what we can do in TL we can do with ACGsConjecture: what we can do with ACGs we can do with TL

Also can apply to TAGWhat are the relations with other TL type constructor (q(x , y , z), ↑, ↓)


CVG: a (Weakly) Syntactocentric Formalism

SyntaxInference

rules

SemanticsInference

rules

InterfaceInference rules

Example (Chris liked Sandy)

` liked, like’ : np (c np (s s, ι ( ι ( π a ` Sandy, Sandy’ : np, ι a`

ˆliked Sandy c , like’ Sandy’np (s s, ι ( π a ` Chris, Chris’ : np, ι a

`ˆsChris

ˆliked Sandy c ˜

, like’ Sandy’ Chris’ : s, π a


In Situ OperatorsThe Syntax-Semantics Mismatch

Example (Chris liked Sandy)

π =` topin-situ, top’ : np (a np, ι ( ιπ

π a ` Sandy, Sandy’ : np, ι a`

ˆSandy topin-situ

a , top’ Sandy’ : np, ιππ a

...π

...` λot.

ˆsChrisˆliked t c ˜

, λox .like’ x Chris’ : np ( s, ι ( π a`

ˆsChrisˆliked

ˆSandy topin-situ

a c ˜, (top’ Sandy’) (λox .like’ x Chris’) : s, π a

The G Rule...

Γ ` a, b : A,BDC a ∆

...t, x : A,B , Γ′ ` e, c : E ,C a ∆′

GΓ; Γ′ ` e[t/a], b (λox .c) : E ,D a ∆,∆′


The G Rule Revisited

...Γ ` a, b : A,BD

C a ∆

...t, x : A,B , Γ′ ` e, c : E ,C a ∆′

GΓ; Γ′ ` e[t/a], b (λox .c) : E ,D a ∆,∆′

G〈A,BDC 〉 : 〈A,BD

C 〉 ( (〈A,B〉 ( 〈E ,C 〉) ( 〈E ,D〉

〈A,BDC 〉 :=Syn A

〈A,B〉 ( 〈E ,C 〉 :=Syn A ( E〈E ,D〉 :=Syn EG〈A,BD

C 〉 :=Syn λotu.u t

LSyn

〈A,BDC 〉 :=Sem (B ( C ) ( D

〈A,B〉 ( 〈E ,C 〉 :=Sem B ( C〈E ,D〉 :=Sem DG〈A,BD

C 〉 :=Sem λotu.t u

LSem

LSyn(u) = aLSyn(v) = eLSyn(G〈A,BD

C 〉 u (λot.v)) = (λotu.u t) a (λot.e)

→β e[t/a]

LSem(u) = bLSem(v) = cLSem(G〈A,BD

C 〉 u (λot.v)) = (λotu.t u) b (λox .c)

→β b (λox .c)


A New Perspective on the CVG Architecture

Architectural motivation:

CVG = ACG

Syntax SemanticsInterface

Interface

Overt movement as higher order term in the syntax:Γ ` a, d : AC

B ,DFE a ∆ t, x : A,D; Γ′ ` b, e : B,E a ∆′

GΓ; Γ′ ` at b, dx e : C ,F a ∆; ∆′

2nd order except when trigger by a ACB type


Conclusion

Encoding in a same framework: sharing and comparing analysisACG composition modes: flexible and open architectures“Syntax”, “function”, “relation”, “compositionality”, “rule-to-rule” intuitions maybe realized by different mathematical notions


C. Barker.Continuations and the nature of quantification.Natural Language Semantics, 10:211–242, 2002.

P. W. Culicover and R. Jackendoff.Simpler Syntax.Oxford University Press, 2005.

H. B. Curry.Some logical aspects of grammatical structure.In R. Jakobson, editor, Structure of Language and its Mathematical Aspects:Proceedings of the Twelfth Symposium in Applied Mathematics, pages 56–68.American Mathematical Society, 1961.

P. de Groote.Towards a montagovian account of dynamics.In Proceedings of Semantics and Linguistic Theory XVI, 2006.http://research.nii.ac.jp/salt16/proceedings/degroote.new.pdf.

http://research.nii.ac.jp/salt16/proceedings/degroote.new.pdf


D. Dowty.Grammatical relations and montague grammar.In P. Jacobson and G. K. Pullum, editors, The Nature of SyntacticRepresentation. Reidel, Dordrecht, 1982.

C. Gardent and L. Kallmeyer.Semantic construction in feature-based tag.In Proceedings of the 10th Meeting of the European Chapter of the Associationfor Computational Linguistics (EACL), 2003.

R. Jackendoff.Foundations of Language: Brain, Meaning, Grammar, Evolution.Oxford University Press, 2002.

L. Kallmeyer and M. Romero.Scope and situation binding for ltag.Research on Language and Computation, 2007.

M. Kracht.The Mathematics of Language.Number 63 in Studies in Generative Grammar. Mouton de Gruyter, Berlin, 2003.


R. Montague.The proper treatment of quantification in ordinary english.In Formal Philosophy: Selected Papers of Richard Montague. Yale UniversityPress, 1974.Re-edited in "Formal Semantics: The Essential Readings", Paul Portner andBarbara H. Partee, editors. Blackwell Publishers, 2002.

R. Muskens.Lambda Grammars and the Syntax-Semantics Interface.In R. van Rooy and M. Stokhof, editors, Proceedings of the ThirteenthAmsterdam Colloquium, pages 150–155, Amsterdam, 2001.

R. Muskens.Lambdas, Language, and Logic.In G.-J. Kruijff and R. Oehrle, editors, Resource Sensitivity in Binding andAnaphora, Studies in Linguistics and Philosophy, pages 23–54. Kluwer, 2003.


D. Oehrle.Some 3-Dimensional Systems of Labelled Deduction.Logic Jnl IGPL, 3(2-3):429–448, 1995.doi: 10.1093/jigpal/3.2-3.429.URL http://jigpal.oxfordjournals.org.

R. T. Oehrle.Term-labeled categorial type systems.Linguistic and Philosophy, 17(6):633–678, December 1994.doi: 10.1007/BF00985321.

C. Pollard.High-order categorical grammars.In Proceedings of Categorial Grammars 2004, Montpellier, pages 340–361, June2004.

C. Pollard.An introduction to convergent grammar.Presentation at Calligramme Seminar, 2008.http://www.ling.ohio-state.edu/~scott/cvg/.

http://jigpal.oxfordjournals.org

http://www.ling.ohio-state.edu/~scott/cvg/


A. Ranta.Type Theoretical Grammar.Oxford University Press, 1994.

K. Vijay-Shanker.Using descriptions of trees in a tree adjoining grammar.Computational Linguistics, 18(4):481–518, December 1992.

D. J. Weir.Characterizing Mildly Context-Sensitive Grammar Formalisms.PhD thesis, University of Pennsylvania, 1988.

background acg example: cfg semantics for tag cg cvg ... · background acg example: cfg semantics...

Documents