Computational Semantics Lorraine University NLP Master, 2013

Home , Computational semantics

Computational Semantics Lorraine University NLP Master, 2013 – 2014

Sylvain Pogodalla1

1mailto:[email protected] http://www.loria.fr/~pogodall Oﬃce: C. 302 LORIA/INRIA Nancy–Grand Est France

2013 – 2014

1 / 157 Outline

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography

2 / 157 Introduction

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography

3 / 157 Introduction Computational Semantics?

What is computational semantics? A suitable interpretation level A way to relate semantic structures and syntax (interpretation or realization)

4 / 157 Meaning

1 Introduction

2 Meaning Sense and Denotation Models and Truth-Conditionality Criterion [Winter, 2010] Building Denotations Structural Ambiguity

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography 5 / 157 Meaning Sense and Denotation Sense and Denotation

Gottlob Frege Sense = mode of presentation Denotation = the object it refers to Ex.: 1 + 1 and 2 have the same denotation but not the same sense

6 / 157 Meaning Sense and Denotation Meaning as an Observable Property [Winter, 2010]

Select a speciﬁc property of language, stable across speakers and situations Extra-linguistic and intra-linguistic semantic phenomena

Example (Extra-linguistic)

What is common to the objects that people categorize as being red? What the eﬀect of asking please pick a blue card from this pack?

Example (Intra-Linguistic)

How do speaker identify relations between pairs of words like red-color, dog-animal? What are the relations between the sentences please pick a blue card from this pack and please pick a card from this pack?

Contrasts Example

Red is a color / ?Red is an animal Every red thing has a color / ?Every red thing has an animal

7 / 157 Meaning Sense and Denotation Entailment

Example Tina is tall and thin Tina is thin

Example Tina is tall and thin ⇒ Tina is thin A dog entered the room ⇒ An animal entered the room John picked a blue card from this pack ⇒ John picked a card from this pack

Stability of judgments Indefeasible reasoning (vs. Tina is a bird. Tina can ﬂy) Entailment judgements ≡ grammaticality judgments Models and denotation

8 / 157 Meaning Models and Truth-Conditionality Criterion [Winter, 2010] Models and Truth-Conditionality Criterion

Aim Establishing a relation between language expressions and objects in models. Formal semantics Applied semantics Linking exp and exp M where M is a model. J K Denotations of sentences are truth-values 1 (for “true”) and 0 (for “false”)

Deﬁnition (TCC)

A semantic theory T satisﬁes the truth-conditionality criterion (TCC) if for all sentences S1 and S2 the following conditions are equivalent:

1 Sentences S1 intuitively entails sentence S2 M M 2 For all models M in T S1 ≤ S2 J K J K 1 is an empirical statement 2 is a theoretical statement

9 / 157 Meaning Models and Truth-Conditionality Criterion [Winter, 2010] Models and Truth-Conditionality Criterion (cont’d)

Models S M1 ≤ S M1 Entailment 1 2 TCC J KM2 J KM2 ←→ S1 ≤ S2 S1 ⇒ S2 J KM3 J KM3 S1 ≤ S2 J K ... J K

10 / 157 Meaning Building Denotations Building Denotations

For every model M:

In addition to truth-values, there exists EM an arbitrary non-empty set of entities in M M Tina denotes an arbitrary entity tina = Tina in EM J KM M tall and thin denote arbitrary sets tall = tall and thin = thin of entities in EM J K J K What does it mean that Tina is thin is true? tina ∈ thin 1 if x ∈ A is(x, A) = 0 if x 6∈ A Tina is thin M = is(tina, thin) J K Constants is is constant across models!

Exercise: What’s the denotation of Tina is tall and thin?

11 / 157 Meaning Building Denotations Tina is tall and thin

and(A, B) = A ∩ B Tina is tall and thin M = is(tina, and(tall, thin)) J K Denotations Expressions Category Type Abstract denotation Tina NP entity tina tall A set of entities tall thin A set of entities thin tall and thin AP set of entities and(tall, thin) Tina is thin S truth-value is(tina) Tina is tall and thin S truth-value is(tina, and(tall, thin))

Example We assume our semantic theory satisfies the TCC. Show that the interpretations given so far account for the following facts: Tina is tall and thin ⇒ Tina is thin Tina is thin 6⇒ Tina is tall and thin See [Hopcroft et al., 2003, first chapter, and in particular Section 1.2.2 “Reduction to Definitions”] for an introduction to rigorous (formal) proofs. 12 / 157 Meaning Building Denotations Involving Syntactic Structures

Example All pianists are composers and Tina is a pianist. All composers are pianists and Tina is a pianist. ⇒? Tina is a composer.

Compositionality The denotation of a complex expression is determined by the denotations of its immediate parts and the ways they combine with each other.

13 / 157 Meaning Building Denotations

Example S is(tina, and(tall, thin)) NP is AP tina is and(tall, thin) Tina A and A tall and thin tall thin

About Compositionality Direct compositionality Some syntactic assumptions How the denotations of functions know what their arguments are?

14 / 157 Meaning Structural Ambiguity Structural Ambiguity

Example (Tina is not tall and thin)

Tina is not tall and thin ⇒? Tina is thin

S S

NP is AP NP is AP

Tina AP and A Tina not AP

not AP thin A and A

tall tall thin

What makes the syntactic ambiguity a semantic ambiguity? Compositionality What’s the denotation of not? What are the denotations of the two structures?

15 / 157 Meaning Structural Ambiguity Structural Ambiguity (cont’s)

S S

NP is AP NP is AP

Tina AP and A Tina not AP

not AP thin A and A

tall tall thin is(tina, and(not(tall), thin)) is(tina, not(and(tall, thin)))

tina is and(not(tall), thin) tina is not(and(tall, thin))

not(tall) and thin not and(tall, thin))

not tall tall and thin

16 / 157 Meaning Structural Ambiguity Exercise [Winter, 2010]I

Two sentences are called equivalent if they entail each other as in the following example: Tina is tall and thin ⇔ Tina is both tall and thin

1 Give more examples for pairs of sentences that you consider intuitively for equivalent sentences 2 State the condition that the TCC requires for equivalent sentences M 3 Assuming the structure [both[tall and thin]], how can we deﬁne both ? Is it a constant? J K 4 Consider the ungrammaticality of the following strings: ∗Tina is both tall ∗Tina is both not tall ∗Tina is both tall or thin Let’s assume that both only appears in adjective phrases as adjacent to and conjunctions. 1 Analyze the truth-values assigned to: Tina is both not tall and thin Tina is not both tall and thin 2 Show that this accounts for: Tina is both not tall and thin ⇒ Tina is thin Tina is not both tall and thin 6⇒ Tina is thin

17 / 157 Meaning Structural Ambiguity Exercise [Winter, 2010]II

5 Consider the following sentences: Tina is not tall and not thin Tina is neither tall nor thin Assume that neither M = both M ? What should then the denotation of nor be to have these twoJ sentencesK equivalent?J K

18 / 157 Meaning Structural Ambiguity Exercise [Winter, 2010]

Two sentences are called contradictory in a given theory if whenever one of them denotes 1 in the theory, the other denotes 0 (e.g. Mary is not tall and Mary is tall). We may also talk about two contradictory readings/structures of sentences, which is especially useful when sentences are structurally ambiguous. 1 Give more examples for contradictory sentences/structures 2 Consider the sentences The bottle is empty and The bottle is full. Can you think o a theoretical assumption that would render these sentences contridactory? 3 Give an entailment using this assumption

19 / 157 Types and Model Structure

1 Introduction

2 Meaning

3 Types and Model Structure Domains Types Type Theory and λ-Calculus Higher-Order Logic

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography 20 / 157 Types and Model Structure Domains Types and Domains

Denotations Expressions Category Type Abstract denotation Tina NP entity tina tall A set of entities tall tall and thin AP set of entities and(tall, thin) Tina is thin S truth-value is(tina,thin) is ? function from entities and set of is entities to truth-values and ? function from pairs of set of enti- and ties to set of entities not ? function from set of entities to set not of entities Systematic relation between expressions of a given syntactic category to a type of denotation Distinction between the objects of the model (entities, set of entities, functions, etc.) ⇒ Domains: parts of the model that gathers objects with the same structure The property for objects to belong to the same domain is expressed by having same type

21 / 157 Types and Model Structure Domains Types and Domains (cont’d)

Deﬁnition (Characteristic Function)

Let A be a subset of B. A function FA from B to {0, 1} the set of truth-values is called the characteristic function of A in B if it satisﬁes for every x ∈ B:

1 if x ∈ A F (x) = A 0 if x 6∈ A

Types and Domains Basic types Complex types Type Domain Type Domain De e E = De e → t Dt t {0, 1} = Dt ......

Example

Let M be a model such that De = {t, j, m}. How many possible denotations for tall are there?

