We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations $\rho_1,\dots,\rho_k$, is the downward closed set Av$(\rho_1,\dots,\rho_k)$ consisting of all equivalence relations which do not contain any of $\rho_1,\dots,\rho_k$: (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?
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We consider the bimodal language, where the first modality is interpreted by a binary relation in the standard way, and the second is interpreted by the relation of inequality. It follows from Hughes (1990), that in this language, non-$k$-colorability of a graph is expressible for every finite $k$. We show that modal logics of classes of non-$k$-colorable graphs (directed or non-directed), and some of their extensions, are decidable.
Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), $m$ sets (resp.~$n$ points) are give $n$ points (resp.~$m$ sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight $O(\log n)$-competitive algorithm. In the same setting for hitting set, we provide a tight $O(\log N)$-competitive algorithm, assuming that all points have integral coordinates in $[0,N)^{2}$. No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems). For both dynamic set cover and hitting set with $d$-dimensional hyperrectangles, we obtain $(\log m)^{O(d)}$-approximation algorithms with $(\log m)^{O(d)}$ worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an \emph{extended quad-tree }approach and a \emph{frequency reduction} technique that reduces geometric set cover instances to instances of general set cover with bounded frequency.
With the increasing complexity of software permeating critical domains such as autonomous driving, new challenges are emerging in the ways the engineering of these systems needs to be rethought. Autonomous driving is expected to continue gradually overtaking all critical driving functions, which is adding to the complexity of the certification of autonomous driving systems. As a response, certification authorities have already started introducing strategies for the certification of autonomous vehicles and their software. But even with these new approaches, the certification procedures are not fully catching up with the dynamism and unpredictability of future autonomous systems, and thus may not necessarily guarantee compliance with all requirements imposed on these systems. In this paper, we identified a number of issues with the proposed certification strategies, which may impact the systems substantially. For instance, we emphasize the lack of adequate reflection on software changes occurring in constantly changing systems, or low support for systems' cooperation needed for the management of coordinated moves. Other shortcomings concern the narrow focus of the awarded certification by neglecting aspects such as the ethical behavior of autonomous software systems. The contribution of this paper is threefold. First, we discuss the motivation for the need to modify the current certification processes for autonomous driving systems. Second, we analyze current international standards used in the certification processes towards requirements derived from the requirements laid on dynamic software ecosystems and autonomous systems themselves. Third, we outline a concept for incorporating the missing parts into the certification procedure.
Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of convex bodies whose union covers $K$ such that a constant factor expansion of each body lies within an $\epsilon$ expansion of $K$. Coverings have been employed in many applications, such as approximations for diameter, width, and $\epsilon$-kernels of point sets, approximate nearest neighbor searching, polytope approximations, and approximations to the Closest Vector Problem (CVP). It is known how to construct coverings of size $n^{O(n)} / \epsilon^{(n-1)/2}$ for general convex bodies in $\textbf{R}^n$. In special cases, such as when the convex body is the $\ell_p$ unit ball, this bound has been improved to $2^{O(n)} / \epsilon^{(n-1)/2}$. This raises the question of whether such a bound generally holds. In this paper we answer the question in the affirmative. We demonstrate the power and versatility of our coverings by applying them to the problem of approximating a convex body by a polytope, under the Banach-Mazur metric. Given a well-centered convex body $K$ and an approximation parameter $\epsilon> 0$, we show that there exists a polytope $P$ consisting of $2^{O(n)} / \epsilon^{(n-1)/2}$ vertices (facets) such that $K \subset P \subset K(1+\epsilon)$. This bound is optimal in the worst case up to factors of $2^{O(n)}$. As an additional consequence, we obtain the fastest $(1+\epsilon)$-approximate CVP algorithm that works in any norm, with a running time of $2^{O(n)} / \epsilon ^{(n-1)/2}$ up to polynomial factors in the input size, and we obtain the fastest $(1+\epsilon)$-approximation algorithm for integer programming. We also present a framework for constructing coverings of optimal size for any convex body (up to factors of $2^{O(n)}$).
Knowledge graph embedding (KGE) is a increasingly popular technique that aims to represent entities and relations of knowledge graphs into low-dimensional semantic spaces for a wide spectrum of applications such as link prediction, knowledge reasoning and knowledge completion. In this paper, we provide a systematic review of existing KGE techniques based on representation spaces. Particularly, we build a fine-grained classification to categorise the models based on three mathematical perspectives of the representation spaces: (1) Algebraic perspective, (2) Geometric perspective, and (3) Analytical perspective. We introduce the rigorous definitions of fundamental mathematical spaces before diving into KGE models and their mathematical properties. We further discuss different KGE methods over the three categories, as well as summarise how spatial advantages work over different embedding needs. By collating the experimental results from downstream tasks, we also explore the advantages of mathematical space in different scenarios and the reasons behind them. We further state some promising research directions from a representation space perspective, with which we hope to inspire researchers to design their KGE models as well as their related applications with more consideration of their mathematical space properties.
When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.
This paper describes a general framework for learning Higher-Order Network Embeddings (HONE) from graph data based on network motifs. The HONE framework is highly expressive and flexible with many interchangeable components. The experimental results demonstrate the effectiveness of learning higher-order network representations. In all cases, HONE outperforms recent embedding methods that are unable to capture higher-order structures with a mean relative gain in AUC of $19\%$ (and up to $75\%$ gain) across a wide variety of networks and embedding methods.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.
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