We are concerned with the micro-macro Parareal algorithm for the simulation of initial-value problems. In this algorithm, a coarse (fast) solver is applied sequentially over the time domain and a fine (time-consuming) solver is applied as a corrector in parallel over smaller chunks of the time interval. Moreover, the coarse solver acts on a reduced state variable, which is coupled with the fine state variable through appropriate coupling operators. We first provide a contribution to the convergence analysis of the micro-macro Parareal method for multiscale linear ordinary differential equations. Then, we extend a variant of the micro-macro Parareal algorithm for scalar stochastic differential equations (SDEs) to higher-dimensional SDEs.

We study the computational problem of rigorously describing the asymptotic behavior of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and we prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constraints and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non-computable.

We investigate natural variations of behaviourally correct learning and explanatory learning—two learning paradigms studied in algorithmic learning theory—that allow us to “learn” equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviourally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\boldsymbol {\Pi }^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

This paper studies the conjecture of Hirschfeldt, Miller, and Podzorov in [13] on the complexity of order-computable sets, where a set A is order-computable if there is a computable copy of the structure $(\mathbb {N}, <,A)$ in the language of linear orders together with a unary predicate. The class of order-computable sets forms a subclass of $\Delta ^{0}_{2}$ sets. Firstly, we study the complexity of computably enumerable (c.e.) order-computable sets and prove that the index set of c.e. order-computable sets is $\Sigma ^{0}_{4}$-complete. Secondly, as a corollary of the main result on c.e. order-computable sets, we obtain that the index set of general order computable sets is $\Sigma ^{0}_{4}$-complete within the index set of $\Delta ^{0}_{2}$ sets. Finally, we continue to study the complexity of more general $\Delta ^{0}_{2}$ sets and prove that the index set of $\Delta ^{0}_{2}$ sets is $\Pi ^{0}_{3}$-complete.

Edge AI is the fusion of edge computing and artificial intelligence (AI). It promises responsiveness, privacy preservation, and fault tolerance by moving parts of the AI workflow from centralized cloud data centers to geographically dispersed edge servers, which are located at the source of the data. The scale of edge AI can vary from simple data preprocessing tasks to the whole machine learning stack. However, most edge AI implementations so far are limited to urban areas, where the infrastructure is highly dependable. This work instead focuses on a class of applications involved in environmental monitoring in remote, rural areas such as forests and rivers. Such applications have additional challenges, including failure proneness and access to the electricity grid and communication networks. We propose neuromorphic computing as a promising solution to the energy, communication, and computation constraints in such scenarios and identify directions for future research in neuromorphic edge AI for rural environmental monitoring. Proposed directions are distributed model synchronization, edge-only learning, aerial networks, spiking neural networks, and sensor integration.

We consider the performance of Glauber dynamics for the random cluster model with real parameter $q\gt 1$ and temperature $\beta \gt 0$. Recent work by Helmuth, Jenssen, and Perkins detailed the ordered/disordered transition of the model on random $\Delta$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $\beta$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $\Delta$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $\beta$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties ‘within the phase’, which are then related to the evolution of the chain.

Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

We study the parameterized complexity of the problem to decide whether a given natural number n satisfies a given $\Delta _0$-formula $\varphi (x)$; the parameter is the size of $\varphi $. This parameterization focusses attention on instances where n is large compared to the size of $\varphi $. We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf {AC}^0$. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $I\Delta _0$ proves the MRDP theorem in a certain weak sense.

Novel prediction methods should always be compared to a baseline to determine their performance. Without this frame of reference, the performance score of a model is basically meaningless. What does it mean when a model achieves an $F_1$ of 0.8 on a test set? A proper baseline is, therefore, required to evaluate the ‘goodness’ of a performance score. Comparing results with the latest state-of-the-art model is usually insightful. However, being state-of-the-art is dynamic, as newer models are continuously developed. Contrary to an advanced model, it is also possible to use a simple dummy classifier. However, the latter model could be beaten too easily, making the comparison less valuable. Furthermore, most existing baselines are stochastic and need to be computed repeatedly to get a reliable expected performance, which could be computationally expensive. We present a universal baseline method for all binary classification models, named the Dutch Draw (DD). This approach weighs simple classifiers and determines the best classifier to use as a baseline. Theoretically, we derive the DD baseline for many commonly used evaluation measures and show that in most situations it reduces to (almost) always predicting either zero or one. Summarizing, the DD baseline is general, as it is applicable to any binary classification problem; simple, as it can be quickly determined without training or parameter tuning; and informative, as insightful conclusions can be drawn from the results. The DD baseline serves two purposes. First, it is a robust and universal baseline that enables comparisons across research papers. Second, it provides a sanity check during the prediction model’s development process. When a model does not outperform the DD baseline, it is a major warning sign.