22 / 157 Types and Model Structure Types Types

Deﬁnition (Types) The set of types T over the basic types e and t is the smallest set such that: {e, t} ⊆ T If τ ∈ T and σ ∈ T then τ → σ ∈ T

Example What’s the type of an adjective? What’s the type of not? What’s the type of is?

23 / 157 Types and Model Structure Types Two-Place Relation

One-Place Predicate Tina smiled is true whenever tina ∈ smiled whenever Fsmiled(tina) = 1 whenever smiled(tina) = 1 De smiled ∈ Fe→t = Dt or smiled : e → t

Two-Place Relation Tina praised Mary is true whenever tina belongs to the set of entities that praised Mary whenever praised Mary (tina) = 1 De J K praised Mary ∈ Dt or praised Mary : e → t D J DKe e J K praised ∈ Dt or praised : e → e → t J K J K Note Tina [praised Mary] = (praised(mary))(tina) J K Characteristic function of a binary relation R:(FR (y))(x) = 1 iﬀ hx, yi ∈ R Currying: f : A × B → C and g : A → B → C such that f (a, b) = (g(a))(b)

24 / 157 Types and Model Structure Types Lexicon [Winter, 2010]

Deﬁnition (Frame)

E Let E = De a set of entities. We deﬁne the frame F as:

E ∆ F = ∪τ∈T Dτ

Deﬁnition (Lexicon)

Let Σ be a ﬁnite vocabulary. A lexical typing function TL of Σ is any function from Σ to T . Given a lexical typing function TL , a corresponding lexical interpretation function over Σ E and a non-empty set of entities E is any function IL from Σ to F such that:

∀w ∈ Σ IL (w): TL (w)

Deﬁnition (Model)

Let Σ be a ﬁnite vocabulary with TL a lexical typing function over Σ. A model over Σ is a pair hE, IL i where E is a non-empty set of entities and IL is a lexical interpretation function over Σ and E.

25 / 157 Types and Model Structure Types Example

Vocabulary Σ = {Tina, Mary, smiled, praised}

TL :Σ −→ T Tina −→ e Typing Mary −→ e smiled −→ e → t praised−→ e → e → t E IL :Σ −→ F Tina −→ tina Mary −→ mary tina −→ 1 smiled −→ Interpretation mary−→ 0  tina −→ 1  tina −→  mary−→ 1 praised−→ tina −→ 1  mary−→  mary−→ 0

Lexicon hΣ, TL i Model hE, IL i What’s the denotation of Mary smiled? What’s the denotation of Mary praised Mary?

26 / 157 Types and Model Structure Types About the Lexicon

Deﬁnition (Restricting Functor)

Let Σ be a ﬁnite vocabulary and TL be a lexical typing function from Σ to T . Let E be a non-empty set. A restricting functor RF E over Σ is a function that maps any word w ∈ Σ to a subset of the domain D . TL (w) For w ∈ Σ: if RF E (w) = D for every set E, we say that the denotation of w is arbitrary TL (w) if RF E (w) is a singleton for every set E, we say that the denotation of w is constant

Have we seen constants? What about “intermediary” kinds of lexical items? E SYM is the set of symmetric functions in De→e→t (f is symmetric iff f (x)(y) = f (y)(x)) E E RMOD is the set of restrictive modifiers in D(e→t)→(e→t) where f ∈ RMOD iff ∀g ∈ De→t ∀x ∈ De , if (f (g))(x) = 1 then g(x) = 1 as well Do you have examples? resemble very charmingly

27 / 157 Types and Model Structure Types Theory and Models

Deﬁnition (Intended Model)

Let Σ be a ﬁnite vocabulary with TL a lexical typing function over Σ. Let E be a E non-empty set and RF a restricting functor over Σ. A model hE, IL i over Σ is an E intended model if for every word w ∈ Σ IL (w) ∈ RF (w).

Deﬁnition (Truth-Conditionality Criterion)

E A semantic theory T that speciﬁes a typing function TL and a restricting functor RF over Σ satisﬁes the truth-conditionality criterion (TCC) if for all structures S1 and S2 the following conditions are equivalent:

1 Structure S1 intuitively entails structure S2 M M 2 For all intended models M in T S1 ≤ S2 J K J K

28 / 157 Types and Model Structure Type Theory and λ-Calculus Reminder

Deﬁnition (λ-Calculus) Syntax V = {x, y,...} and T ::= V|λV.T |(TT ) Free Variables Let t ∈ T FV(x) = {x} FV(λx.u) = FV(u) \{x} FV(t u) = FV(t) ∪ FV(u) If FV(t) = ∅ t is closed.

Example What are the free variables of x y z λx.x y (λx.x x)(λy.y y)

29 / 157 Types and Model Structure Type Theory and λ-Calculus Reminder (cont’d)

Deﬁnition (Substitution) For t, u ∈ T and x ∈ V, the substitution of u for x in t, written t[x := u] ∈ T , is deﬁned as follows (x 6= y): x[x := u] = u y[x := u] = y (v w)[x := u] = v[x := u] w[x := u] (λx.v)[x := u] = λx.v (λy.v)[x := u] = λy.v[x := u] with y 6∈ FV(u)

Example Compute ((λx.x y z)(λy.x y z)(λz.x y z))[x := y]

30 / 157 Types and Model Structure Type Theory and λ-Calculus Reminder (cont’d)

Deﬁnition (Reduction)

Reduction( λx.t) u →β t[x := u] Church-Rosser Theorem For all λ-terms t, u and v such that

∗ ∗ t →β u and t →β v there exists w such that

∗ ∗ u →β w and v →β w

Example Let δ = λx.x x. Reduce Ω = δ δ.

31 / 157 Types and Model Structure Type Theory and λ-Calculus Reminder (cont’d)

Deﬁnition (Simply Typed λ-Calculus)

Γ = x1 : A1,..., xn : An is a context. Axiom Γ, x : A ` x : A

Γ, x : A ` t : B Γ ` t : A → B Γ ` u : A Abs. App. Γ ` λx.t : A → B Γ ` (t u): B

Example What about the types of λx.x? Of λxy.x? Of λx.λy.λz.x z (y z)? Can you type δ = λx.x x?

32 / 157 Types and Model Structure Type Theory and λ-Calculus Syntactic Structures

Deﬁnition (Binary Structure) Given a vocabulary Σ, a binary structure over Σ is one of the following: 1 An occurrence of a word w ∈ Σ.

2 A sequence [S1 S2] where S1 and S2 are binary structures over Σ.

Deﬁnition (Denotation of a Structure)

Let Σ be a vocabulary, E be a non-empty set of entities, TL be a lexical typing function over Σ and IL be a corresponding lexical denotation function over Σ. Then for every binary structure S over Σ, the syntactic typing and denotation functions Ts and Is extend TL and IL as follows:  TL(w) if S is a word w ∈ Σ   β if S = [S1 S2] and Ts(S1) = α → β and Ts(S2) = α Ts(S) =  β if S = [S1 S2] and Ts(S1) = α and Ts(S2) = α → β  undeﬁned otherwise  IL(w) if S is a word w ∈ Σ   (t u) if S = [S1 S2] and Is(S1) = t : α → β and Is(S2) = u : α Is(S) =  (t u) if S = [S1 S2] and Is(S1) = u : α and Is(S2) = t : α → β  undeﬁned otherwise 33 / 157 Types and Model Structure Type Theory and λ-Calculus Examples

Example (Tina praised Mary) [Tina[praised Mary]] praised(mary)(tina): t

Tina tina : e praised(mary): e → t

praised Mary praised : e → e → t mary : e

Example (Tina is tall)

is(tall)(tina): t

Tina tina : e is(tall): e → t

is tall is :(e → t) → (e → t) tall : e → t

34 / 157 Types and Model Structure Type Theory and λ-Calculus Quantiﬁcation

Example (Everyone smiles)

everyone(smiles): t everyone smiles everyone :(e → t) → t smiles : e → t

Example (Every man smiles)

every(man)(smile): t

smiles every(man):(e → t) → t smiles : e → t every man every :(e → t) → (e → t) → t man : e → t

Shake and bake semantics What’s the denotation of everyone? What’s the denotation of every?

35 / 157 Types and Model Structure Type Theory and λ-Calculus Computing Denotations with Functions or Logical Formulas

So far tall as: Set (Characteristic) Function Symbol of relation ⇒ Move to logic

What kind of logic?

36 / 157 Types and Model Structure Higher-Order Logic Higher-Order Logic

HOL Two atomic types: e and t (or ι and o) Logical constants: ⊥ : t ⇒ : t → t → t ∀α:(α → t) → t for each type α

Formulas: well-typed terms of type t

Contrast with ﬁrst order logic Build a HOL formula which is not FOL

37 / 157 Montague Semantics

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography

38 / 157 Montague Semantics Motivations

There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of languages within a single natural and mathematically precise theory. On this point I differ from a number of philosophers (. . . ). [Montague, 1970]

39 / 157 Montague Semantics Montague Semantics [Montague, 1974]

The aim of this paper is to present in a rigorous way the syntax and semantics of a certain fragment of a certain dialect of English. [Montague, 1974]

Fragment Semantic types as homomorphic image of syntactic types Semantic representation as translation of syntactic operations Semantic representation through a “certain artiﬁcial language”, a logical language

40 / 157 Montague Semantics The Original Version

Categories of English Basic types: e and t Type constructors: A/B and A//B Some deﬁnitions: IV, or the category of instransitive verb phrases, is to be t/e. T, or the category of terms, is to be t/IV, TV, or the category of transitive verb phrases, is to be IV/T. CN, or the category of common noun phrases, is to be t//e. ...

