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Functional Semantics for Pregroup Grammars
Gabriel Gaudreault
Concordia University, Montreal
CoCoNat 2015
19 July 2015
Gabriel Gaudreault Concordia University, Montreal CoCoNat 2015Functional Semantics for Pregroup Grammars

Goals
Goal:
Figuring out a way to do semantics with pregroup grammars that
is intuitive and does not require learning higher-level mathematics
What I am presenting:
- A functional calculus that is sensible to the incoming direction of
the inputs and which has functional composition as main reduction
operation
- The 1-1 relation between it and a subset of pregroup grammars
that is relevant for linguistic analysis
- How the system can be used for linguistic analysis

Overview
Categorial Grammars
Formal Semantics
Pregroup Grammars
λ-semantics for Pregroup Grammars?
λ↔-calculus
“Curry-Howard” Correspondence
Syntactic & Semantic Analysis

Categorial Grammars
Main idea: We can assign mathematical types to words and then
check whether sentences are grammatical by looking at the string
of their corresponding types and using our derivation rules.
Types s, t := N, S, ... | s/t | s t
Reduction rules inspired by arithmetic
Butterﬂies like oranges
N (N S) / N N → S
(Noun phrase) (subject verb / object) (Noun phrase)

Semantics for the Lambek Calculus
Semantics through λ-calculus
λx.λy.love(x, y)
Elimination Rule ↔ Function Application
a : A λx.b(x) : A B
b(a) : B
Introduction Rule ↔ Function Abstraction
x : A, Γ b : B
Γ λx.b : A B
In parallel to grammaticality check we do meaning extraction
Alan : NP λw.work(x) : NP S
work(Alan) : S

Pregroups
(P, →, r , l , ·, 1) : Partially ordered monoid over P, in which 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
(think of arithmetic: 2 ∗ 2−1 = 1)
Has properties such as:
a → b ⇔ bl
→ al
⇔ br
→ ar
arl
= alr
= a
(a1...an)l
= al
n...al
1

Pregroup Grammars
In PG, words get assigned pregroup types π, s, n, ¯n, i, etc. which
correspond to syntactic categories
Ordered structure: we can now deﬁne relations such as N → π3,
s2 → s
The role that inverses (A B) and (A / B) played in the original
syntactic calculus is played by adjoints ar b and abl ...
πr s il i ir i il i ol nnnl¯n
s
will dance to save humanitymanA

Pregroup Grammars
Diﬀerence with the Syntactic Calculus:
Those new types are associative — (ab)c = a(bc) — and can
combine in much more ﬂexible ways because they are now
considered as a list of independent information pieces
abl
· bcl
→ acl
a/b · b/c → a/c
(in one step)
We can’t really use the λ-calculus to do semantics anymore, as it
is not clear what kind of functions our types represent:
(a b)/c ⇒ c → a → b but ar
bcl
⇒?

λ↔
-calculus
Termσ Formation Rules
x ∈ Varσ
x ∈ Termσ
c ∈ Conσ
c ∈ Termσ
t ∈ TermΩrφ s ∈ Termφrπ
(t)s ∈ TermΩrπ
t ∈ Termπφl s ∈ TermφΩl
t(s) ∈ TermπΩl
t ∈ Termσ x ∈ Varφ
t.x ∈ Termσφl
x ∈ Varφ t ∈ Termσ
x.t ∈ Termφrσ
Example:
Alan ∈ Terme x.(x)work ∈ Termerp
(Alan)work ∈ Termp

λ↔
-calculus - Function Application
Γ1 t ∈ TermΩrφ Γ2 s ∈ Termφrπ
Γ1, Γ2 (t)s ∈ TermΩrπ
Γ1 t ∈ Termπφl Γ2 s ∈ TermφΩl
Γ1, Γ2 t(s) ∈ TermπΩl
Recall in simply typed λ-calculus:
Γ1 a : A Γ2 λx.f (x) : A → B
Γ1, Γ2 f [a/x]
Example:
Paul ∈ Terme x.(x)run ∈ Termer p
(Paul)run ∈ Termp