We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the necessary and sufficient conditions for the exact recovery of the maximum likelihood estimation, which can give a sharp phase transition for the exact recovery even though the Gaussian perturbations are not identically distributed; see Section 7. Applications include the community detection on hypergraphs.

Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A. We show that for any n, $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K. It was already known that 2-randomness is definable in terms of K (and plain complexity C) as those reals which infinitely often have maximal complexity. We can use our characterization to show that n-randomness is definable purely in terms of K. To do this we extend a certain “limsup” formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta _2^0$ sets of minimal descriptions.

In this thesis, we study the complexity of some mathematical problems: in particular, those arising in computable analysis and algorithmic learning theory for algebraic structures. Our study is not limited to these two areas: indeed, in both cases, the results we obtain are tightly connected to ideas and tools coming from different areas of mathematical logic, including for example descriptive set theory and reverse mathematics.

After giving the necessary preliminaries, we first study the uniform computational strength of the Cantor–Bendixson theorem in the Weihrauch lattice. This work falls into the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. We concentrate on problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. As far as we know, this is the first systematic study of problems at the level of $\mathbf {\Pi }^1_1\mathsf {-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.

We continue considering problems related to (induced) subgraphs. We provide results on the (effective) Wadge complexity of sets of graphs, that are also used to determine the Weihrauch degree of certain decision problems. The decision problems we consider are defined for a fixed graph G, and they take as input a graph H, answering whether G is an (induced) subgraph of H: we also consider the opposite problem (i.e., answering whether H is an induced subgraph of G). We conclude this part on (induced) subgraphs considering the Weihrauch degree of “search problems.”These problems are defined for a fixed graph G, and they take as input a graph H such that G is an (induced) subgraph H: the output is a copy of G in H. In both cases, we highlight differences and analogies between the subgraph and the induced subgraph relation.

We then move our attention to algorithmic learning theory, and we present the framework we use to study the learnability of families of algebraic structures: here, given a countable family of pairwise nonisomorphic structures $\mathfrak {K}$, a learner receives larger and larger pieces of an arbitrary copy of a structure in $\mathfrak {K}$ and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. We say that $\mathfrak {K}$ is learnable if there exists a learner which eventually stabilizes to a correct guess. The framework was lacking a method for comparing the complexity of nonlearnable families, and so we propose a solution to this problem using tools coming from invariant descriptive set theory. To do so, we first prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on infinite binary sequences and then, replacing $E_0$ with Borel equivalence relations of higher complexity, we obtain a new hierarchy of learning problems. This leads to the notion of E-learnability, where a family of structures $\mathfrak {K}$ is E-learnable, for a Borel equivalence relation E, if there is a continuous reduction from the isomorphism relation associated with $\mathfrak {K}$ to E. It is then natural to ask how the notion of E-learnability interacts with “classical” learning paradigms.

Finally, we study the number of mind changes that a learner needs to learn a given family, both from a topological and a combinatorial point of view, and we consider how the complexity of a learner (in terms of Turing reducibility) affects the number of mind changes for learning a given family.

Abstract prepared by Vittorio Cipriani.

E-mail: vittorio.cipriani17@gmail.com.

URL: https://air.uniud.it/handle/11390/1252226.

Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.

We study a version of the Craig interpolation theorem formulated in the framework of the theory of institutions. This formulation proved crucial in the development of a number of key results concerning foundations of software specification and formal development. We investigate preservation of interpolation properties under institution extensions by new models and sentences. We point out that some interpolation properties remain stable under such extensions, even if quite arbitrary new models and sentences are permitted. We give complete characterisations of such situations for institution extensions by new models, by new sentences, as well as by new models and sentences, respectively.

Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $. We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n < 2^{-n}$ for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K-trivial, we construct such an $\alpha $ such that ${K(\alpha \restriction n)>n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial.

In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.

Cook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:

• • There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.

$\bigvee $

$\bigvee $

$\mathcal {N}\mathcal {P}$

$\mathcal {N}\mathcal {P}$

Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${\mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems).

A propositional version of the construction shows that at least one of the following three statements is true:

1. 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm).

2. 2. $E \not \subseteq P/poly$.

3. 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${\text {NP}}$ sets.

We study the problem of detecting the community structure from the generalized stochastic block model with two communities (G2-SBM). Based on analysis of the Stieljtes transform of the empirical spectral distribution, we prove a Baik–Ben Arous–Péché (BBP)-type transition for the largest eigenvalue of the G2-SBM. For specific models, such as a hidden community model and an unbalanced stochastic block model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.