BA the set of basic expressions of the category A. PA is the set of phrases of the category A. love ∈ BTV Mary ∈ BT, he0 ∈ BT man ∈ BCN

41 / 157 Montague Semantics Syntactic Rules

Basic Rules

S1 BA ⊂ PA for every category A.

S2 If ζ ∈ PCN, then F0(ζ), F1(ζ), F2(ζ) ∈ PT where:

F0(ζ) = every ζ F1(ζ) = the ζ F2(ζ) is a ζ or an ζ according as the ﬁrst word in ζ takes a or an ...

42 / 157 Montague Semantics Syntactic Rules

Rules of functional application 0 0 S4 If α ∈ Pt/IV and δ ∈ PIV, then F4(α, δ) ∈ Pt , where F4(α, δ) = αδ and δ is the result of replacing the ﬁrst verb in δ by its third person singular present

S5 If δ ∈ PIV/T and β ∈ PT, then F5(δ, β) ∈ PIV, where F5(δ, β) = δβ if β does not have the form hen and F5(δ, hen) = δ himn ...

43 / 157 Montague Semantics Syntactic Rules

Rules of quantiﬁcation

S14 If α ∈ PT and φ ∈ Pt , then F10,n(α, φ) ∈ Pt , where either:

1 α does not have the form hek , and F10,n(α, φ) comes from φ by replacing the ﬁrst occurence of hen or himn by α and all other      he   him  occurrences of hen or himn by she or her respectively,  she   it     masc.  according as the gender of the ﬁrst BCN or BT in α is fem. , or  neuter  2 α = hek , and F10,n(α, φ) comes from φ by replacing all occurrences of hen or himn by hek or himk respectively ...

44 / 157 Montague Semantics Every man loves Mary

every man loves Mary (t,S14, F10,0(every man, he0 loves Mary)) every man (T, S2, F0(man)) he0 loves Mary (t, S4, F4(he0, love Mary))

man (CN) he0 (T = t/IV) love Mary (IV,S5,F5(love, Mary))

love (TV = IV/T) Mary (T)

45 / 157 Montague Semantics Translating English into (Intensional) Logic

Categories to Semantic Types f is a function such that f (e) = e f (t) = t f (A/B) = f (A//B) = f (B) → f (A) where A, B are categories

Translation rules: the · function T1 If α is in the domain of g, then α = g(α) [interpretation of constants]. hen = λP.P xn...

T2 if ζ ∈ PCN then every ζ = λP.∀x.ζ(x) ⇒ P(x), ......

T4 if δ ∈ Pt/IV, β ∈ PIV then F4(δ, β) = δ(β)

T5 if δ ∈ PIV/T, β ∈ PT then F5(δ, β) = δ(β) ...

T14 If α ∈ PT, φ ∈ Pt then F10,n(α, φ) = α(λxn.φ) ...

46 / 157 (λP.∀x.man(x) ⇒ P(x))(λx0.love(x0, mary)) →β ∀x.man(x) ⇒ (λx0.love(x0, mary))(x) →β ∀x.man(x) ⇒ love(x, mary)

(λP.P x0)(λx.love(x, mary)) λP.∀x.man(x) ⇒ P(x) →β (λx.love(x, mary)) x0 →β love(x0, mary)

(λox.o (λy.love(x, y)))(λP.P mary) →β (λP.P mary)(λy.love(x, y)) →β (λy.love(x, y)) mary) →β λx.love(x, mary)

Montague Semantics Every man loves Mary

every man loves Mary (t,S14, F10,0(every man, he0 loves Mary))

he0 loves Mary (t, S4, F4(he0, love Mary)) every man (T, S2, F0(man))

man (CN)

man love Mary (IV,S5,F5(love, Mary))

he0 (T = t/IV) λP.P x0

love (TV = IV/T) Mary (T) λox.o (λy.love(x, y)) λP.P mary

47 / 157 Montague Semantics Remarks

Type homomorphism Translation ⇒ What about widespread syntactic formalisms?

48 / 157 Phenomena at the Syntax-Semantics Interface

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography

49 / 157 Phenomena at the Syntax-Semantics Interface Conjunction I

Example (Mary and every boy smiles)

smiles

Mary

and

every boy

50 / 157 Phenomena at the Syntax-Semantics Interface Conjunction II

Example (Mary and every boy smiles)

(and (every boy)(Mary)) smiles : t (∀e x.(boy x ⇒ smile x)) ∧ (smile m)

and (every boy)(mary):(e → t) → t smiles : e → t λr.(∀e x.(boy x ⇒ r x)) ∧ ((λP.P m)) r) smile →β λr.(∀e x.(boy x ⇒ r x)) ∧ (r m)

mary :(e → t) → t and (every boy) : ((e → t) → t) → (e → t) → t λP.P m λQ.λr.((λQ.∀e x.(boy x ⇒ Q x)) r) ∧ (Q r) →β λQ.λr.(∀e x.(boy x ⇒ r x)) ∧ (Q r)

and : every boy :(e → t) → t ((e → t) → t) → ((e → t) → t) → (e → t) → t λQ.∀ x.(boy x ⇒ Q x) λPQ.λr.(P r) ∧ (Q r) e

every :(e → t) → (e → t) → t boy : e → t λPQ.∀e x.(P x ⇒ Q x) boy 51 / 157 Phenomena at the Syntax-Semantics Interface Quantiﬁcation and Object Position I

How can we have both the NP subject and the NP object be arguments of a a transitive verb? Allow for abstraction (see later) Change the denotation of transitive verbs: e = e IVJ K= t/e IV = e → t JT K= (e → t) → t JTVK= IV/T TV = ((e → t) → t) → e → t JIV0 =K t/T TV0 = IV/T = (t/T)/T TV0 = ((e → t) → t) → ((e → t) → t) → t JsmilesK = λs.s(λx.smile x) Jloves K= λos.s(λx.o(λy.love x y)) J K

52 / 157 Phenomena at the Syntax-Semantics Interface Quantiﬁcation and Object Position II

CFG Based Approach S → NP VP S = VP NP VP → tV NP JVPK = JtV KJNP K NP → Det N JNPK = JDetKJ N K Det → a JDetK = JλPQKJ.∃x.KP x ∧ Q x Det → every JDetK = λPQ.∀x.P x ⇒ Q x tV → loves JtV K = λos.s(λx.o(λy.love x y)) N → man JN K = man N → woman JNK = woman NP → Mary JNPK = λP.P m NP → John JNPK = λP.P j NP → NP1 Conj NP2 JNPK = λr. Conj ( NP1 r)( NP2 r) Conj → and JConjK = λs1Js2.s1 ∧K sJ2 K J K J K

John loves a woman =? J K Every man loves some woman =? J K How do you get an object wide scope reading?

53 / 157 Phenomena at the Syntax-Semantics Interface Adjectives

Grammar S → NP VP S = VP NP NP → Det N JNPK = JDetKJ N K Det → a JDetK = JλPQKJ.∃x.KP x ∧ Q x Det → every JDetK = λPQ.∀x.P x ⇒ Q x VP → smiles JVP K = λs.s(λx.smile x) N → man JN K = man NP → Mary JNPK = λP.P m NP → John JNPK = λP.P j N → Adj N JN K = λx.( Adj N ) x Adj → big JAdjK = λN.λJx.bigKJx K∧ N x J K

big man = λx.(big x) ∧ (man x) intersective adjectives J K A big man smiles =? J K beautiful dancer =? subsective adjectives J K former student =? non-intersective and non-subsective adjectives J K

54 / 157 Abstract Categorial Grammars

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars Architecture of Grammatical Formalisms λ-terms in the Syntax... and Everywhere Principles and Deﬁnition ACG Composition: The Picture About Word Order Providing a Syntax-Semantics Interface to Context-Free Grammars Modularity of the Components A Functional View on TAG TAG as ACG The CG Approach to Scope Ambiguity Removing Ambiguity From Syntax 55 / 157 Conclusion on ACG

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography Abstract Categorial Grammars The Parallel Architecture: a New Hypothesis Some Observations on Various Grammatical Formalisms

Syntactic Objects (trees, proofs, f-structures) are somehow prior and semantics must be parasitic on those syntactic objects [Muskens, 2001]

