The document discusses modelling Petri nets using Dialectica categories. It begins by introducing Petri nets and Dialectica categories. It then shows how special types of Petri nets called elementary nets can be modeled as a Dialectica category. Finally, it discusses how usual Petri nets can also be modeled as Dialectica categories by considering the pre and post-conditions of transitions as maps into a linear object rather than a binary relation. The modeling allows categorical tools and results to be transferred to the study of Petri nets.
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.
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 for the women+@DCS Sheffield University, UK
Title: Natural Language Inference for Humans
Valeria de Paiva,
Topos Institute, Berkeley, USA
Abstract: One hears much about the incredible results of recent neural nets methods in NLP. In particular much has been made of the results on the Natural Language Inference task using the huge new corpora SNLI, MultiNLI, SciTail, etc, constructed since 2015. Wanting to join in the fun, we decided to check the results on the corpus SICK (Sentences Involving Compositional Knowledge), which is two orders of magnitude smaller than SLNI and presumably easier to deal with.
We discovered that there were many results that did not agree with our intuitions. As a result, we have written so far five papers on the subject (with another one submitted to COLING2020).
I want to show you a potted summary of this work, to explain why we think this work is not near completion yet and how we're planning to tackle it.
This is work with Katerina Kalouli, Livy Real, Annebeth Buis and Martha Palmer. The papers are
Explaining Simple Natural Language Inference. Proceedings of the 13th Linguistic Annotation Workshop (LAW 2019), 01 August 2019. ACL 2019,
WordNet for “Easy” Textual Inferences. Proceedings of the Globalex Workshop, associated with LREC 2018
Graph Knowledge Representations for SICK. informal Proc of 5th Workshop on Natural Language and Computer Science, Oxford, UK, 08 July 2018
Textual Inference: getting logic from humans. Proc of the 12th International Conference on Computational Semantics (IWCS), 22 September 2017
Correcting Contradictions. Proc of Computing Natural Language Inference Workshop (CONLI 2017) @IWCS 2017
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 for the women+@DCS Sheffield University, UK
Title: Natural Language Inference for Humans
Valeria de Paiva,
Topos Institute, Berkeley, USA
Abstract: One hears much about the incredible results of recent neural nets methods in NLP. In particular much has been made of the results on the Natural Language Inference task using the huge new corpora SNLI, MultiNLI, SciTail, etc, constructed since 2015. Wanting to join in the fun, we decided to check the results on the corpus SICK (Sentences Involving Compositional Knowledge), which is two orders of magnitude smaller than SLNI and presumably easier to deal with.
We discovered that there were many results that did not agree with our intuitions. As a result, we have written so far five papers on the subject (with another one submitted to COLING2020).
I want to show you a potted summary of this work, to explain why we think this work is not near completion yet and how we're planning to tackle it.
This is work with Katerina Kalouli, Livy Real, Annebeth Buis and Martha Palmer. The papers are
Explaining Simple Natural Language Inference. Proceedings of the 13th Linguistic Annotation Workshop (LAW 2019), 01 August 2019. ACL 2019,
WordNet for “Easy” Textual Inferences. Proceedings of the Globalex Workshop, associated with LREC 2018
Graph Knowledge Representations for SICK. informal Proc of 5th Workshop on Natural Language and Computer Science, Oxford, UK, 08 July 2018
Textual Inference: getting logic from humans. Proc of the 12th International Conference on Computational Semantics (IWCS), 22 September 2017
Correcting Contradictions. Proc of Computing Natural Language Inference Workshop (CONLI 2017) @IWCS 2017
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
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2. 2/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Goals
Thank you, ACT Summer School organizers!
I want talk about the application of Dialectica categories to
modelling Petri Nets.
I start by describing Petri nets and Dialectica categories.
Then I show how to think of special Petri nets as a kind of
Dialectica category (following Brown and Gurr) and how to think
of usual Petri nets as dialectica categories.
Then we discuss if this is good modelling or not, and why
Valeria de Paiva ACT 2020
3. 3/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Outline
Petri nets
Dialectica categories
Elementary nets
Petri nets as double dialectica constructions
Time permitting I will try to compare with new work on Petri nets.
Valeria de Paiva ACT 2020
4. 4/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Using Categories
Category: a collection of objects and of morphisms, satisfying
obvious laws for composition of morphisms;
there is an identity for each object and morphisms of the correct
domain and co-domain compose, associatively.
Functors: the natural notion of morphism between categories
Natural transformations: the natural notion of morphisms between
functors
Constructors: products, sums, functions spaces, limits, duals....
Adjunctions: an abstract version of equality
Valeria de Paiva ACT 2020
5. 5/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Categorical models of Petri Nets
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 CT to CSci, etc
Valeria de Paiva ACT 2020
10. 10/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Why Petri Nets?
Modeling is hard. People seem to like Petri nets.
Concurrency is hard. People seem to like Petri nets
(non-determinism vs causal independence).
Huge number of types: colored, stochastic, higher-order, etc..
Huge number of papers/books/systems
Natural application of Linear Logic
Recent work on Baez/Master/Moeller and independently of
Lopes/Hauesler/Benevides
Valeria de Paiva ACT 2020
12. 12/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
What’s the problem?
