This document presents a derivational event semantics system that can compositionally derive semantic representations of natural language expressions from pregroup grammars. The semantics uses a conjunctivist approach where semantic values are monadic predicates that combine using conjunction. The system shows correspondences between syntactic operations in pregroup derivations and semantic operations that handle event predicates and variables. This allows the semantic system to closely follow the structure of pregroup grammars while compositionally deriving layered event representations. The document outlines challenges like handling multiple event variables and introducing thematic roles, and proposes solutions like encoding variable reference types and extending the syntactic type hierarchy to semantic alterations.

1. Derivational Event Semantics for Pregroup
Grammars
Gabriel Gaudreault
Logical Aspects of Computational Linguistics
December 5, 2016

2. Goals
Today I am presenting a derivational system in which event
semantic representations of natural language expressions can
be compositionally derived
The structure of those derivations will be dictated by a
pregroup grammar
Multiple neat correspondences between the syntactic
operations used in pregroup derivations and the semantic ones
used to handle the meaning predicates and event variables are
shown
Those correspondences allow this new semantic appendange
to stay close to the original simplicity of the structure of the
original pregroup grammars

3. Outline
Pregroup Grammars
Event Semantics
Conjunctivism
Pregroup Grammars + Conjunctivist Semantics
Conclusion

4. Pregroup Grammars
Pregroup Grammars (Lambek 1999)
Main idea: We can assign mathematical types to words and then
check whether sentences are grammatical by looking at their
corresponding strings of types and using specific derivation rules.
Types α, β := n, s, o, π, ... | αr
β | αβl
John likes oranges
π πr sol o → s

5. Formal Definition of Pregroups
Pregroup (P, →, r , l , ·, 1) :
Partially ordered monoid over a set P, where every element a ∈ P
has a right and left adjoint — ar ∈ P, al ∈ P respectively —
subject to
a · ar
→ 1 → ar
· a al
· a → 1 → a · al
More precisely:
Associativity: a(bc) = (ab)c
Existence of an identity: 1a = a1 = a
Reflexitivity: a → a
Antisymmetry: if a → b and b → a then a = b
Transitivity: if a → b and b → c then a → c
Reminiscent of arithmetic properties of exponents: 3 ∗ (3−14) = 4

6. Formal Definition of Pregroups
Fun properties of pregroups:
a → b ⇔ bl
→ al
⇔ br
→ ar
arl
= alr
= a
(a1...an)l
= al
n...al
1
Types can be defined for anything you want; you can bypass the
fact that there’s no disjunctive or conjunctive type by simply
adding new types, e.g.
s2 ≈ declarative ∧ past
In general the types do not form a lattice, e.g.
¯n, N → π, o
¯n ∨ N undefinied

7. Pregroup Grammars
The syntactic types in a pregroup grammars correspond to strings
of pregroup elements, e.g.
likes : πr sol at : ir i ¯nl two : ¯nnl
Pregroup Grammars being ordered structure, it is possible to define
ordering relations over syntactic types such as N → π, s2 → s
For instance, a possible reduction for John likes Mary could be:
John likes Mary
N πr sol N
N(πr
sol
)N → π(πr
sol
)o → sol
o → s

8. Pregroup Grammar Derivations
Derivations look really good. The order of the contractions do not
really matter for this kind of grammar.The contraction links
between types show how information flows throughout derivations
πr s il i ir i il i ol nnnl¯n
s
will dance to save humanitymanA

9. Pregroup Parsing
Pregroup grammar parsing has lower complexity than traditional
categorial grammar parsing. Compare this derivation tree:
He
NP
likes
(NP S)/NP
the
NP/N
big
N/N
red
N/N
cat
N
N
N
NP
NP S
S

10. Pregroup Parsing
With this much simpler one
He
π3
likes
πr
3sol
sol
the
¯nnl
onl
snl
big
nnl
snl
red
nnl
snl
cat
n
s
Work can start on the contractions as soon as types start being
put together in this case. When we reach the last lexical item, we
know whatever comes next will have to be of the type of a noun.

11. Structure of Syntactic Pregroup Types
Pregroup grammars differ from most formal syntactic systems as
their syntactic types are, in a way, simple pieces of information
concatenated in a string and can be combined independently from
either side.
Traditional Categorial Grammars: NP / N
Minimalist Grammars: =N D
Pregroup Grammars: ¯nnl
πnl · n ¯n
¯nnl · n
π
r
rrr
rrj
¨
¨¨¨
¨¨%
r
rrr
rrj
¨
¨¨¨
¨¨%

12. Formal Semantics
Traditional set-theoretic characterisation of verbs as relations
between 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 predicate
that takes two arguments as input
λx.λy.kiss(x, y) : (N S) / N

13. Event Semantics
Also possible to provide an analysis in terms of events (Davidson
1967)
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 by
other 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)

14. 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 when
she wore that dress
54 3 2
6
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e
ee…
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


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


s
e
e
ee…
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15. 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 entail
John punches Barry in Chicago
∃e.punch(John, Barry) ∧ Loc(e, Chicago)

16. Event Semantics
Possible to go even further and decompose complementation in
terms 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 using
events 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?