Changing the syntactic analysis to simplify one mapping makes the other mapping more complex. A third possibility is to keep both correspondences simple by localizing the complexity in the syntactic component itself.(...) [T]here is a mismatch between phonology and meaning, which has to be encoded somewhere in the mapping among the levels of structure. If this mismatch is eliminated at one point in the system, it pops up elsewhere. [Jackendoﬀ, 2002, p.15]

56 / 157 Abstract Categorial Grammars The Parallel Architecture: a New Hypothesis Mainstream Architectures On the Place of the Syntactic Component

Three Components

Syntax syntactic structures Phonology 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 the other ones

57 / 157 Interface

Abstract Categorial Grammars The Parallel Architecture: a New Hypothesis A Tripartite Parallel Architecture

Three Components

Syntax, Syntactic Interface structures Phonology Interface Semantics

Interface

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

Language comprises a number of independent combinatorial systems which are 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]

58 / 157 Abstract Categorial Grammars λ-terms in the Syntax... and Everywhere λ-terms in the Syntax... and Everywhere

What are λ-terms useful for? Montague-like semantics Generalization of trees and strings Any kind of signatures (atomic types and typed constants): FOL propositions, descriptions (LFG f-structures, URL), other logics Very well studied generative system Variable 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 Theory TAG → MCTAG

60 / 157 Abstract Categorial Grammars λ-terms in the Syntax... and Everywhere The Tectogrammatical and Phenogrammatical Distinction

On the Grammar Architecture [Curry, 1961] Tectogrammatical: abstract combinatorial structure of the grammar Phenogrammatical: concrete operations on syntactic data structures (strings, trees, descriptions) Contrary to the view that: Syntactic objects are the main objects Semantics (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]. . .

61 / 157 Abstract Categorial Grammars Principles and Deﬁnition ACG: a Grammatical Framework

Main Features ACG is a (grammatical) framework An 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

62 / 157 Abstract Categorial Grammars Principles and Deﬁnition ACG: Formal Properties

Generative Power [de Groote and Pogodalla, 2004, Salvati, 2006, Kanazawa and Salvati, 2007, Kanazawa, 2009] String language Tree language ACG(1,n) ﬁnite ﬁnite ACG(2,1) regular regular ACG(2,2) context-free linear contex-free non-duplicating macro ACG ⊂ 1-visit attribute grammar (2,3) well-nested multiple context-free mildly context-sensitive ACG hyperedge replacement gram. (2,4) (multiple context-free) ACG(2,4+n) ACG(2,4) ACG(2,4) ACG(3,n) MELL decidability MELL decidability

Complexity

ACG(2,n) parsing is polynomial, equivalent to datalog querying [Salvati, 2007, Kanazawa, 2007] Reduces to best cases with standard techniques (magic set rewriting) with correct preﬁx Earley algorithms [Kanazawa, 2008]

63 / 157 o λ x.met Chris x : NP ( S

o λ P.P met : ((NP ( NP ( S) ( S) ( S

o λ x.x + met + Chris : σ ( σ

o o λ P.P (λ os.s + met + o) : ((σ ( σ ( σ) ( σ) ( σ

Abstract Categorial Grammars Principles and Deﬁnition

ACG Deﬁnition G = hΣa, Σo, L, Si

Σa NP, S : type A(G ) Λ(Σ ) Chris : NP = {t| ` t : S} a met : NP ( NP ( S

L(NP) = σ (S) = σ L L(α β) = L(α) L(β) ( ) = Chris ( ( L Chris L(t u) = L(t) L(u) ( ) = λos.s + met + o o o L met L(λ x.t) = λ x.L(t)

Σo σ : type

Chris : σ O(G ) Λ(Σo ) met : σ + : σ ( σ ( σ

64 / 157 met Sandy Chris (NP\S)/NP NP NP NP\S S

skip reduction

Abstract Categorial Grammars Principles and Deﬁnition Example My First “Chris met Sandy” ACG Program

Σa : NP, S : type Chris, Sandy : NP met Sandy Chris : S met : NP ( NP ( S

:= Chris NP := σ Chris Sandy := Sandy S := σ o met := λ os.s + met + o

Σo : L(met Sandy Chris) σ : type = L(met) L(Sandy) L(Chris) + : σ ( σ ( σ = (λo os .s + met + o)(Sandy)(Chris) Chris, Sandy : σ = (λo s.s + met + Sandy)(Chris) met : σ = Chris + met + Sandy

65 / 157 G1 ◦ G2

Abstract Categorial Grammars ACG Composition: The Picture ACG Architecture Composition Ability

Functional composition Abstract language sharing (bimorphism) (TAG derivation trees) Parsing and generation (syntactic realization) in the G3 usual sense: function inversion G2

(TAG derived trees)

(TAG string languages)

66 / 157 Abstract Categorial Grammars Word order Intermediate Conclusion

So far... Discussion on possible architectures of grammatical formalisms Discussion on function-compositional properties of ACG and modularity: Abstract structures are mapped to object structures Object structures: Strings, Simple semantic objects, Complex semantic objects: Continuized semantics: S := (t ( t) ( t Results of ACG composition Dynamic semantics: S := γ ( (γ ( t) ( t [de Groote, 2006] ARC CAuLD: Construction Automatique de représentationsLogiques du Discours, http://www.loria.fr/~pogodalla/cauld/ Underspecified representations Algorithms for parsing and generation (in the usual sense) are essentially the same: ACG parsing : finding the abstract antecedent of an object Abstract structures?

69 / 157 Abstract Categorial Grammars Providing a Syntax-Semantics Interface to Context-Free Grammars CFG into ACG Encoding

Example (CFG)

ρ0 : S ρ0 : S → NP VP ρ1 : VP → tV NP ρ2 : NP ρ1 : VP ρ2 : NP → John ρ3 : NP → Mary ρ5 : tV ρ3 : NP ρ4 : VP → left John ρ5 : tV → saw saw Mary

CFG as ACG

ΣRules LCFG ΣStrings ρ0 : NP ( VP ( S := λxy.x + y : σ ( σ ( σ ρ1 : tV ( NP ( VP := λxy.x + y : σ ( σ ( σ ρ2 : NP := John : σ ρ3 : NP := Mary : σ ρ4 : VP := left : σ ρ5 : tV := saw : σ

70 / 157 skip reduction

Abstract Categorial Grammars Providing a Syntax-Semantics Interface to Context-Free Grammars CFG into ACG Encoding (cont’d)

CFG as ACG

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)

71 / 157 Abstract Categorial Grammars Providing a Syntax-Semantics Interface to Context-Free Grammars CFG Encoding

CFG derivation trees

Gsem

GCFG

CFG string languages CFG direct semantics

72 / 157 Abstract Categorial Grammars Providing a Syntax-Semantics Interface to Context-Free Grammars A Direct Semantics Sharing Abstract Languages

CFG syntax as ACG

CFG (direct) semantics as ACG

ΣRules Lsem ΣLog ρ0 : NP ( VP ( S := λsP.P s : e ( (e ( t) ( t ρ1 : tV ( NP ( VP := λPos.P s o :(e ( e ( t) ( e ( e ( t ρ2 : NP := John : e ρ3 : NP := Mary : e ρ4 : VP := left : e ( t ρ5 : tV := saw : e ( e ( t

73 / 157 skip reduction

Abstract Categorial Grammars Providing a Syntax-Semantics Interface to Context-Free Grammars A Direct Semantics (cont’d)

CFG (direct) semantics as ACG

Lsem(ρ0 ρ2 (ρ1 ρ5ρ3): S) = (λsP .Ps )John((λPos .Pso )saw Mary) →β (λP.P John)((λos .saw so )Mary) o →β (λP.P John)(λ s.saw s Mary) o →β (λ s.saw s Mary)John →β saw John Mary

74 / 157 Abstract Categorial Grammars Modularity of the Components A Continuized (Higher–Order) Semantics

CFG continuized semantics as ACG [Barker, 2002]

ΣRules Lsem ΣLog S := (t ( t) ( t NP := (e ( t) ( t VP := ((e ( t) ( t) ( t tV := ((e ( e ( t) ( t) ( t N := ((e ( t) ( t) ( t Det := (((e ( t) ( t) ( t) ( (e ( t) ( t o o o ρ0 : NP ( VP ( S := λ svp.v(λ P.s(λ x.p(Px))) o o o ρ1 : tV ( NP ( VP := λ voP.v(λ R.o(λ y.P(Ry))) o ρ2 : NP := λ P.P John o ρ5 : tV := λ P.P saw o o ρevery : Det := λ KP.K(λ Q.∀x.(Q x) ⇒ (P x))

Scope ambiguity Scope displacement NP as a scope island

75 / 157 Abstract Categorial Grammars Modularity of the Components CFG Encoding

CFG derivation trees

Gcont. sem

GCFG

Gsem

CFG string languages CFG continuized semantics

CFG direct semantics

76 / 157 Abstract Categorial Grammars A Functional View on TAG Trees as λ-Terms

Trees Build on a Ranked Alphabet S S NP VP NP VP John likes NP John sleeps Mary

S2(NP1John)(VP2likes (NP1Mary)) S2(NP1John)(VP1sleeps)

