Derivational Event Semantics for PregroupGrammars

Gabriel Gaudreault

Logical Aspects of Computational Linguistics

December 5, 2016

Goals

I Today I am presenting a derivational system in which eventsemantic representations of natural language expressions canbe compositionally derived

I The structure of those derivations will be dictated by apregroup grammar

I Multiple neat correspondences between the syntacticoperations used in pregroup derivations and the semantic onesused to handle the meaning predicates and event variables areshown

I Those correspondences allow this new semantic appendangeto stay close to the original simplicity of the structure of theoriginal pregroup grammars

Outline

I Pregroup Grammars

I Event Semantics

I Conjunctivism

I Pregroup Grammars + Conjunctivist Semantics

I Conclusion

Pregroup Grammars

Pregroup Grammars (Lambek 1999)

Main idea: We can assign mathematical types to words and thencheck whether sentences are grammatical by looking at theircorresponding strings of types and using specific derivation rules.

Types α, β := n, s, o, π, ... | αrβ | αβl

John likes orangesπ πr so l o → s

Formal Definition of Pregroups

Pregroup (P,→, r , l , ·, 1) :Partially ordered monoid over a set P, where every element a ∈ Phas a right and left adjoint — ar ∈ P, al ∈ P respectively —subject to

a · ar → 1→ ar · a al · a→ 1→ a · al

More precisely:

I Associativity: a(bc) = (ab)c

I Existence of an identity: 1a = a1 = a

I Reflexitivity: a→ a

I Antisymmetry: if a→ b and b → a then a = b

I Transitivity: if a→ b and b → c then a→ c

Reminiscent of arithmetic properties of exponents: 3 ∗ (3−14) = 4

Formal Definition of Pregroups

Fun properties of pregroups:

a→ b ⇔ bl → al ⇔ br → ar

arl = alr = a

(a1...an)l = aln...al1

Types can be defined for anything you want; you can bypass thefact that there’s no disjunctive or conjunctive type by simplyadding new types, e.g.

s2 ≈ declarative ∧ past

In general the types do not form a lattice, e.g.

n̄,N → π, o

n̄ ∨ N undefinied

Pregroup Grammars

The syntactic types in a pregroup grammars correspond to stringsof pregroup elements, e.g.

likes : πr so l at : i r i n̄l two : n̄nl

Pregroup Grammars being ordered structure, it is possible to defineordering relations over syntactic types such as N → π, s2 → s

For instance, a possible reduction for John likes Mary could be:

John likes MaryN πr so l N

N(πr so l)N → π(πr so l)o → so lo → s

Pregroup Grammar Derivations

Derivations look really good. The order of the contractions do notreally matter for this kind of grammar.The contraction linksbetween types show how information flows throughout derivations

πr s i l i i r i i l i o l nnnln̄

s

will dance to save humanitymanA

Pregroup Parsing

Pregroup grammar parsing has lower complexity than traditionalcategorial grammar parsing. Compare this derivation tree:

HeNP

likes(NP \ S)/NP

theNP/N

big

N/N

redN/N

catN

N

N

NP

NP \ S

S

Pregroup Parsing

With this much simpler one

Heπ3

likes

πr3so l

so l

the

n̄nl

onl

snl

big

nnl

snl

red

nnl

snlcatn

s

Work can start on the contractions as soon as types start beingput together in this case. When we reach the last lexical item, weknow whatever comes next will have to be of the type of a noun.

Structure of Syntactic Pregroup Types

Pregroup grammars differ from most formal syntactic systems astheir syntactic types are, in a way, simple pieces of informationconcatenated in a string and can be combined independently fromeither side.

Traditional Categorial Grammars: NP / NMinimalist Grammars: =N D

Pregroup Grammars: n̄nl

πnl · n n̄

n̄nl · n

π

HHHH

HHj

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HHHH

HHj

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Formal Semantics

Traditional set-theoretic characterisation of verbs as relationsbetween explicit arguments, e.g.