17. Conjuntivism
Conjunctivism: Forget about argument passing and functional
application, semantic values are monadic predicates and they
combine 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 right
logic to do the interpretation. In this case, we use Plural Logic
(Boolos 1984, Schein 1993, Pietroski 2005).

18. Plurality
Intuitive notion of plurality in natural language:
There is a cake on the table
There are cakes on the table
Plural predicates, i.e. do not admit a distributive reading:
The pens form a square, BUT a single pen does not form a
square by itself
The cats gather at night, BUT a single cat cannot gather by
itself
Two different relations:
x ∈ X := x is an element of X
x X := x is one of the X

19. Conjunctivism
Example
[[Cats danced ]] = ∃E.∃X.Agent(E, X)∧Cat(X)∧Plur(X)∧danced(E)
There are a possibly plural event E and possibly plural entity
X
The agents of the values of the event are values of the entity
The values of the entity are cats
The values of the entity are plural (more than one)
The values of the event are events of dancing

20. 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)
= ∃e.Agent(e, John) ∧ kiss(e) ∧ ∀x.girl(x) → Patient(e, x)

21. Quantification
Conjunctivism comes with a distributivity axiom for singular
predicates
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 are
agents of the values of the event

22. Event Semantics in Pregroup Grammars
Goal:
We want to be able to go from the leaves to the full
representation in the simplest way.
We want to only use ∧ as meaning combination operator, as
it represents the essence of Conjunctivism
∃e.dance(e) ∧ Agent(e, John)
β(e )α(e )

23. 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)

24. Event Semantics in Pregroup Grammars
The first step to define is not very hard, we know that meaning
predicates 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 the
beginning 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)

25. Conjunctivism
One of the first real problem comes from the multiple layers of
hidden event and entity variables that can be present in a single
sentence.
e
y
Saturn(y)on(e, y)
e
danced(e)X
Agent(e, X)cats(X)

26. Conjunctivism
Question: How to derivationally explain the event representation
and layers?
A change of variable needs to happen, so that we do not end up
with:
[[ John knows Sara died]]
= ∃e.Agent(e, John) ∧ know(e) ∧ Agent(e, Sara) ∧ died(e)

27. 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 cat
sol 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 contract
with the noun before contracting with the verb.
How should we then analyse this?

28. Handling Multiple Event Variables
A potential solution is to encode a reference to the event layer a
basic type might be acting upon.
John likes the cat
seol
e oenr
x nx
Agent(e, John) ∧ like(e) Patient(e, x) ∧ the(e, x) cat(x)
Now we set the rule that whenever two types contract, the
variables they point to must be unified. This also has the
advantage to ”close” a layer as soon as it is not referenced by any
basic type.

29. Event Semantics in Pregroup Grammars
One last problem we will mention is the introduction of a thematic
role.
The following lexical items cannot be either contracted
syntactically as they are, nor can their meanings be conjoined and
event variables unify, as they are not acting on the same layer.
John dances
Nx πr
ese
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.

30. Syntax-Semantics Hierarchy
Remember that as pregroups are partially ordered we defined some
relations on those types, e.g. s2 → s, ¯n → π. This ordering can be
extended to accomodate changes in variables
αx → βy
A[x] → A[x] ∧ θ(x, y)
For instance
Nx → πe
A[x] → A[x] ∧ Agent(x, e)

31. Syntax-Semantics Hierarchy
Note that it could be tempted to try to close the x variable at the
same time, but this will not work as it could still present in another
of the basic type.
my cat
¯nx nr
x nx ⇒ πenr
x 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 unified
with the cat entity.

32. Note
This is another reason why using a single event variable at a time like Pietroski
did, instead of one for each simple type, does not work for pregroup grammars.
N π
x ⇒ 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)
¯nnl n
¯nπnl n
π
©dd‚
dd‚ ©
x1 x2
x1 (x1 = x2)e x2
e (e = x2)?
e (x1 = x2)?
©dd‚
dd‚ ©

33. Pregroup Grammars + Conjunctivism
My solution:
Syntax Semantics
Pregroup Grammars Conjunctivism
Concatenation of syntactic types Predicate conjunction
Basic types Event variables
Contraction of types Unification of event variables
Type ordering Alteration of truth conditions
The full syntactico-semantic representation of an expression is now
of the form:
((a1, x1), ..., (an, xn), A)
where ai is a simple pregroup type, xi is an event variable, A is a
logical expression with free variables xi ’s. Variables can be
identical, in which case they will stay the same throughout the
derivation no matter what gets assigned to them.

34. Conclusion
The goal of this project was twofold.
First, to give a new semantics for pregroup grammars, which differ
in many respects from other functional grammatical formalisms.
Second, to show how a very restricted version of event semantics
could be derived compositionally using this special kind of
grammar.
It’s been a pleasure working on this project, I really hope I managed
to convey why this kind of work is interesting and worth doing.

35. Thank you!
Thank you for your attention!