S of arity 2 (non-terminal) S2 : τ ( τ ( τ NP of arity 1 (non-terminal) NP1 : τ ( τ VP? VP of arity 1 and VP of arity 2 1 2 VP : τ τ, VP : τ τ τ (non-terminals) 1 ( 2 ( ( John : τ John of arity 0 (terminal)

78 / 157 Abstract Categorial Grammars A Functional View on TAG A Functional View on TAG

Tree Adjunction:

X X −→ X

X X ∗

79 / 157 Abstract Categorial Grammars A Functional View on TAG Auxiliary Trees as Functions

Example S S VP NP VP NP VP −→ apparently VP∗ apparently VP likes NP likes NP

  VP VP VP  o  λ x.  →β apparently VP apparently x likes NP likes NP   S       VP   o λa. VP  λ x.   NP a          apparently x likes NP

80 / 157 Abstract Categorial Grammars A Functional View on TAG Adjunction as Functional Application

0 0 γlikes γapparently =   S      VP  VP λo a.  λo x.   NP a        apparently x likes NP S      VP VP →β   o   NP  λ x.    apparently x likes NP S

NP VP →β apparently VP

likes NP

81 / 157 Abstract Categorial Grammars A Functional View on TAG Substitution Operation

−→ X X X

82 / 157 Abstract Categorial Grammars A Functional View on TAG Substitution as Functional Application

Example S S

NP VP NP VP −→ NP likes NP NP Chris likes NP

Chris Sandy Sandy

  S S         NP NP NP VP  o      λ os. s VP      →β     Chris Sandy Chris likes NP likes o Sandy

83 / 157 Abstract Categorial Grammars A Functional View on TAG Putting Everything Together

Σtrees :

τ : type   VP o   γapparently = λ ax.a   :(τ ( τ) ( τ ( τ apparently x o I = λ x.x : τ ( τ NP γJohn = : τ John   S   (τ τ)  VP  ( γ = λo Saso.S   : (τ τ) likes  s a    ( (   τ τ τ likes o ( ( (

84 / 157 Abstract Categorial Grammars A Functional View on TAG TAG Derivation as Term Application

Example

γlikes II γJohn γMary =    S         VP  NP NP λo Saso.S  (λo x.x)(λo x.x)     s a          Mary likes o John   S        VP  NP NP → λo so.    β  s          Mary likes o John S

NP VP

→β John likes NP

Mary

85 / 157 Abstract Categorial Grammars A Functional View on TAG Yield as an ACG

Derived Tree Signature S

S2 : τ ( τ ( τ NP VP NP1 : τ ( τ VP1 : τ ( τ, VP2 : τ ( τ ( τ John likes NP John : τ John S2(NP1John)(VP2likes (NP1Mary))

String signature (as before): GYield τ := σ John := John σ : type X := λx.x X := λxy.x + y John, likes ... : σ 1 2 ···

S2(NP1John)(VP2likes (NP1Mary)) := John + (likes + Mary)

86 / 157 Abstract Categorial Grammars A Functional View on TAG TAG as ACG The Current Picture

TAG derivation trees

Λ(Σderivations ) Gtyped trees

TAG derived trees

Λ(Σtrees ) Gyield

TAG string languages

Λ(Σstring)

87 / 157 Abstract Categorial Grammars A Functional View on TAG

A(Gyield ) = TAG Derived Trees?

Σtrees : τ : type   VP o   γapparently = λ ax.a   :(τ ( τ) ( τ ( τ apparently x o I = λ x.x : τ ( τ NP γJohn = : τ John

γapparently I γJohn = apparently NP

John

88 / 157 Abstract Categorial Grammars TAG as ACG TAG as ACG

Category Induced Constraints The site of an adjunction has the same category as the root (and foot) node of the auxiliary tree The site of a substitution has the same category as the root node of the substituted tree

Ltyped trees Σderivations −→ Σtrees cJohn : NP := γJohn : τ capparently :(VP ( VP) ( VP ( VP := γapparently :(τ ( τ) ( τ ( τ NP, VP, S ... : types := σ

Example VP There is no t : VP ∈ Λ(Σderivations ) such that t := apparently NP John

89 / 157 Abstract Categorial Grammars TAG as ACG Control on the Derived Trees

Gtyped trees = hΣderivations , Σtrees , Ltyped trees, Si

NP, VP, S ... := σ NP cJohn : NP := John ! o VP capparently :(VP VP) VP VP := λ ax.a ( ( ( apparently x   S ! clikes :(S ( S) ( (VP ( VP) o VP := λ Saso.S  s a  ( NP ( NP ( S likes o

S NP VP clikes (capparently IVP)cJohncMary : S = John apparently VP likes NP Mary

90 / 157 Abstract Categorial Grammars TAG as ACG TAG Derivation Trees as Abstract Terms

clikes 0 3 1 2

clikes IS(capparently IVP)cJohncMary : S = IS capparently cJohn cMary

IVP clikes :(S ( S) ( (VP ( VP) ( NP ( NP ( S | {z } | {z } |{z} |{z} 0 1 2 3

91 / 157 Abstract Categorial Grammars TAG as ACG TAG as ACG Intermediate Picture

TAG derivation trees

Λ(Σderivations ) Gtyped trees

TAG derived trees

Λ(Σtrees ) Gyield

TAG string languages

Λ(Σstring)

92 / 157 Abstract Categorial Grammars TAG as ACG Let’s Build Some Semantic Representation

Forgetting few seconds about TAG, we have: VP, NP, S : types cjohn, cmary : NP A higher-order signature Σderivations : capparently : VP ( VP clikes :(VP ( VP) ( NP ( S Some knowledge about Montague-like semantics?

A standard interpretation

S := t NP := (e ( t) ( t VP := e ( t o o cjohn := λ P.P j capparently := λ aP.a(λx.apparently(P x)) IVP := λx.x clikes := λaos.s(a(λx.o(λy.like x y)))

How to get the object wide scope reading?

93 / 157 Abstract Categorial Grammars TAG as ACG TAG with Semantics

Λ(Σderivations )

TAG derivation trees

GLog

Gtyped trees

TAG derived trees

Λ(Σtrees ) Λ(ΣLog) Gyield

TAG string languages

Λ(Σstring)

94 / 157 Abstract Categorial Grammars TAG as ACG Intermediate Conclusion

So far Trees as λ-terms Yield as an ACG Typing control: ACG from derivation trees to derived trees Some semantics added. Is it a function from syntax?

Questions? Any TAG feature missing?

Order of Gtyped trees ?(clikes :(VP ( VP) ( NP ( S)

95 / 157 Abstract Categorial Grammars TAG as ACG The Actual Picture

Λ(ΣTAG )

GTAG

GLog

Gtyped trees

Gyield

99 / 157 Abstract Categorial Grammars TAG as ACG The Final Picture

Λ(ΣMCTAG ) GMCTAG Λ(Σtuple ) Gtuple Λ(ΣTAG )

GTAG

GLog

Gtyped trees Gyield

103 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguities

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

CFG-like systems

loves = λos.s(λx.o(λy.love(x, y))) J K loves = λos.o(λy.s(λx.love(x, y))) J K

Underspeciﬁed framework (see the section on underspectiﬁcation) ∀x.man x → (∃y.woman y ∧ love x y) every man loves some woman → → → ? ∃y.woman y ∧ (∀x.man x → love x y)

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

105 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Strengths and Weaknesses

Underspeciﬁed framework Cons: Pros: Description language One syntactic analysis Ambiguity in the semantic recipe, Expressivity not in the interface

TL framework Pros: Cons: No intermediate language Syntactic ambiguity Ambiguity handled by the process

Question Is there an ACG way providing a proof-theoretic approach with only one syntactic structure and no intermediate language?