[[ kiss ]] = {(John,Mary), (Charles,Dana), (Katy ,Paul)}

then

kiss(a, b) = > ⇐⇒ a kisses b ⇐⇒ (a, b) ∈ [[ kiss ]]

Nice syntax/semantic correspondence:

Syntactically kiss is divalent AND its semantic value is a predicatethat takes two arguments as input

λx .λy .kiss(x , y) : (N \ S) / N

Event Semantics

Also possible to provide an analysis in terms of events (Davidson1967)

kiss(e, x , y) := x kisses y at event e

[[ John kissed Mary ]] ⇐⇒ ∃e.kissed(e, John,Mary)

Those implicit event variables can also be taken scope over byother expressions

[[ John kissed Mary yesterday ]]

⇐⇒ ∃e.kissed(e, John,Mary) ∧ yesterday(e)

Similar to the way adjectives combine with nouns

[[ passionate dance ]] = passionate(x) ∧ dance(x)

[[ dance passionately ]] = dance(e) ∧ passionately(e)

Event Semantics

Nice for entailments

1. John pinched Sarah

2. John pinched Sarah intensely

3. John pinched Sarah in the afternoon

4. John pinched Sarah when she wore that dress

5. John pinched Sarah at school

6. John pinched Sarah intensely at school in the afternoon whenshe wore that dress

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6

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AAAAU

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Event Semantics

while avoiding some bad ones

John kisses Maria in Chicago

∃e.kiss(e, John,Maria) ∧ Loc(e,Chicago)

John punches Barry on the nose

∃e′.kiss(e′, John,Barry) ∧ Loc(e′, nose)

Does not entailJohn punches Barry in Chicago

∃e.punch(John,Barry) ∧ Loc(e,Chicago)

Event Semantics

Possible to go even further and decompose complementation interms of thematic relations over shared event

[[ John dances ]] = ∃e.Agent(e, John) ∧ dance(e)

Don’t forget that tense and aspect can also be analysed usingevents in this sort of way:

[[ had kissed ]](e) = ∃e ′.e ′ < now ∧ Culminate(e, e ′)

The question now becomes: Where does it stop?

Does it have to stop?

Conjuntivism

Conjunctivism: Forget about argument passing and functionalapplication, semantic values are monadic predicates and theycombine using conjunctions.

[[ Cats danced on Saturn ]]

= ∃E .∃X .∃y .(Agent(E ,X ) ∧ cat(X ) ∧ Plural(X ))

∧ (danced(E ) ∧ E < now) ∧ location(E , y) ∧ Saturn(y)

Looks nonsensical, but actually makes sense if you pick the rightlogic to do the interpretation. In this case, we use Plural Logic(Boolos 1984, Schein 1993, Pietroski 2005).

Plurality

Intuitive notion of plurality in natural language:

I There is a cake on the table

I There are cakes on the table

Plural predicates, i.e. do not admit a distributive reading:

I The pens form a square, BUT a single pen does not form asquare by itself

I The cats gather at night, BUT a single cat cannot gather byitself

Two different relations:

I x ∈ X := x is an element of X

I x ≺ X := x is one of the X

Conjunctivism

Example

[[Cats danced ]] = ∃E .∃X .Agent(E ,X )∧Cat(X )∧Plur(X )∧danced(E )

I There are a possibly plural event E and possibly plural entityX

I The agents of the values of the event are values of the entity

I The values of the entity are cats

I The values of the entity are plural (more than one)

I The values of the event are events of dancing

Quantification

Mixing events and quantification, example:(Champollion 2010-2015, de Groote & Winter 2015)

[[ John kisses every girl ]]

= ∃e.∀x .girl(x)→ Agent(e, John) ∧ kiss(e) ∧ Patient(e, x)

6= ∃e.Agent(e, John) ∧ kiss(e) ∧ ∀x .girl(x)→ Patient(e, x)

Quantification

Conjunctivism comes with a distributivity axiom for singularpredicates

Psg (Tpl) := ∀x .x ≺ Tpl ↔ Psg (x)

[[ Every girl danced ]] =

∃E .∃X .Everyag (E ,X ) ∧ Agent(E ,X )

∧∀x .x ≺ X ↔ girl(x) ∧ ∀e.e ≺ E ↔ danced(E )

In this case, Everyag makes sure that the values of the entity areagents of the values of the event

Event Semantics in Pregroup Grammars

Goal:

I We want to be able to go from the leaves to the fullrepresentation in the simplest way.

I We want to only use ∧ as meaning combination operator, asit represents the essence of Conjunctivism

∃e.dance(e) ∧ Agent(e, John)

β(e ′′)α(e ′)

Functional Event Semantics

In functional semantics, there are different possibilities, e.g.

