In 2020, Dajani and Kalle investigated invariant measures and frequencies of digits of signed binary expansions arising from a parameterised family of piecewise linear interval maps of constant slope 2. Central to their study was a property called ‘matching,’ where the orbits of the left and right limits of discontinuity points agree after some finite number of steps. We obtain analogous results for a parameterised family of ‘symmetric golden maps’ of constant slope $\beta$, with $\beta$ the golden mean. Matching is again central to our methods, though the dynamics of the symmetric golden maps are more delicate than the binary case. We characterize the matching phenomenon in our setting, present explicit invariant measures and frequencies of digits of signed $\beta$-expansions, and—time permitting—show further implications for a family of piecewise linear maps which arise as jump transformations of the symmetric golden maps.

Joint with Karma Dajani.

Correlation clustering is a classic model for clustering with several applications in data mining and unsupervised machine learning. In this problem, we are given a complete graph where each edge is labeled + or -; and the goal is to find a partition of the vertex set so as to minimize the number of + edges across the parts of the partition plus the number of – edges within the parts of the partition.

We will first present a new result showing that a constant number of rounds of the Sherali-Adams hierarchy yields a strict improvement over the natural LP: we present the first better-than-two approximation for the problem, improving upon the previous approximation of 2.06 of Chawla, Makarychev, Schramm, Yaroslavtsev based on rounding the natural LP (which is known to have an integrality gap of 2). We will then review several recent ideas which have led to practical constant factor approximations to Correlation Clustering in various setups: distributed and parallel environments, differentially-private algorithms, dynamic algorithms, or sublinear time algorithms.

This is a combination of joint works with Euiwoong Lee, Alantha Newman, Silvio Lattanzi, Slobodan Mitrovic, Ashkan Norouzi-Fard, Nikos Parotsidis, Jakub Tarnawski, Chenglin Fan and Andreas Maggiori.

https://semflajolet.math.cnrs.fr/

Denotational semantic is a field which consists in studying computation without computing. That is, a program P of type int → int can be seen on a high level as a function from the integers to the integers.

After a quick review of the idea behind denotational semantic, we will show that the word “denotational” can in fact be replaced by the word “categorical”. A program will not be interpreted as a mere function, but rather as a morphism in some category. This enlargement of perspective allows us to consider a lot more interesting interpretations for programs: relations, games, probability kernels…

We will then cover the basic categorical ingredients that makes a category a good candidate for interpreting computation. That is, a category must handle tupling and curryfication. If the time allows for it, we will then quickly introduce how those categorical ideas lead to the development of linear logic.

This talk *will not* assume any prior knowledge in category theory nor in denotational semantic.

https://www.irill.org/events/2023/event20230202.html

Bien que le théorème de cohérence de MacLane pour les catégories monoïdales ait été prouvé initialement par une méthode proche de la réécriture, il présente un caractère topologique. C’est ce qui a poussé Kapranov à suggérer en 1993 une « preuve instantanée » de ce théorème basée sur l’existence d'une famille de polytopes qu’il nomme permutoassociaèdres. Dans la première partie, on formalisera cette idée, montrant que la cohérence de MacLane est une conséquence directe de la simple connexité des permutoassociaèdres. Cela suggère un lien combinatoire plus général entre cohérence n-catégorique et n-connexité de certains espaces, avatar « strict » de travaux infini-catégoriques récents de Shaul Barkan.

Dans la deuxième partie, on s'intéressera à l'instanciation du théorème général à la classe des “hypergraph polytopes” (aussi connus sous le nom de nestoèdres), dont les faces (et en particulier les sommets et les arêtes) sont décrits par des objets combinatoires appelés “constructs”. Les arêtes sont naturellement orientées à l'aide d'un ordre appelé ordre de Tamari généralisé, et lorsqu'on instancie encore davantage, à la classe des opéraèdres, on obtient un système de réécriture de termes convergent (sous-jacent au problème de cohérence des opérades catégorifiées de Dosen et Petric), ce qui permet dans ce cas d'obtenir la cohérence par des méthodes de réécriture (complétion de Squier). Les diagrammes obtenus sont légèrement différents de ceux de Dosen et Petric.