106 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity in Categorial grammars

The standard way loves loves everyone NP\S/NP [NP] [NP] NP\S/NP someone S/(NP\S) NP\S S/NP (NP/S)\S S someone everyone S S/NP (NP/S)\S S/(NP\S) NP\S S S o o o o Csomeone (λ y.Ceveryone (λ x.Cloves y x)) Ceveryone (λ x.Csomeone (λ y.Cloves y x))

The ACG way Replace \ and / by ( Ceveryone :(NP ( S) ( S

107 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity in ACG: The Semantics

ΣCG Σlog Cloves : NP ( NP ( S loves : e ( e ( t Ceveryone :(NP ( S) ( S ∀, ∃ :(e → t) ( t Csomeone :(NP ( S) ( S ∧, ⇒ : t ( t ( t Lamb-log NP := e S := t o Cloves := λ os.loves(s.o) o Ceveryone := λ P.∀x.human(x) ⇒ P x o Csomeone := λ P.∃y.human(y) ∧ P y

o o Ceveryone (λ x.Csomeone (λ y.Cloves x y)) o o o o :=amb-log (λ P.∀x.human(x) ⇒ P x)((λ x.(λ P.∃y.human(y) ∧ P y)(λ y.loves(x, y)))) o o →β (λ P.∀x.human(x) ⇒ P x)((λ x.(∃y.human(y) ∧ loves(x, y)))) →β (∀x.human(x) ⇒ (∃y.human(y) ∧ loves(x, y)))

108 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity in ACG: The Semantics (cont’d)

o o Csomeone (λ y.Ceveryone (λ x.Cloves x y)) o o o o :=amb-log (λ P.∃y.human(y) ∧ P y)((λ x.(λ P.∀x.human(x) ⇒ P y)(λ y.loves(x, y)))) o o →β (λ P.∃y.human(y) ∧ P x)((λ x.(∀x.human(x) ⇒ loves(x, y)))) →β (∃y.human(y) ∧ (∀x.human(x) ⇒ loves(x, y)))

109 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity in ACG

ΣCG Σstring Cloves : NP ( NP ( S loves : σ Ceveryone :(NP ( S) ( S everyone : σ Csomeone :(NP ( S) ( S someone : σ Lamb-string o Cloves := λ os.s + loves + o o Ceveryone := λ P.P everyone o Csomeone := λ P.P someone

o o Ceveryone (λ x.Csomeone (λ y.Cloves x y)) o o o o o :=amb-string (λ P.P everyone)((λ x.λ P.P someone)(λ y.(λ os.s + loves + o) y x)) o o o o →β (λ P.P everyone)((λ x.λ P.P someone)(λ y.x + loves + y)) o o o →β (λ P.P everyone)(λ x.(λ y.x + loves + y) someone) o o →β (λ P.P everyone)(λ x.x + loves + someone) o →β (λ x.x + loves + someone) everyone →β everyone + loves + someone

110 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity in ACG (cont’d)

o o Csomeone (λ y.Csomeone (λ x.Cloves x y)) o o o o o :=amb-string (λ P.P someone)((λ y.λ P.P everyone)(λ x.(λ os.s + loves + o) y x)) o o o o →β (λ P.P someone)((λ y.λ P.P everyone)(λ x.x + loves + y)) o o o →β (λ P.P someone)(λ y.(λ x.x + loves + y) everyone) o o →β (λ P.P someone)(λ y.everyone + loves + y) o →β (λ y.everyone + loves + x) someone →β everyone + loves + someone

111 / 157 Gamb

Non-functional relation

Λ(ΣSimpleSyn) Σsurface

Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity Non Injective Lexicon

Λ(ΣCG )

GLog

Gamb-string

Λ(ΣLog)

Λ(Σstring )

112 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity in ACG (cont’d)

ΣCG ΣSimpleSyn Cloves : NP ( NP ( S cloves : NP ( NP ( S Ceveryone :(NP ( S) ( S ceveryone : NP Csomeone :(NP ( S) ( S csomeone : NP

Lamb o o Cloves := λ os.loves o sλ os.cloves o s o Ceveryone := λ P.P everyoneceveryone o Csomeone := λ P.P someonecsomeone

o o Ceveryone (λ x.Csomeone (λ y.Cloves x y)) :=amb cloves csomeone ceveryone o o Csomeone (λ y.Csomeone (λ x.Cloves x y)) :=amb cloves csomeone ceveryone

113 / 157 Gamb-string

Abstract Categorial Grammars Removing Ambiguity From Syntax Scope Ambiguity Non Injective Lexicon

Λ(ΣCG )

GLog

Gamb

Non-functional relation

Λ(ΣSimpleSyn) Λ(ΣLog) Σsurface

Λ(Σstring )

114 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax Conjunction John and every kid ran

o o o Cand := λ PQR.P(λ x.Q(λ y.R(cand x y))) Crun := crun Ckid := ckid Crun : NP ( S crun : NP ( S CJohn := cJohn Ckid : N ckid : N CJohn : NP cJohn : NP Cand : ((NP ( S) ( S) cand : NP ( NP ( NP ( ((NP ( S) ( S) crun(cand cJohn(cevery ckid )) ( (NP ( S) ( S o Cand (λ P.PCJohn)(cevery ckid )Crun

o crun := λ s.s + ran Crun := run ckid := kid Ckid := kid cJohn := John CJohn := j o o cand := λ xy.x + and + y Cand := λ PQ. λR.(PR) ∧ (QR)

Σstring ΣLog (John + and + (every + kid)) + ran (run j) ∧ (∀x.kid x ⇒ run x) 115 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax De re and De dicto Readings

o o Cseek := λ xP.P(λ y.cseek x y) Cbook := cbook Cseek : NP ( ((NP ( S) ( S) ( S cseek : NP ( NP ( S Cbook : N cbook : N Cseek CJohn(CaCbook ) cseek cJohn(cacbook ) o (CaCbook )(λ y.Cseek CJohn (λo Q.Q y))

C := λo xo. c := λo xy.x + seeks + y seek seek try x (λo z.o c := book book (λo y.ﬁnd z y)) Cbook := book

Σ Σ Log string ∃y.(book y) ∧ (try j (λo x.ﬁnd x y)) John + seeks + (a + book) try j (λo x.∃y.(book y) ∧ (ﬁnd x y))

116 / 157 Abstract Categorial Grammars Removing Ambiguity From Syntax And More...

So far Conjunction De re and de dicto readings Coordination of quantified and non-quantified NPs VP ellipsis John saw a kid and so did Bill de re and de dicto readings Quantification and negation (every kid did’nt run)

Generalization: the Scoping Constructor

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

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

syn sem Given a lexical entre w : a ∈ TL(A), we have cw : a and Cw : a such that: if a ∈ A then asyn = a and asem = a syn syn syn sem sem sem if a = α ( β then a = α ( β and a = α ( β . syn syn sem sem sem sem if a = α ⇑ β then a = α and a = (α ( β ) ( β 117 / 157 Abstract Categorial Grammars Conclusion on ACG Conclusion on ACG

A large number of grammatical formalisms can be encoded into (2nd order) ACG Semantic ambiguity in type-logical grammars arises from higher-order (3rd order) types Type-logical grammars can be provided a “simple syntactic” level of 2nd order We can apply the same higher-order technics to any 2nd order ACG! Encoding in a same framework: sharing and comparing analysis ACG composition modes: ﬂexible and open architectures “Syntax”, “function”, “relation”, “compositionality”, “rule-to-rule” intuitions may be realized by diﬀerent mathematical notions Shallow opposition between syntactocentric and parallel formalisms

119 / 157 Underspeciﬁcation

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography

120 / 157 Underspeciﬁcation Main Features [Egg, 2010]

Principles Principle To omit some information from linguistic descriptions Aim To capture alternative realisations in one single representation Aim To avoid enumeration of the alternatives

Example Morphological features in grammar rules:

S −→ NPnom VP NPnom −→ John NPacc −→ John

Uniﬁcation grammars:

S −→ NP[case = nom] VP NP −→ John

121 / 157 Underspeciﬁcation

Features [Bos, 1995] Two levels of description: an object-level of linguistic representations a meta-level of describing these representations Fragments of object-level semantic representations Glue points (hole) in the fragments Relation between glue points and fragments

∀x.man(x) ⇒ ∃y.woman(y)∧

loves(x, y)

122 / 157 Underspeciﬁcation Description and Models

Getting the Readings Each hole gets ﬁlled by a fragment Respecting the description We get a modela description

123 / 157 Underspeciﬁcation Semantic Ambiguity: Example

∀x.man(x) ⇒ ∃y.woman(y)∧

loves(x, y)

∀x.man(x) ⇒ ∃y.woman(y)∧

∃y.woman(y)∧ ∀x.man(x) ⇒

loves(x, y) loves(x, y)

124 / 157 Discourse

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse Challenges for Compositionality Accessibility According to DRT Accessibility According to Discourse Hierarchy Dynamic Logic Continuation Semantics

125 / 157 9 Selected Bibliography Discourse Challenges for Compositionality Challenging the Compositionality Principle From Sentence to Discourse

Example (Anaphoric Pronouns and their Antecedents) She is extraordinarily beautiful Vincent looks at Mia. She dances. After he lost the match, Butch left town.

Challenges How to resolve the pronouns How to represent the pronouns

126 / 157 Discourse Challenges for Compositionality Pronoun Resolution

Highly Visible Constraints Agreement (gender, number. . . ) Enough for Resolution?

Example Butch threw aTV at the window. It broke. Butch threw a vase at the wall. It broke.

127 / 157 Discourse Challenges for Compositionality Pronoun Resolution More Examples

Example George Burns and Gracie Allen: The Salesgirl Gracie: Oh, yeah. . . and then Mr. and Mrs. Jones were having matromonial trouble, and my brother was hired to watch Mrs. Jones. George: Well, I imagine she was a very attractive woman. Gracie: She was, and my brother watched her day and night for six months. George: Well, what happened? Gracie: She ﬁnally got a divorce. George: Mrs. Jones? Gracie: No, my brother’s wife.

