Seminars

titre: Problèmes algorithmiques en théorie des groupes infinis
resumé:
Malgré le titre très général, il s'agira uniquement de problèmes concernant les sous-groupes de groupes infinis, et même juste les sous-groupes de groupes libres. Les résultats et méthodes que je présenterai sont issus de près de 40 ans de littérature et sont dûs à un grand nombre d'auteurs.

Je commencerai par poser le paysage, y compris pour ceux qui ne savent plus ce qu'est le groupe libre -- où l'on verra qu'on est, du point de vue algorithmique, dans une variante de la combinatoire des mots. Je présenterai ensuite l'outil central de la plupart des algorithmes efficaces sur les sous-groupes du groupe libre : la représentation de chaque sous-groupe finiment engendré par un graphe étiqueté et enraciné (disons : d'un automate :-)…) unique et facilement calculable à partir d'un ensemble de générateurs du sous-groupe considéré, qu'on appelle le graphe de Stallings.

Le jeu consiste ensuite à traduire les problèmes algorithmiques sur les sous-groupes en problèmes algorithmiques sur les graphes de Stallings, et à résoudre ces problèmes de la façon la plus efficace possible.

On considèrera notamment les problèmes suivants -- bon, juste le début de cette longue liste.
- Le problème du mot généralisé : étant donnés k+1 éléments du groupe libre (ce sont des mots), le dernier appartient-il au sous-groupe engendré par les k premiers ?
- Le problème de l'indice : étant donné un tuple d'éléments du groupe libre, le sous-groupe qu'ils engendrent est-il d'indice fini ?
- Le problème de la base : étant donné un tuple d'éléments du groupe libre, trouver le rang, et une base du sous-groupe qu'ils engendrent.
- Le problème de l'intersection : étant donnés deux tuples d'éléments du groupe libre, calculer l'intersection des sous-groupes qu'ils engendrent (ou calculer une base de cette intersection).
- Le problème de la conjugaison : étant donnés deux tuples d'éléments du groupe libre, engendrent-ils le même sous-groupe ? deux sous-groupes conjugués ?
- Et de nombreux autres problèmes (mots clés : minimalité de Whitehead, facteur libre, malnormalité, clôture par radical, clôture au sens de la topologie pro-p, etc…)


title: Algorithmic problems in the theory of infinite groups
abstract:
In spite of the very general title, we will talk only about problems on subgroups of infinite groups, and in fact, only on subgroups of free groups . The results and methods I will present have been obtained over the past 40 years and are due to many researchers.

I will start by setting the landscape, including for those who forgot what the free group is --- and we will see that we are dealing here, from the algorithmic point of view, with a variant of combinatorics on words. I will then present the tool that is central to most efficient algorithms on subgroups of free groups: the representation of each finitely generated subgroup by a labeled rooted graph (shall we say… an automaton?) which is unique and easily computable when a tuple of generators of the subgroup under consideration is given. This graph is called the Stallings graph.

The game consists, then, in translating algorithmic problems on subgroups into algorithmic problems on Stallings graphs, and in solving these problems as efficiently as possible.

We will discuss in particular the following problems (clearly: just the beginning of this long list).
- The generalized word problem: given k+1 elements of the free group (these are words), does the last one belong to the subgroup generated by the k first ones?
- The index problem: given a tuple of elements of the free group, does the subgroup they generate have finite index?
- The basis problem: given a tuple of elements of the free group, find the rank and a basis of the subgroup they generate.
- The intersection problem: given two tuples of elements of the free group, compute the intersection of the subgroups they generate (compute a basis of this intersection).
- The conjugacy problem: given two tuples of elements of the free group, are the subgroups they generate equal? conjugated?
- And many other problems (keywords: Whitehead minimality, free factors, malnormality, closure under radicals, closure in the sense of the pro-p topology, etc…)

Title: Theory and practice of learning-based compressed data structures

Presenter: Giorgio Vinciguerra

Abstract:
We revisit two fundamental and ubiquitous problems in data structure design:
predecessor search and rank/select primitives. We show that real data present a
peculiar kind of regularity based on geometric considerations. We name it
“approximate linearity”.
We thus expand the horizon of compressed data structures by presenting two
solutions for the problems above that discover, or “learn”, in a principled
algorithmic way, these approximate linearities. We provide a walkthrough of
these new theoretical achievements, also with a focus on open-source libraries
and their experimental improvements. We conclude by discussing the plethora of
research opportunities that these new learning-based approaches to data
structure design open up.

Zoom link: univ-lille-fr.zoom.us/j/95419000064

Title: Foundations of Languages for Interpretability.

Abstract:
The area of interpretability in Machine Learning aims for the design of algorithms that we humans can understand and trust. One of the fundamental questions of interpretability is: given a classifier M, and an input vector x, why did M classify x as M(x)? In order to approximate an answer to this "why" question, many concrete queries, metrics and scores have emerged as proxies, and their complexity has been studied over different classes of models. Many of these analyses are ad-hoc, but they tend to agree on the fact that these queries and scores are hard to compute over Neural Networks, but easy to compute over Decision Trees. It is thus natural to think of a more general approach, like a query language in which users could write an arbitrary number of different queries, and that would allow for a generalized study of the complexity of interpreting different ML models. Our work proposes foundations for such a language, tying to First Order Logic, as a way to have a clear understanding of its expressiveness and complexity. We manage to define a minimalistic structure over FO that allows expressing many natural interpretability queries over models, and we show that evaluating such queries can be done efficiently for Decision Trees, in data-complexity.

Zoom link: univ-lille-fr.zoom.us/j/95419000064

Titile: The BOLDR project
Abstract: I
n this presentation, I will give an account of the BOLDR project and
perspectives in the field of language integrated queries.

