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A central quest in explainable AI relates to understanding the decisions made by (learned) classifiers. There are three dimensions of this understanding that have been receiving significant attention in recent years. The first dimension relates to characterizing conditions on instances that are necessary and sufficient for decisions, therefore providing abstractions of instances that can be viewed as the "reasons behind decisions." The next dimension relates to characterizing minimal conditions that are sufficient for a decision, therefore identifying maximal aspects of the instance that are irrelevant to the decision. The last dimension relates to characterizing minimal conditions that are necessary for a decision, therefore identifying minimal perturbations to the instance that yield alternate decisions. We discuss in this tutorial a comprehensive, semantical and computational theory of explainability along these dimensions which is based on some recent developments in symbolic logic. The tutorial will also discuss how this theory is particularly applicable to non-symbolic classifiers such as those based on Bayesian networks, decision trees, random forests and some types of neural networks.

It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains multiple notions of mode and various examples of pathological measures that have no mode in any sense. Since the masses of balls induce natural orderings on the points of $X$, this article aims to shed light on some of the problems in non-parametric MAP estimation by taking an order-theoretic perspective, which appears to be a new one in the inverse problems community. This point of view opens up attractive proof strategies based upon the Cantor and Kuratowski intersection theorems; it also reveals that many of the pathologies arise from the distinction between greatest and maximal elements of an order, and from the existence of incomparable elements of $X$, which we show can be dense in $X$, even for an absolutely continuous measure on $X = \mathbb{R}$.

We investigate logics and classes of problems below Fagin's existential second-order logic (ESO) and above Feder and Vardi's logic for constraint satisfaction problems (CSP), the so called monotone monadic SNP without inequality (MMSNP). It is known that MMSNP has a dichotomy between P and NP-complete but that the removal of any of these three restrictions imposed on SNP yields a logic that is Ptime equivalent to ESO: so by Ladner's theorem we have three stronger sibling logics that are nondichotomic above MMSNP. In this paper, we explore the area between these four logics, mostly by considering guarded extensions of MMSNP, with the ultimate goal being to obtain logics above MMSNP that exhibit such a dichotomy.

Object Storage Systems (OSS) inside a cloud promise scalability, durability, availability, and concurrency. However, open-source OSS does not have a specific approach to letting users and administrators search based on the data, which is contained inside the object storage, without involving the entire cloud infrastructure. Therefore, in this paper, we propose Sherlock, a novel Content-Based Searching (CoBS) architecture to extract additional information from images and documents. Here, we store the additional information in an Elasticsearch-enabled database, which helps us to search for our desired data based on its contents. This approach works in two sequential stages. First, the data will be uploaded to a classifier that will determine the data type and send it to the specific model for the data. Here, the images that are being uploaded are sent to our trained model for object detection, and the documents are sent for keyword extraction. Next, the extracted information is sent to Elasticsearch, which enables searching based on the contents. Because the precision of the models is so fundamental to the search's correctness, we train our models with comprehensive datasets (Microsoft COCO Dataset for multimedia data and SemEval2017 Dataset for document data). Furthermore, we put our designed architecture to the test with a real-world implementation of an open-source OSS called OpenStack Swift. We upload images into the dataset of our implementation in various segments to find out the efficacy of our proposed model in real-life Swift object storage.

Harnessing logical reasoning ability is a comprehensive natural language understanding endeavor. With the release of Generative Pretrained Transformer 4 (GPT-4), highlighted as "advanced" at reasoning tasks, we are eager to learn the GPT-4 performance on various logical reasoning tasks. This report analyses multiple logical reasoning datasets, with popular benchmarks like LogiQA and ReClor, and newly-released datasets like AR-LSAT. We test the multi-choice reading comprehension and natural language inference tasks with benchmarks requiring logical reasoning. We further construct a logical reasoning out-of-distribution dataset to investigate the robustness of ChatGPT and GPT-4. We also make a performance comparison between ChatGPT and GPT-4. Experiment results show that ChatGPT performs significantly better than the RoBERTa fine-tuning method on most logical reasoning benchmarks. With early access to the GPT-4 API we are able to conduct intense experiments on the GPT-4 model. The results show GPT-4 yields even higher performance on most logical reasoning datasets. Among benchmarks, ChatGPT and GPT-4 do relatively well on well-known datasets like LogiQA and ReClor. However, the performance drops significantly when handling newly released and out-of-distribution datasets. Logical reasoning remains challenging for ChatGPT and GPT-4, especially on out-of-distribution and natural language inference datasets. We release the prompt-style logical reasoning datasets as a benchmark suite and name it LogiEval.