128 / 157 Discourse Challenges for Compositionality Pronoun Resolution Approaches

See for instance [Jurafsky and Martin, 2000] Heuristic and statistical approaches Focus-base approaches Centering

But how to represent them?

A Bit of History Beginning of the 80’s Discourse Representation Theory (DRT) [Kamp, 1981, Kamp and Reyle, 1993] and File Change Semantics [Heim, 1983]

129 / 157 Discourse Challenges for Compositionality First-Order Logic and Discourse

Example Mia is a woman woman(Mia) She loves Vincent. love(x, Vincent) Mia is a woman. She loves Vincent. woman(Mia) ∧ love(x, Vincent) ∧ x = Mia

Example A woman snorts. She collapses. ∃z.(woman z ∧ snort z) ∧ collapse(x) ∧ x = z)

130 / 157 Discourse Challenges for Compositionality Intra-Sentential Anaphora

Example Donkey Sentences If John owns a donkey, he beats it. ∃x.(donkey(x) ∧ owns(John, x) ⇒ beats(John, y) ∧ x = y) ∀x.(donkey(x) ∧ owns(John, x) ⇒ beats(John, y) ∧ x = y)

131 / 157 Discourse Accessibility According to DRT Accessibility Anaphoric pronouns and their antecedents

Example (Existentials, proper nouns, and negation) John owns a car. It is red. ∃x car x ∧ own j x ∧ red x John doesn’t own a car. ∗It is red. ¬(∃x car x ∧ own j x) ∧ red x John doesn’t own a car. He is ecology-minded. ¬(∃x car x ∧ own j x) ∧ ecolo j

132 / 157 Discourse Accessibility According to DRT DRT and Context Change Potential

Taking the Context into Account A woman snorts Statement about the world A referent is made available in the context for further reference

133 / 157 Discourse Accessibility According to DRT Discourse Representation Structures

Example x, y woman(x) A woman snorts. She collapses. snort(x) collapse(y) y = x

DRSs Discourse Referents Conditions Conditions can be complex (with logical connectives, except quantiﬁers, and other DRSs)

134 / 157 Discourse Accessibility According to DRT Discourse Representation Structures

1 If x1,..., xn(n ≥ 0) are discourse referents and γ1, . . . , γm(m > 0) are conditions then x1,..., xn γ1 . is a DRS . γn

2 If R is a relation symbol of arity n and x1,..., xn are some discourse referents, then R(x1,..., xn) is a condition

3 if t1 and t2 are discourse referents or constants, then t1 = t2 is a condition

4 if K1 and K2 are DRSs, then K1 ⇒ K2 is a condition

5 if K1 and K2 are DRSs, then K1 ∨ K2 is a condition 6 if K is a DRSs, then ¬K is a condition 7 Nothing else is a condition or a DRS

135 / 157 Discourse Accessibility According to DRT Accessibility Constraint

Example x, y x = John John has a car car(y) owns(x, y)

136 / 157 Discourse Accessibility According to DRT Accessibility Constraint

Example z

x, y x = John John doesn’t have a car. It is red ¬ car(y) owns(x, y) red(z) z =??

Accessibility and Subordination

1 K1 is immediately subordinate to K2 iff ¬K1 (or K1 ⇒ K for some K) is a condition of K2, or if K2 ⇒ K1 is a condition of some K 2 K1 is subordinate to K2 (K1 < K2) iff either 1 K1 is immediately subordinate to K2, or 2 there is K3 such that K3 < K2 and K1 is immediately subordinate to K3 3 For K a DRS, x a referent, and γ a condition, x is accessible from γ in K iff there are K2 ≤ K1 ≤ K such that x is in the universe of K1 and γ in the conditions of K2

137 / 157 Discourse Accessibility According to DRT Interpreting DRSs

Several Possibilities Kamp’s original embedding semantics Groenendijk and Stokhof’s dynamic semantics [Groenendijk and Stokhof, 1991] Translation into ﬁrst-order logic [Blackburn and Bos, 2005] via λ-DRT [Muskens, 1991, Muskens, 1996]

And several drawbacks: destructive assignments.

Available tools BOXER[Bos, 2008] and C and C tools Grail[Moot, 1999]

138 / 157 Discourse Accessibility According to Discourse Hierarchy Accessibility Anaphoric Pronouns and Their Antecedents

Example (Hierarchical structure of the discourse [Busquets et al., 2001]) John went to the hospital. Mary broke his nose. John went to the hospital Peter broke his arm Mary broke his nose Peter broke his arm HeShe even bit him He∗She even bit him.

What we’ve learned from theories on rhetorical structure Segments of the discourse stand in relation to each other Depending on the relation (coordinating, subordinating), discourse markers are accessible or not

139 / 157 Discourse Dynamic Logic Dynamic Logics

Technical Issues Non-standard interpretation of formulas: Interpretation as relations between assignment functions φ → ψ = {hg, hi|h = g&∀k.h φ k ⇒ ∃j.k ψ j} (J∃x.φ) ∧K ψ ⇔ ∃x.(φ ∧ ψ) (scopeJ theorem)K J K Destructive assignment and variable clash

φ ⇒ ψ =? J K

Formal Semanticist or Logician? What are the useful data to feed the context with? How do discourse and sentences combine? What are the semantic recipes of the lexical items Should I design a new logic? Continuation semantics

140 / 157 Discourse Continuation Semantics Continuation Semantics Accessibility: The Context (Accessible Discourse Referents) as an Argument

Principles [de Groote, 2006]

S = γ → (γ → t) → t =∆ Ω JNPK = (e → S ) → S JN K = e → JS K J K 0 0 JS1K.S2 = λe.λφ.J KS1 e (λe . S2 e φ) J K J K J K

Interpretation of sentences

e:γ φ:γ→t z }| { S z }| { | {z } t

Composition of sentences

e:γ φ:γ→t z }| { z }| { S1. S2 | {z } 0 0 λe . S2 e φ:γ→t J K

141 / 157 Discourse Continuation Semantics CS: an Example

Example A man is sleeping. He is snoring. λe.λφ.∃x. (man x) ∧ (sleeping x) ∧ (φ (x :: e)) λe.λφ.(snoring (sel e)) ∧ (φ e)

Composition of sentences

T .S = λe.λφ. T e (λe0. S e0 φ) J K J K J K

λe φ.[λe φ.∃x.(man x) ∧ (sleeping x) ∧ φ (x :: e)] e (λe0.(λe φ.(snoring (sel e)) ∧ (φ e)) e0 φ)

→β λe φ.[λφ.∃x.(man x) ∧ (sleeping x) ∧ (φ (x :: e))] (λe0.(snoring (sel e0)) ∧ (φ e0)) 0 0 0 →β λe φ.[∃x.(man x) ∧ (sleeping x) ∧ ((λe .(snoring (sel e )) ∧ (φ e ))( x :: e))]

→β λe φ.[∃x.(man x) ∧ (sleeping x) ∧ ((snoring (sel (x :: e)) ∧ (φ (x :: e))))]

142 / 157 Discourse Continuation Semantics Accessibility The Context (Accessible Discourse Referents) as an Argument

Composition of sentences

0 0 S1.S2 = λeφ. S1 e (λe . S2 e φ) J K J K J K

Example j y John owns a car car y λeφ.∃y.car y ∧ own j y ∧ φ (y :: e) own j y it z=? λPeφ.P (sel e) e φ z it is red red z λeφ.red(sel e) ∧ φ e z =? j y z car y John owns a car λeφ.∃y.car y ∧ own j y ∧ red(sel y :: e) own j y it is red ∧φ (y :: e) red z z = y

143 / 157 Discourse Continuation Semantics Lexical Semantics

Lexicon John = λPeφ.P j e φ JownsK = λOS.S(λx.O(λye0φ0.own x y ∧ φ0 e0)) Ja K = λPQeφ.∃y.P y (y :: e) φ ∧ Q y (y :: e) φ JcarK = λxeφ.car x J K

Example

a car = λQeφ.∃y.car y ∧ Q y (y :: e) φ JownsKJ (K a car ) = λS.S(λx.(λQeφ.∃y.car y ∧ Q y (y :: e) φ) J K J KJ K (λye0φ0.own x y ∧ φ0 e0)) = λS.S(λx.(λeφ.∃y.car y ∧ (λye0φ0.own x y ∧ φ0 e0) y (y :: e) φ)) = λS.S(λx.(λeφ.∃y.car y ∧ (own x y ∧ φ (y :: e)))) owns ( a car ) John = (λPeφ.P j e φ)(λx.(λeφ.∃y.car y ∧ (own x y ∧ φ (y :: e)))) J K J KJ K J K = (λeφ.(λx.(λeφ.∃y.car y ∧ (own x y ∧ φ (y :: e)))) j e φ) = λeφ.∃y.car y ∧ (own j y ∧ φ (y :: e))

