This document provides an overview of modeling the Lambek Calculus using Dialectica Categories. It begins with introductions to the Lambek Calculus and Dialectica Categories. It then discusses how Dialectica Categories can be used to model proof theory categorically. Specifically, it presents the non-commutative Dialectica category DialM(Sets) which can model the non-commutative multiplicative fragment of the Lambek Calculus. It concludes by noting the advantages of this approach over previous work and areas for future work.
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
Benchmarking Linear Logic Proofs, Valeria de PaivaValeria de Paiva
Talk at the 1st Joint Meeting Brazil-France in Mathematics, https://impa.br/eventos-do-impa/eventos-2019/1st-joint-meeting-brazil-france-in-mathematics/
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
Benchmarking Linear Logic Proofs, Valeria de PaivaValeria de Paiva
Talk at the 1st Joint Meeting Brazil-France in Mathematics, https://impa.br/eventos-do-impa/eventos-2019/1st-joint-meeting-brazil-france-in-mathematics/
Dialectica Categories Surprising Application: mapping cardinal invariantsValeria de Paiva
Talk at 2nd Set Theory and General Topology Week in Salvador, Bahia, Brazil, March 2012
Abstract: Goethe famously said that "Mathematicians are like Frenchmen:
whatever you say to them they translate into their own language and forthwith it is something entirely different." True. Even more true of category theorists. Following this great tradition of appropriating other people's work, I want to tell you how I learned about "cardinalities of the continuum" from Blass and Morgan and da Silva
and how I want to rock their boat, just a little, in the direction of my kind of mathematics.
Or: Representing symmetries
This talk is a summary and a motivation why one should study representation theory and its categorification.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Slides: dtubbenhauer.com/talks.html
This is a friendly Lambda Calculus Introduction by Dustin Mulcahey. LISP has its syntactic roots in a formal system called the lambda calculus. After a brief discussion of formal systems and logic in general, Dustin will dive in to the lambda calculus and make enough constructions to convince you that it really is capable of expressing anything that is "computable". Dustin then talks about the simply typed lambda calculus and the Curry-Howard-Lambek correspondence, which asserts that programs and mathematical proofs are "the same thing".
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1. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Dialectica Categories for the Lambek Calculus
Valeria de Paiva
Nuance Communications, CA
(joint work with Harley Eades III, Augusta, GA)
April, 2017
Valeria de Paiva APA2017 – Seattle, WA
2. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
I want to talk to you about modeling the Lambek Calculus,
using Dialectica Categories.
(dedicated to Jim Lambek, 1922–2014)
Valeria de Paiva APA2017 – Seattle, WA
4. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Introduction
Lambek Calculus
Dialectica Categories
putting things together...
Valeria de Paiva APA2017 – Seattle, WA
5. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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
Valeria de Paiva APA2017 – Seattle, WA
6. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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)
Valeria de Paiva APA2017 – Seattle, WA
8. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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...
Valeria de Paiva APA2017 – Seattle, WA
9. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
Categorical Proof Theory
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
Valeria de Paiva APA2017 – Seattle, WA
10. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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
Valeria de Paiva APA2017 – Seattle, WA
14. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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 ”!”.
Valeria de Paiva APA2017 – Seattle, WA
15. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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.
Valeria de Paiva APA2017 – Seattle, WA
16. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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’.
Valeria de Paiva APA2017 – Seattle, WA
17. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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?
Valeria de Paiva APA2017 – Seattle, WA
18. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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.
Valeria de Paiva APA2017 – Seattle, WA
19. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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...
Valeria de Paiva APA2017 – Seattle, WA
21. Introduction
Lambek Calculus
Categorical Proof Theory
Dialectica Categories
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.
Valeria de Paiva APA2017 – Seattle, WA