Several classes of solutions allow programming languages to express
queries: specific APIs such as JDBC, Object-Relational Mappings (ORMs)
such as Hibernate, and language-integrated query frameworks such as
Microsoft's LINQ. However, most of these solutions do not allow for
efficient cross-databases queries, and none allow the use of complex
application logic from the programming language in queries.

We study the design of a new language-integrated query
framework called BOLDR that allows the evaluation in databases of
queries written in general-purpose programming languages containing
application logic, and targeting several databases following different
data models. In this framework, application queries are translated to
an intermediate representation. Then, they are typed with a type
system extensible by databases in order to detect which database
language each subexpression should be translated to. This type system
also allows us to detect a class of errors before execution. Next,
they are rewritten in order to avoid query avalanches and make the
most out of database optimizations. Finally, queries are sent for
evaluation to the corresponding databases and the results are
converted back to the application. Our experiments show that the
techniques we implemented are applicable to real-world database
applications, successfully handling a variety of language-integrated
queries with good performances.

This talk will give an overview of what has been achieved so far (mainly
in the context of Julien Lopez' PhD Thesis) and will glimpse at preliminary
work that is being done in the context of a collaboration with Oracle Labs.

Speaker: Nofar Carmeli (nofar.carme.li/)

Zoom link: univ-lille-fr.zoom.us/j/95419000064

Title: The Complexity of Answering Unions of Conjunctive Queries.

Abstract:
We discuss the fine-grained complexity of enumerating the answers to a query over a relational database. With the ideal guarantees, linear time is required before the first answer to read the input and determine its existence, and then we need to print the answers one by one. Consequently, we wish to identify the queries that can be solved with linear preprocessing time and constant or logarithmic delay between answers. A known dichotomy classifies CQs into those that admit such enumeration and those that do not. The computationally expensive component of query answering is joining tables, which can be done efficiently if and only if the join query is acyclic. However, the join query usually does not appear in a vacuum; for example, it may be part of a larger query, or it may be applied to a database with dependencies. We inspect how the complexity changes in these settings and chart the borders of tractability within. In addition, we consider the task of enumerating query answers with a uniformly random order, and we propose to do so using an efficient random-access structure for representing the set of answers. We also prove conditional lower bounds showing that our algorithms capture all tractable queries in some cases. Among our results, we show that a union of tractable conjunctive queries may be intractable w.r.t. random access; on the other hand, a union of intractable conjunctive queries may be tractable w.r.t. enumeration.

Title: Extracting nested relational queries from implicit definitions

Abstract:
arxiv.org/pdf/2005.06503.pdf

In this talk, I will present results obtained jointly with Michael
Benedikt establishing a connection between the Nested Relational
Calculus (NRC) and sets implicitly definable using Δ₀ formulas.

Call a formula φ(I,O) an implicit definition of the relation O(x,...) in
terms of I(y,...) if O is functionally determined by I: for every I, O,
O', if both φ(I,O) and φ(I,O') hold, then we have O ≡ O'. When φ is
first-order and I and O are relations over base sorts, then Beth's
definability theorem states that there is a first-order formula
ψ(I,x,...) corresponding to O whenever φ(I,O) holds. Further, this
explicit definition ψ can be effectively be computed from a sequent
calculus proof witnessing that φ is functional. This allows for
practical use of implicit definitions in the context of database
programming, as there is a well-established link between fragments of
explicitly FO definable relations and relational calculi.

NRC is a conservative extension of relational calculi from database
theory with limited powerset types in addition to tupling and anonymous
base types. NRC expressions thus not only encompass flat relations over
primitive datatypes like SQL but also nested collections, while
remaining useful in practice.

We extend the above correspondence between first-order logic and flat
relational queries to NRC and implicit definitions using set-theoretical
Δ₀ formulas over (typed) nested collection. Our proof of the equivalence
goes through a notion of Δ₀-interpretation and a generalization of Beth
definability for multi-sorted structures. This proof is non-constructive
and thus does not yield any useful algorithm for converting implicit
definitions into NRC terms. Using an approach more closely related to
proof-theoretic interpolation, we give a constructive proof of the
result restricted to intuitionistic provability, i.e, when the input
functionality proof π of φ(I,O) is carried out in intuitionistic logic.
Further, if π is cut-free, this can be done efficiently. Whether or not
there exists a polynomial-time procedure working with classical proofs
of functionality is still an open problem.

I will focus on the effective result for the talk, and if time allows,
discuss the difficulties with extending it to classical logic. I will
not assume any background in either database or model theory.

Title: The Complexity of Counting Problems over Incomplete Databases

Abstract: In this presentation, I will talk about various counting problems that naturally
arise in the context of query evaluation over incomplete databases. Incomplete
databases are relational databases that can contain unknown values in the form
of labeled nulls. We will assume that the domains of these unknown values are
finite and, for a Boolean query $q$, we will consider the following two
problems: given as input an incomplete database $D$, (a) return the number of
completions of $D$ that satisfy $q$; or (b) return or the number of valuations
of the nulls of $D$ yielding a completion that satisfies $q$.


We will study the computational complexity of these problems when $q$ is a
self-join--free conjunctive query, and study the impact on the complexity of
the following two restrictions: (1) every null occurs at most once in $D$ (what
is called *Codd tables*); and (2) the domain of each null is the same. Roughly
speaking, we will see that counting completions is much harder than counting
valuations, and that both (1) and (2) can reduce the complexity of our
problems.

I will also talk about the approximability of these problems and prove that,
while counting valuations can efficiently be approximated, in most cases
counting completions cannot.

On our way, we will encounter the counting complexity classes #P, Span-P and
Span-L.

The presentation will be based on joint work with Marcelo Arenas and Pablo
Barcelo; see arxiv.org/abs/1912.11064