A context-free language is inherently ambiguous if any grammar recognizing it is ambiguous, i.e. there is a word that is generated in two different ways. Deciding the inherent ambiguity of a context-free language is a difficult problem, undecidable in general. The first examples of inherently ambiguous languages were discovered in the 1960s, using iteration techniques on derivation trees. They belonged to a particular subfamily of context-free languages, called bounded context-free languages.

    Although they made it possible to prove the inherent ambiguity of several languages, as for example the language L = a^n b^m c^p with n=m or m=p, iteration techniques are still very laborious to implement, are very specific to the studied language, and seem even sometimes unadapted. For instance, the relative simplicity of the proof of inherent ambiguity of L completely collapses by replacing the constraint "n=m or m=p" by "n≠m or m≠p". 
    In this talk, I will present two useful criteria based on generating series to prove easily the inherent ambiguity of some bounded context-free languages. These languages, which have a rational generating series, resisted both the classical iteration techniques developed in the 1960’s and the analytic methods introduced by Philippe Flajolet in 1987.

The performance and reliability of Cyber-Physical Systems are increasingly aided through the use of digital twins, which mirror the static and dynamic behaviour of a Cyber-Physical System (CPS) in software. Digital twins enable the development of self-adaptive CPSs which reconfigure their behaviour in response to novel environments. It is crucial that these self-adaptations are formally verified at runtime, to avoid expensive re-certification of the reconfigured CPS. In this talk, I discuss formally verified self-adaptation in a digital twinning system, by constructing a non-deterministic model which captures the uncertainties in the system behaviour after a self-adaptation. We use Signal Temporal Logic to specify the safety requirements the system must satisfy after reconfiguration and employ formal methods based on verified monitoring over Flow* flowpipes to check these properties at runtime. This gives us a framework to predictively detect and mitigate unsafe self-adaptations before they can lead to unsafe states in the physical system.

Informally, obfuscation aims to compile programmes into unintelligible ones while preserving functionality. As a first approximation, this can be seen as trying to convert a programme to a “black-box representation”, which cannot be used for anything beyond obtaining its input/output behaviour. Learning, on the other hand, aims to generate the representation of a function given oracle access to it. In this talk, we will investigate new links between the two fields.

No specialised knowledge of cryptography or computational complexity is required, and the talk is aimed at a general TCS audience. For completeness however, we provide below a more technical description of our results.

Most recent works on cryptographic obfuscation focus on the high-end regime of obfuscating general circuits while guaranteeing computational indistinguishability between functionally equivalent circuits. Motivated by the goals of simplicity and efficiency, we initiate a systematic study of “low-end” obfuscation, focusing on simpler representation models and information-theoretic notions of security. We obtain the following results. 1) Positive results via “white-box” learning. We present a general technique for obtaining perfect indistinguishability obfuscation from exact learning algorithms that are given restricted access to the representation of the input function. We demonstrate the usefulness of this approach by obtaining simple obfuscation for decision trees and multilinear read-k arithmetic formulas. 2) Negative results via PAC learning. A proper obfuscation scheme obfuscates programs from a class C by programs from the same class. Assuming the existence of one-way functions, we show that there is no proper indistinguishability obfuscation scheme for k-CNF formulas for any constant k ≥ 3; in fact, even obfuscating 3-CNF by k-CNF is impossible. This result applies even to computationally secure obfuscation, and makes an unexpected use of PAC learning in the context of negative results for obfuscation. 3) Separations. We study the relations between different information-theoretic notions of indistinguishability obfuscation, giving cryptographic evidence for separations between them.

This is joint work with Elette Boyle, Yuval Ishai, Robert Robere, and Gal Yehuda, which was presented at ITCS 2023.