∃e.Agent(e, John) ∧ dance(e)

λP.∃e.P(e) ∧ dance(e)λe.Agent(e, John)

or

∃e.dance(e) ∧ Agent(e, John)

λP.∃e.dance(e) ∧ ∃e.Agent(e, x) ∧ P(e)λx .John(x)

Event Semantics in Pregroup Grammars

The first step to define is not very hard, we know that meaningpredicates should conjoin.

red cat and hei dances

red(x ′) ∧ cat(x ′′)

cat(x ′′)red(x ′)

Agent(e ′, i) ∧ dances(e ′′)

dances(e ′′)Agent(e ′, i)

Making predicates take scope over the same value from thebeginning is problematic, and we so make use of unification.

red(x ′) ∧ cat(x ′′) ∧ x ′ = x ′′

cat(x ′′)red(x ′)

Agent(e ′, i) ∧ dances(e ′′) ∧ e ′ = e ′′

dances(e ′′)Agent(e ′, i)

Conjunctivism

One of the first real problem comes from the multiple layers ofhidden event and entity variables that can be present in a singlesentence.

e

y

Saturn(y)on(e, y)

e

danced(e)X

Agent(e,X )cats(X )

Conjunctivism

Question: How to derivationally explain the event representationand layers?

A change of variable needs to happen, so that we do not end upwith:

[[ John knows Sara died ]]

= ∃e.Agent(e, John) ∧ know(e) ∧ Agent(e,Sara) ∧ died(e)

Handling Multiple Event Variables

The main issue here is that pregroup grammars are not functional,in the sense that the other of the contractions is not set.

For instance, in

John likes the catso l onr n

Agent(e, John) ∧ like(e) Patient(e, x) ∧ the(e, x) cat(x)

There is no restriction on the determiner to force it to contractwith the noun before contracting with the verb.

How should we then analyse this?

Handling Multiple Event Variables

A potential solution is to encode a reference to the event layer abasic type might be acting upon.

John likes the catseo l

e oenrx nx

Agent(e, John) ∧ like(e) Patient(e, x) ∧ the(e, x) cat(x)

Now we set the rule that whenever two types contract, thevariables they point to must be unified. This also has theadvantage to ”close” a layer as soon as it is not referenced by anybasic type.

Event Semantics in Pregroup Grammars

One last problem we will mention is the introduction of a thematicrole.

The following lexical items cannot be either contractedsyntactically as they are, nor can their meanings be conjoined andevent variables unify, as they are not acting on the same layer.

John dancesNx πrese

John(x) dance(e)

How can we jump from one layer to another?

One possibility is to use some of the machinery already in place:the syntactic hierarchy.

Syntax-Semantics Hierarchy

Remember that as pregroups are partially ordered we defined somerelations on those types, e.g. s2 → s, n̄→ π. This ordering can beextended to accomodate changes in variables

αx → βyA[x ] → A[x ] ∧ θ(x , y)

For instance

Nx → πeA[x ] → A[x ] ∧ Agent(x , e)

Syntax-Semantics Hierarchy

Note that it could be tempted to try to close the x variable at thesame time, but this will not work as it could still present in anotherof the basic type.

my catn̄xnr

x nx ⇒ πenrx nx

my(x) cat(x) ∃e.my(x) ∧ Agent(x , e) cat(x)

The x in the first term is now inaccessible and cannot be unifiedwith the cat entity.

Note

This is another reason why using a single event variable at a time like Pietroskidid, instead of one for each simple type, does not work for pregroup grammars.

N πx ⇒ e

John(x) Agent(e, x) ∧ John(x)

Take the case of a determiner and noun that get used as subject. Following the

left path would give you the wrong final representation, as the variable from

the noun would get unified with the one from the transformed determiner, e.g.

Agent(e, x) ∧ two(e) ∧ cat(e)

n̄nl n

n̄πnl n

π��@@R

@@R��

x1 x2

x1 (x1 = x2)e x2

e (e = x2)?e (x1 = x2)?

��@@R

@@R��

Pregroup Grammars + Conjunctivism

My solution:

Syntax Semantics

Pregroup Grammars ConjunctivismConcatenation of syntactic types Predicate conjunction

Basic types Event variablesContraction of types Unification of event variables

Type ordering Alteration of truth conditions

The full syntactico-semantic representation of an expression is nowof the form:

((a1, x1), ..., (an, xn),A)

where ai is a simple pregroup type, xi is an event variable, A is alogical expression with free variables xi ’s. Variables can beidentical, in which case they will stay the same throughout thederivation no matter what gets assigned to them.

Conclusion

The goal of this project was twofold.

First, to give a new semantics for pregroup grammars, which differin many respects from other functional grammatical formalisms.

Second, to show how a very restricted version of event semanticscould be derived compositionally using this special kind ofgrammar.

It’s been a pleasure working on this project, I really hope I managedto convey why this kind of work is interesting and worth doing.

Thank you!

Thank you for your attention!