Modern shock-capturing schemes often suffer from numerical shock anomalies if the flow field contains strong shocks, which may limit their further application in hypersonic flow computations. In the current study, we devote our efforts to exploring the primary numerical characteristics and the underlying mechanism of shock instability for second-order finite-volume schemes. To this end, we, for the first time, develop the matrix stability analysis method for the finite-volume MUSCL approach. Such a linearized analysis method allows to investigate the shock instability problem of the finite-volume shock-capturing schemes in a quantitative and efficient manner. Results of the stability analysis demonstrate that the shock stability of second-order scheme is strongly related to the Riemann solver, Mach number, limiter function, numerical shock structure, and computational grid. Unique stability characteristics associated with these factors for second-order methods are revealed quantitatively with the established method. Source location of instability is also clarified by the matrix stability analysis method. Results show that the shock instability originates from the numerical shock structure. Such conclusions pave the way to better understand the shock instability problem and may shed new light on developing more reliable shock-capturing methods for compressible flows with high Mach number.

Natural language generation from structured data mainly focuses on surface-level descriptions, suffering from uncontrollable content selection and low fidelity. Previous works leverage logical forms to facilitate logical knowledge-conditioned text generation. Though achieving remarkable progress, they are data-hungry, which makes the adoption for real-world applications challenging with limited data. To this end, this paper proposes a unified framework for logical knowledge-conditioned text generation in the few-shot setting. With only a few seeds logical forms (e.g., 20/100 shot), our approach leverages self-training and samples pseudo logical forms based on content and structure consistency. Experimental results demonstrate that our approach can obtain better few-shot performance than baselines.

It is not difficult to think of applications that can be modelled as graph problems in which placing some facility or commodity at a vertex has some positive or negative effect on the values of all the vertices out to some distance, and we want to be able to calculate quickly the cumulative effect on any vertex's value at any time or the list of the most beneficial or most detrimential effects on a vertex. In this paper we show how, given an edge-weighted graph with constant-size separators, we can support the following operations on it in time polylogarithmic in the number of vertices and the number of facilities placed on the vertices, where distances between vertices are measured with respect to the edge weights: Add (v, f, w, d) places a facility of weight w and with effect radius d onto vertex v. Remove (v, f) removes a facility f previously placed on v using Add from v. Sum (v) or Sum (v, d) returns the total weight of all facilities affecting v or, with a distance parameter d, the total weight of all facilities whose effect region intersects the ``circle'' with radius d around v. Top (v, k) or Top (v, k, d) returns the k facilities of greatest weight that affect v or, with a distance parameter d, whose effect region intersects the ``circle'' with radius d around v. The weights of the facilities and the operation that Sum uses to ``sum'' them must form a semigroup. For Top queries, the weights must be drawn from a total order.

A triangulation of a polytope into simplices is refined recursively. In every refinement round, some simplices which have been marked by an external algorithm are bisected and some others around also must be bisected to retain regularity of the triangulation. The ratio of the total number of marked simplices and the total number of bisected simplices is bounded from above. Binev, Dahmen and DeVore proved under a certain initial condition a bound that depends only on the initial triangulation. This thesis proposes a new way to obtain a better bound in any dimension. Furthermore, the result is proven for a weaker initial condition, invented by Alk\"amper, Gaspoz and Kl\"ofkorn, who also found an algorithm to realise this condition for any regular initial triangulation. Supposably, it is the first proof for a Binev-Dahmen-DeVore theorem in any dimension with always practically realiseable initial conditions without an initial refinement. Additionally, the initialisation refinement proposed by Kossaczk\'y and Stevenson is generalised, and the number of recursive bisections of one single simplex in one refinement round is bounded from above by twice the dimension, sharpening a result of Gallistl, Schedensack and Stevenson.

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.