Introduction
Lambek Calculus
Introduction
I want to talk to you about modeling the Lambek Calculus,
using Dialectica Categories.
(dedicated to Jim Lambek, 1922–2014)

Introduction
Lambek Calculus
Introduction
Lambek Calculus (1958, 1988, 1993, 2012)

Introduction
Lambek Calculus
Introduction
Lambek Calculus
putting things together...

Introduction
Lambek Calculus
What is the Lambek Calculus?
One of the several “type grammars”in use in Linguistics.
A long history: Ajdukiewicz [1935], Bar-Hillel [1953], Lambek
[1958, 1961], Ades-Steedman [1982], etc.
It provides a syntactic account of sentencehood.
Two classes of type grammars:
1. Combinatory Categorial Grammar: Szabolcsi [1992],
Steedman-Baldridge [2011], etc..
2. Categorial Type Logics: van Benthem, Morrill [1994], Moortgat
[1994], etc..
Combinators/Lambda-calculus distiction.
Both classes worked on nowadays

Introduction
Lambek Calculus
What is the Lambek Calculus?
Here a purely a logical system, like usual propositional logic, but
with no structural rules at all.
Recall the basic logic ‘equation’:
A → (B → C) ⇐⇒ A ∧ B → C ⇐⇒ B → (A → C)
Now make your conjunction non-commutative, so that
A ⊗ B = B ⊗ A
Then you end up with two kinds of ‘implication’ ( , ):
A → (B C) ⇐⇒ A ⊗ B → C ⇐⇒ B → (A C)

Introduction
Lambek Calculus
What is the Lambek Calculus? sequent calculus

Introduction
Lambek Calculus
Why Dialectica Categories?
For G¨odel (1958): a way to prove consistency of higher order
arithmetic
For Girard (1987): a way to show that Linear Logic had serious
pedigree
For Hyland (1987):
An intrinsic way modelling G¨odel’s Dialectica,
Proof theory in the abstract (Hyland, 2002)
Should produce a CCC, it wouldn’t.
For me: a Swiss army knife...

Introduction
Lambek Calculus
Types are formulae/objects in appropriate category,
Terms/programs are proofs/morphisms in the category,
Logical constructors are appropriate categorical constructions.
Most important: Reduction is proof normalization (Tait)
Outcome: Transfer results/tools from Logic to Categories to
Computing

Introduction
Lambek Calculus
Curry-Howard for Implication
Natural deduction rules for implication (without λ-terms)
A → B A
B
[A]
·
·
·
·
π
B
A → B
Natural deduction rules for implication (with λ-terms)
M : A → B N : A
M(N): B
[x : A]
·
·
·
·
π
M : B
λx.M : A → B
function application abstraction

Introduction
Lambek Calculus

Introduction
Lambek Calculus
The challenges of modeling Linear Logic
Traditional categorical modeling of intuitionistic logic:
formula A object A of appropriate category
A ∧ B A × B (real product)
A → B BA (set of functions from A to B)
But these are real products, so we have projections (A × B → A)
and diagonals (A → A × A) which correspond to deletion and
duplication of resources.
Not linear!!!
Need to use tensor products and internal homs in Category Theory.
Hard to deﬁne the “make-everything-as-usual”operator ”!”.

Introduction
Lambek Calculus
The simplest Dialectica Category
The Dialectica category Dial2(Sets) objects are triples, an object is
A = (U, X, R), where U and X are sets and R ⊆ U × X is a
set-theoretic relation. A morphism from A to B = (V , Y , S) is a
pair of functions f : U → V and F : Y → X such that
uRFy → fuSy.
Theorem 1: You just have to ﬁnd the right structure. . .
(de Paiva 1989) The category Dial2(Sets) has a symmetric mo-
noidal closed structure, and involution which makes it a model of
(exponential-free) multiplicative linear logic.
Theorem 2 (Hard part): You still want usual logic. . .
There is a comonad ! which models exponentials/modalities, hence
recovers Intuitionistic and Classical Logic.

Introduction
Lambek Calculus
Can we give some intuition for these categories?
Blass makes the case for thinking of problems in computational
complexity. Intuitively an object of Dial2(Sets)
(U, X, R)
can be seen as representing a problem.
The elements of U are instances of the problem, while the
elements of X are possible answers to the problem instances.
The relation R says whether the answer is correct for that instance
of the problem or not.
The morphisms between these problems have two components:
while f maps instances of a problem to instances of another, F
maps solutions ‘backwards’.

Introduction
Lambek Calculus
What’s diﬀerent for the Lambek calculus?
Need to have a non-commutative tensor ⊗.
Need to have two (left and right) implications.
Can we have these disturbing minimally the (admitedly)
complicated structures?

Introduction
Lambek Calculus
The non-commutative Dialectica Category
(de Paiva 1992, Amsterdam Colloquium) Category
DialM(Sets), objects are A = (U, X, R), where U and X are sets
and U × X → M is a M-valued relation. A morphism from A to
B = (V , Y , S) is a pair of functions f : U → V and F : Y → X
such that R(u, Fy) ≤M S(fu, y).
Theorem 3: have the right strux. . .
The category DialM(Sets) has a non-symmetric monoidal closed
structure, hence it is a model of (exponential-free) non-commutative
multiplicative linear logic.
Theorem 4 (Hard part): You still want usual logic. . .
There is a comonad ! which models exponentials/modalities, and a
comonad κ (Yetter) that brings back commutativity. Putting the
two together we recover Intuitionistic and Classical Logic.

Introduction
Lambek Calculus
Conclusions
Introduced you to the Lambek calculus, as a relative of Linear
Logic
Introduced you to Dialectica categories
(there’s much more to say...)
Described one example of Dialectica categories DialM(Sets),
a non-commutative case. Should’ve shown you the modalities that
make it work.
Advantages over previous work:
1. Proved syntax works as expected.
2. Working on implementation in Agda.
Hinted at why one might want to use this system for PLs.
To do: comparison with pregroups...

Introduction
Lambek Calculus
Thank you!

Introduction
Lambek Calculus
Some References
J. Lambek, The Mathematics of Sentence Structure. American
Mathematical Monthly, pages 154–170, 1958.
de Paiva, The Dialectica Categories, Cambridge University DPMMS PhD
thesis, Technical Report 213, 1991.
de Paiva, A Dialectica Model of the Lambek Calculus, In Proc Eighth
Amsterdam Colloquium, December 17–20, 1991. Proceedings edited by
Martin Stokhof and Paul Dekker, Institute for Logic, Language and
Computation, University of Amsterdam, 1992, pp. 445-462.
Hyland, J. Martin E. Proof theory in the abstract, Annals of pure and
applied logic 114.1-3, 2002, pp. 43-78.

Dialectica Categories for the Lambek Calculus

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Dialectica Categories for the Lambek Calculus