A long-standing problem in the theory of Petri nets has
been the lack of any clear methodology for producing a
compositional theory of nets (and indeed the lack of even
a clear notion as to what a map between nets should be).
Brown and Gurr, 1990
Winskel suggests that Petri nets and other models of concurrency
can be profitably cast into an algebraic framework.
Many people followed this suggestion with several categorical
models of Petri nets proposed.
Our proposal is to use Dialectica morphisms, so let us check the
Dialectica construction for Linear Logic
Valeria de Paiva ACT 2020
13. 13/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Linear Logic
A proof theoretic logic described by Girard in 1986.
Basic idea: assumptions cannot be discarded or duplicated.
They must be used exactly once – just like dollar bills (except
when they’re marked by a modality !)
Other approaches to accounting for logical resources before.
Relevance Logic!
Great win of Linear Logic:
Account for resources when you want to, otherwise fall back
to traditional logic via translation A → B iff !A −◦ B
Valeria de Paiva ACT 2020
14. 14/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Linear Implication and (Multiplicative) Conjunction
Traditional implication: A, A → B B
A, A → B A ∧ B Re-use A
Linear implication: A, A −◦ B B
A, A −◦ B A ⊗ B Cannot re-use A
Traditional conjunction: A ∧ B A Discard B
Linear conjunction: A ⊗ B A Cannot discard B
Of course: !A !A⊗!A Re-use
!(A) ⊗ B B Discard
Valeria de Paiva ACT 2020
15. 15/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
The challenges of modeling Linear Logic
Traditional 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) and
they relate via adjunction
A ∧ B → C ⇐⇒ A → Hom(B, C) = CB
These are real products, have projections (A × B → A)
and diagonals (A → A × A) corresponding to deletion and
duplication of resources.
Not linear!!!
Need to use tensor products and internal homs.
Hard to define the “make-everything-usual”operator ”!”.
Valeria de Paiva ACT 2020
16. 16/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Dialectica Categories
The category Dial2(Sets) has as objects triples A = (U, X, α),
where U, X are sets and α is an ordinary relation between U and
X. (so either u and x are α related, α(u, x) = 1 or not.)
A map from A = (U, X, α) to B = (V , Y , β) is a pair of functions
(f , F), where f : U → V and F : X → Y such that
U
α
X
⇓ ∀u ∈ U, ∀y ∈ Y α(u, Fy) implies β(fu, y)
V
f
?
β
Y
6
F
or α(u, F(y)) ≤ β(f (u), y)
Valeria de Paiva ACT 2020
17. 17/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Dialectica Categories Constructions
On the board, lost slides.
Do internal-hom and tensor. and maybe adjunction.
Theorem: The category Dial2(Sets) is a sound model for
(intuitionistic and classical) Linear Logic.
And Petri Nets...
Valeria de Paiva ACT 2020
18. 18/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Dialectica Categories as (safe) Petri Nets
A category of safe nets SNet objects are of the form
(E, B, pre, post). (pre, post are relations E × B → 2) Morphisms
are pairs of functions (f , F) making the two diagrams below
commute.
E
pre
B
⇓
E
f
?
pre
B
6
F
E
post
B
⇑
E
f
?
post
B
6
F
where as before ∀e ∈ E, ∀b ∈ B pre(e, Fb ) implies pre (fe, b )
and also, the corresponding reversed implication for post:
∀e ∈ E, ∀b ∈ B post(e, Fb ) is implied by post (fe, b ).
Brown and Gurr 1990
Valeria de Paiva ACT 2020
19. 19/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Dialectica Categories as Petri Nets
Why is this good? Structurally safe nets, say Brown and Gurr.
Can construct refinement morphisms of nets.
Can lift all the categorical structure of Dial2(Sets) to a given
category of nets. Are these nets any good?
Similar to Winskel’s nets, constructed for different reasons.
Valeria de Paiva ACT 2020
20. 20/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Dialectica Categories as (general) Petri Nets
A category of safe nets SNet objects are of the form
(E, B, pre, post), BUT where pre, post are maps into N, a lineale.
Morphisms are pairs of functions (f , F) making the two diagrams
below commute.
E
pre
B
⇓
E
f
?
pre
B
6
F
E
post
B
⇑
E
f
?
post
B
6
F
where as before ∀e ∈ E, ∀b ∈ B pre(e, Fb ) implies pre (fe, b )
and also, the corresponding reversed implication for post
conditions:
∀e ∈ E, ∀b ∈ B post(e, Fb ) is implied by post (fe, b )
Valeria de Paiva ACT 2020
21. 21/21
Introduction
Petri Nets
Linear Logic
Dialetica cats
Dialectica and Petri
Conclusions
Dialectica morphisms correspond to simulations between
Petri nets, using markings.
CCS constructors can be simulated too.
Nielsen and Winskel have a collection of work on categories of
concurrency models, how much of that can be reproduced in
our setting?
Is this good modelling? Clearly people like Petri nets,
especially probabilistic ones. What is the theorem that we
want to prove?
Can this work shed some light on how to model probabilistic
theories using category theory?
at least it gives me the impulse to read about it!
Valeria de Paiva ACT 2020