144 / 157 Discourse Continuation Semantics A New Dynamic Logic [de Groote, 2007]

Logical connectives

Conjunction: A u B =∆ λeφ.A e(λe0.B e0φ) Existential quantification: Σx.Px =∆ λeφ.∃x.P x (x :: e) φ Negation: Ã =∆ λeφ.¬(A e (λe.>)) ∧ φ e Universal quantification: Πx.Px =∆ ˜(Σx.˜(P x)

Example Lexical semantics Montague semantics Dynamics semantics a, some λPQ.∃x.P x ∧ Q x λPQ.Σx.P x u Q x every λPQ.∀x.P x ⇒ Q x λPQ.Πx.P x A Q x

145 / 157 Discourse Continuation Semantics Discourse

See Philippe de Groote’s slide

146 / 157 Selected Bibliography

1 Introduction

2 Meaning

3 Types and Model Structure

4 Montague Semantics

5 Phenomena at the Syntax-Semantics Interface

6 Abstract Categorial Grammars

7 Underspeciﬁcation

8 Discourse

9 Selected Bibliography

147 / 157 Selected Bibliography

Barker, C. (2002). Continuations and the nature of quantification. Natural Language Semantics, 10:211–242. Blackburn, P. and Bos, J. (2005). Representation and Inference for Natural Language. A First Course in Computational Semantics. CSLI. Bos, J. (1995). Predicate logic unplugged. In Proceedings of the Tenth Amsterdam Colloquium. Bos, J. (2008). Wide-coverage semantic analysis with boxer. In Bos, J. and Delmonte, R., editors, Semantics in Text Processing. STEP 2008 Conference Proceedings, Research in Computational Semantics, pages 277–286. College Publications. Busquets, J., Vieu, L., and Asher, N. (2001). La SDRT : une approche de la cohérencedu discours dans la tradition de la sémantiquedynamique. Verbum, 23(1).

148 / 157 Selected Bibliography

Culicover, P. W. and Jackendoﬀ, R. (2005). Simpler Syntax. Oxford University Press. Curry, H. B. (1961). Some logical aspects of grammatical structure. In Jakobson, R., editor, Structure of Language and its Mathematical Aspects: Proceedings of the Twelfth Symposium in Applied Mathematics, pages 56–68. American Mathematical Society. de Groote, P. (2006). Towards a montagovian account of dynamics. In Gibson, M. and Howell, J., editors, Proceedings of Semantics and Linguistic Theory (SALT) 16. http://elanguage.net/journals/index.php/salt/article/view/16.1/1791. de Groote, P. (2007). Yet another dynamic logic. Presentation at the 4th Lambda Calculus and Formal Grammar workshop. http://www.loria.fr/equipes/calligramme/acg/workshops/lcfg-04/slides/ lcfg04-degroote.pdf.

149 / 157 Selected Bibliography de Groote, P. and Pogodalla, S. (2004). On the expressive power of Abstract Categorial Grammars: Representing context-free formalisms. Journal of Logic, Language and Information, 13(4):421–438. Dowty, D. (1982). Grammatical relations and montague grammar. In Jacobson, P. and Pullum, G. K., editors, The Nature of Syntactic Representation. Reidel, Dordrecht. Egg, M. (2010). Semantic underspeciﬁcation: Concepts and applications. Lecture at JSM’10. http://jsm.loria.fr/jsm10/documents/lectures/egg.pdf. Gardent, C. and Kallmeyer, L. (2003). Semantic construction in feature-based tag. In Proceedings of the 10th Meeting of the European Chapter of the Association for Computational Linguistics (EACL). Groenendijk, J. and Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy, 14(1):39–100.

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Heim, I. R. (1983). File change semantics and the familiarity theory of deﬁniteness. In [Portner and Partee, 2002], pages 164–190. Reprinted in [Portner and Partee, 2002]. Hopcroft, J. E., Motwani, R., and Ullman, J. D. (2003). Introduction to Automata Theory, Languages, and Computation. Addison Wesley, 2nd edition. Jackendoﬀ, R. (2002). Foundations of Language: Brain, Meaning, Grammar, Evolution. Oxford University Press. Jurafsky, D. and Martin, J. H. (2000). Speech and Language Processing: An introduction to Natural Language Processing, Computational Linguistics, and Speech Recognition. Prentice Hall. Kallmeyer, L. and Romero, M. (2008). Scope and situation binding for ltag. Research on Language and Computation, 6(1):3–52.

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Kamp, H. (1981). A theory of truth and semantic representation. In Groenendijk, J. A., Janssen, T., and Stokhof, M., editors, Formal Methods in the Study of Language. Foris, Dordrecht. Kamp, H. and Reyle, U. (1993). From Discourse to Logic. Kluwer Academic Publishers. Kanazawa, M. (2007). Parsing and generation as datalog queries. In Proceedings of the 45th Annual Meeting of the Association of Computational Linguistics (ACL), pages 176–183, Prague, Czech Republic. Association for Computational Linguistics. http://www.aclweb.org/anthology/P/P07/P07-1023. Kanazawa, M. (2008). A preﬁx-correct earley recognizer for multiple context-free grammars. In Proceedings of the Ninth International Workshop on Tree Adjoining Grammars and Related Formalisms, pages 49–56, Tuebingen, Germany.

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Kanazawa, M. (2009). Second-order abstract categorial grammars as hyperedge replacement grammars. Journal of Logic, Language, and Information, 19(2):137–161. http://www.springerlink.com/content/p644605651088uv6/. Kanazawa, M. and Salvati, S. (2007). Generating control languages with abstract categorial grammars. In Penn, G., editor, Proceedings of The 12th conference on Formal Grammar FG 2007. CSLI Publications. http://www.dei.unipd.it/~fgrammar/fg07/fg07preproc/0005/paper_10.pdf. Kracht, M. (2003). The Mathematics of Language. Number 63 in Studies in Generative Grammar. Mouton de Gruyter, Berlin. Montague, R. (1970). Universal grammar. Theoria, 36:373–398.

153 / 157 Selected Bibliography

Montague, R. (1974). The proper treatment of quantiﬁcation in ordinary english. In Formal Philosophy: Selected Papers of Richard Montague. Yale University Press. Re-edited in ”Formal Semantics: The Essential Readings”, Paul Portner and Barbara H. Partee, editors. Blackwell Publishers, 2002. Moot, R. (1999). Grail: an interactive parser for categorial grammars. In Proceedings of VEXTAL’99, pages 255–261. http://www.let.uu.nl/~Richard.Moot/personal/grail.html. Muskens, R. (1991). Anaphora and the Logic of Change. In van Eijck, J., editor, Logics in AI, Proceedings of JELIA ’90, volume 478 of Lecture Notes in Computer Science, pages 412–427. Springer-Verlag, Berlin. Muskens, R. (1996). Combining montague semantics and discourse representation. Linguistics and Philosophy, 19(2).

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Muskens, R. (2001). Lambda Grammars and the Syntax-Semantics Interface. In van Rooy, R. and Stokhof, M., editors, Proceedings of the Thirteenth Amsterdam Colloquium, pages 150–155, Amsterdam. Muskens, R. (2003). Lambdas, Language, and Logic. In Kruijﬀ, G.-J. and Oehrle, R., editors, Resource Sensitivity in Binding and Anaphora, Studies in Linguistics and Philosophy, pages 23–54. Kluwer. Oehrle, D. (1995). Some 3-Dimensional Systems of Labelled Deduction. Logic Jnl IGPL, 3(2-3):429–448. Oehrle, R. T. (1994). Term-labeled categorial type systems. Linguistic and Philosophy, 17(6):633–678. Pollard, C. (2004). High-order categorical grammars. In Proceedings of Categorial Grammars 2004, Montpellier, pages 340–361.

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Pollard, C. (2008). An introduction to convergent grammar. Presentation at Calligramme Seminar. http://www.ling.ohio-state.edu/~scott/cvg/. Portner, P. and Partee, B. H., editors (2002). Formal Semantics: The Essential Readings. Blackwell Publishers. Ranta, A. (1994). Type Theoretical Grammar. Oxford University Press. Salvati, S. (2006). Encoding second order string ACG with Deterministic Tree Walking Transducers. In Shuly Wintner, editor, Proceedings of The 11th conference on Formal Grammar FG 2006, FG Online Proceedings, pages 143–156, Malaga Espagne. CSLI Publications. http://cslipublications.stanford.edu/FG/2006/salvati.pdf.

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Salvati, S. (2007). On the complexity of Abstract Categorial Grammars. In Proceedings of the 10th Conference on Mathematics of Language, MOL 10. http: //wwwhomes.uni-bielefeld.de/mkracht/mol10/abstracts/acg_complexity.pdf. Weir, D. J. (1988). Characterizing Mildly Context-Sensitive Grammar Formalisms. PhD thesis, University of Pennsylvania. Winter, Y. (2010). Elements of Formal Semantics. To appear in Edimburgh University Press.

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