λ↔
-calculus - Function Abstraction
Γ, x ∈ Varφ t ∈ Termσ
Γ t.x ∈ Termσφl
x ∈ Varφ, Γ t ∈ Termσ
Γ x.t ∈ Termφrσ
Recall lambda abstraction:
Γ, x : A b : B
Γ λx.b(x) : A → B
Pregroup expansion rule:
x ∈ Vara x ∈ Vara
x.x ∈ Termara

Reformulation of Pregroup Types
We redeﬁne pregroup types as a deductive system.
init
A A
Γ, A B l I
Γ BAl
Γ1 ABl Γ2 BΣl
l E
Γ1, Γ2 AΣl
A, Γ B r I
Γ Ar B
Γ1 Σr A Γ2 Ar B r E
Γ1, Γ2 Σr B
We change the way of looking at Ar BCl : it now behaves more like
an non-commutative linear functional type A B C rather than
a cartesian product or sum A⊥ ⊕ B ⊕ C⊥
Σl
and Σr
stand for a sequences of left and right adjoint types Cl
1...Cl
n,
Cr
1 ...Cr
n respectively.

Pregroup−λ↔
Grammars
Easy correspondence between the functional calculus and those
new pregroup types:
init
x : A x : A
Γ, x : A b : B l I
Γ b.x : BAl
Γ1 a : ABl Γ2 b : BΣl
l E
Γ1, Γ2 a(b) : AΣl
x : A, Γ b : B r I
Γ x.b : Ar B
Γ1 a : Σr A Γ2 b : Ar B r E
Γ1, Γ2 (a)b : Σr B

β-reduction
One of the major diﬀerences between that system and something
like a λ-calculus with 2 “directional”-λ’s is the β-reduction rules:
t|x (b|x1 ...|xn ) =β (t)[x := b] |x1 ...|xn
when t|x ∈ Termσφl and b|x1 ...|xn ∈ Termφπl
1...πl
n
This is the main reason why an untyped λ↔-calculus is not possible

β-reduction
t|x (b|x1 ...|xn ) =β (t)[x := b] |x1 ...|xn
1...πl
n
E.g.
the
ı(x).x : ¯nnl
green
green(y).y : nnl
ı(green(y)).y : ¯nnl
Compare this to
the
λx.ı(x) : NP/N
green
λy.green(y) : N/N z : N
green(z) : N
ı(green(z)) : NP
λz.ı(green(z)) : NP/N

β-reduction
t|x (b|x1 ...|xn ) =β (t)[x := b] |x1 ...|xn
1...πl
n
We still need to be able to pass abstracted predicates though
Someone
∃x.(x)y.y : s(πr
3s)l
runs
z.(z)run : πr
3s
∃x.(x) z.(z)run : s
∃x.(x)run : s

New Typing Structure
The pregroup types we will end up using in our analyses only form
a subset of the ones possible in the algebraic formulation. For
instance, there are
No more types of the form Ar , Arr B, Al B, AB, etc.
No more relations such as (AB)l ↔ Bl Al
Only one kind of contraction is possible: between
“disconnected” types
It is totally ﬁne: we will end up not needing those relations and
types at all, and our contraction rule is suﬃcient

Predictions:
give never needs its equivalent type i(op)l
give a star to Bob
ipl ol ¯nnl n p¯nl N
→ ipl ol ¯n p¯nl ¯n
→ ipl ol o p
→ ipl p → i
for does not need φ(o¯j)l either
John wants for Mary to live
N πr
3sφl φ¯jl ol N ¯jil i
→ π3 πr
3sφl φ¯jl ol o ¯j
→ sφl φ¯jl ¯j
→ sφl φ → s

More examples:
somebody
ssl π
?
πr
Edward likes pie
s
s
John
N
who
¯nr ¯nsl πl
Carl
π
I like ice cream
s
¯n
John
N
who
¯nr ¯n(πs)l
?
πs
¯n
John
N
whom
¯nr ¯nol lsl
I like ice cream
s
?
ol
¯n

Future
More in-depth analysis of semantic power of the new calculus,
i.e. what does an associative & compositional semantic layer
imply for semantic analysis?
Proving Church-Rosser Property
Category theoretical analysis of the new calculus

The End
Thank you for listening
Special thanks to: Alan Bale, Claudia Casadio, and Robert Seely
for their help in this project

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