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In recent years, there has been increasing interest in explanation methods for neural model predictions that offer precise formal guarantees. These include abductive (respectively, contrastive) methods, which aim to compute minimal subsets…
This paper is about reachability analysis in a restricted subclass of multi-pushdown automata. We assume that the control states of an automaton are partially ordered, and all transitions of an automaton go downwards with respect to the…
We show how up-to techniques for (bi-)similarity can be used in the setting of weighted systems. The problems we consider are language equivalence, language inclusion and the threshold problem (also known as universality problem) for…
We consider the computational power of silent transitions in one-way automata with storage. Specifically, we ask which storage mechanisms admit a transformation of a given automaton into one that accepts the same language and reads at least…
In many cases the convexity of the image of a linear map with range is $R^n$ is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of ``automatic convexity''. To make…
We study \emph{multiplicity equivalence} testing of automata over partially commutative monoids (pc monoids) and show efficient algorithms in special cases, exploiting the structure of the underlying non-commutation graph of the monoid.…
We investigate the complexity of the containment problem "Does $L(A)\subseteq L(B)$ hold?", where $B$ is an unambiguous register automaton and $A$ is an arbitrary register automaton. We prove that the problem is decidable and give upper…
Typos in the abstract have been corrected. Let $\rho_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/\pi^n)$. Here $\mathcal O$ is a ramified DVR with uniformiser $\pi$. If $\rho_n$…
The $\sigma$-ideal $(v^0)$ is associated with the Silver forcing, see \cite{bre}. Also, it constitutes the family of all completely doughnut null sets, see \cite{hal}. We introduce segments and $*$-segments topologies, to state some…
We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable…
The concept of causal abstraction got recently popularised to demystify the opaque decision-making processes of machine learning models; in short, a neural network can be abstracted as a higher-level algorithm if there exists a function…
In this survey paper we present recent results obtained by Khare, Wintenberger and the author that have led to a proof of Serre's conjecture, such as existence of compatible families, modular upper bounds for universal deformation rings and…
We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by…
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or…
Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim conjectured that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset partition of $G$, then…
We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…
Since the seminal work by Angluin and the introduction of the L*-algorithm, active learning of automata by membership and equivalence queries has been extensively studied to learn various extensions of automata. For weighted automata,…
We present a conjecture on the average number of Galois orbits of newforms when fixing the weight and varying the level. This conjecture implies, for instance, that the central L-values (resp. L-derivatives) are nonzero for 100% of even…
We consider the value 1 problem for probabilistic automata over finite words: it asks whether a given probabilistic automaton accepts words with probability arbitrarily close to 1. This problem is known to be undecidable. However, different…
Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This…