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We explore a modification of QCD sum rules where, instead of Borel transforms of current correlators, one considers the correlators in coordinate space as functions of Euclidean time. Taking the nucleon channel as an example, we derive such Euclidean time sum rules and compare them with the traditional Borel sum rules. We show that a rough estimate of nucleon mass and residue is also possible working in coordinate space, but such sum rules are much more affected by the uncertainties in power corrections and continuum contribution than the Borel ones: the fiducial interval is practically absent.

Characteristic rules have been advocated for their ability to improve interpretability over discriminative rules within the area of rule learning. However, the former type of rule has not yet been used by techniques for explaining predictions. A novel explanation technique, called CEGA (Characteristic Explanatory General Association rules), is proposed, which employs association rule mining to aggregate multiple explanations generated by any standard local explanation technique into a set of characteristic rules. An empirical investigation is presented, in which CEGA is compared to two state-of-the-art methods, Anchors and GLocalX, for producing local and aggregated explanations in the form of discriminative rules. The results suggest that the proposed approach provides a better trade-off between fidelity and complexity compared to the two state-of-the-art approaches; CEGA and Anchors significantly outperform GLocalX with respect to fidelity, while CEGA and GLocalX significantly outperform Anchors with respect to the number of generated rules. The effect of changing the format of the explanations of CEGA to discriminative rules and using LIME and SHAP as local explanation technique

People are commonly interested in predicting a statistical property of a random event such as mean and variance. Proper scoring rules assess the quality of predictions and require that the expected score gets uniquely maximized at the precise prediction, in which case we call the score directly elicits the property. Previous research work has widely studied the existence and the characterization of proper scoring rules for different properties, but little literature discusses the choice of proper scoring rules for applications at hand. In this paper, we explore a novel task, the indirect elicitation of properties with parametric assumptions, where the target property is a function of several directly-elicitable sub-properties and the total score is a weighted sum of proper scoring rules for each sub-property. Because of the restriction to a parametric model class, different settings for the weights lead to different constrained optimal solutions. Our goal is to figure out how the choice of weights affects the estimation of the target property and which choice is the best. We start it with simulation studies and observe an interesting pattern: in most cases, the optimal estimation o

The goal of inductive logic programming (ILP) is to find a set of logical rules that generalises training examples and background knowledge. We introduce an ILP approach that identifies pointless rules. A rule is pointless if it contains a redundant literal or cannot discriminate against negative examples. We show that ignoring pointless rules allows an ILP system to soundly prune the hypothesis space. Our experiments on multiple domains, including visual reasoning and game playing, show that our approach can reduce learning times by 99% whilst maintaining predictive accuracies.

A major challenge in inductive logic programming is learning big rules. To address this challenge, we introduce an approach where we join small rules to learn big rules. We implement our approach in a constraint-driven system and use constraint solvers to efficiently join rules. Our experiments on many domains, including game playing and drug design, show that our approach can (i) learn rules with more than 100 literals, and (ii) drastically outperform existing approaches in terms of predictive accuracies.

We derive a complete set of Feynman rules in the general two-Higgs doublet model effective field theory where the effects of additional new physics are parametrized by operators up to mass dimension-six. We calculate the physical Higgs spectrum, contributions to the couplings and masses of electroweak gauge bosons and fermions, and all contact interactions arising from dimension-six operators. We also present results in specific limits and types, which include the $CP$-conserving limit, alignment limit, and the four types of two-Higgs doublet models: type-I, -II, -X, and -Y. We discuss the differences between the two-Higgs doublet model effective field theory and the renormalizable two-Higgs doublet model or the standard model effective field theory. We create a FeynRules model package for calculating all Feynman rules in the general effective field theory and its specific limits and types.

A framework is presented to compute approximations of an integral $I(f)=\displaystyle \int_a^b f(x) dx$ from a pair of companion rules and its associate rule. We show that an associate rule is a weighted mean of two companion rules. In particular, the trapezoidal (T) and Simpson (S) rules are weighted means of the companion pairs (L,R) and (T,M) respectively, with L the left rectangle, R the right rectangle and M the midpoint rules. As L,R,T and M reproduce exactly the number $π=\displaystyle \int_0^π2\, \sin^2(x) dx$, we named them the four "{beautiful}" \ rules. For this example the geometrical interpretation of the rules suggest possible applications of the transcendental number $π$ in architectural design, justifying the attribute beautiful given to the mentioned rules. As a complement we consider other appropriate integrand functions $f$, applying composite rules in order to obtain good approximations of $π$, as shown in the worked numerical examples.

In many real-world applications, sequential rule mining (SRM) can provide prediction and recommendation functions for a variety of services. It is an important technique of pattern mining to discover all valuable rules that belong to high-frequency and high-confidence sequential rules. Although several algorithms of SRM are proposed to solve various practical problems, there are no studies on target sequential rules. Targeted sequential rule mining aims at mining the interesting sequential rules that users focus on, thus avoiding the generation of other invalid and unnecessary rules. This approach can further improve the efficiency of users in analyzing rules and reduce the consumption of data resources. In this paper, we provide the relevant definitions of target sequential rule and formulate the problem of targeted sequential rule mining. Furthermore, we propose an efficient algorithm, called targeted sequential rule mining (TaSRM). Several pruning strategies and an optimization are introduced to improve the efficiency of TaSRM. Finally, a large number of experiments are conducted on different benchmarks, and we analyze the results in terms of their running time, memory consumpti

RRULES is presented as an improvement and optimization over RULES, a simple inductive learning algorithm for extracting IF-THEN rules from a set of training examples. RRULES optimizes the algorithm by implementing a more effective mechanism to detect irrelevant rules, at the same time that checks the stopping conditions more often. This results in a more compact rule set containing more general rules which prevent overfitting the training set and obtain a higher test accuracy. Moreover, the results show that RRULES outperforms the original algorithm by reducing the coverage rate up to a factor of 7 while running twice or three times faster consistently over several datasets.

Nanson's and Baldwin's voting rules select a winner by successively eliminating candidates with low Borda scores. We show that these rules have a number of desirable computational properties. In particular, with unweighted votes, it is NP-hard to manipulate either rule with one manipulator, whilst with weighted votes, it is NP-hard to manipulate either rule with a small number of candidates and a coalition of manipulators. As only a couple of other voting rules are known to be NP-hard to manipulate with a single manipulator, Nanson's and Baldwin's rules appear to be particularly resistant to manipulation from a theoretical perspective. We also propose a number of approximation methods for manipulating these two rules. Experiments demonstrate that both rules are often difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study.

The goal of this paper is to propose and study properties of multiwinner voting rules which can be consider as generalisations of single-winner scoring voting rules. We consider SNTV, Bloc, k-Borda, STV, and several variants of Chamberlin--Courant's and Monroe's rules and their approximations. We identify two broad natural classes of multiwinner score-based rules, and show that many of the existing rules can be captured by one or both of these approaches. We then formulate a number of desirable properties of multiwinner rules, and evaluate the rules we consider with respect to these properties.

Rules based approaches for data quality solutions often use business rules or integrity rules for data monitoring purpose. Integrity rules are constraints on data derived from business rules into a formal form in order to allow computerization. One of challenges of these approaches is rules discovering, which is usually manually made by business experts or system analysts based on experiences. In this paper, we present our rule-based approach for data quality analyzing, in which we discuss a comprehensive method for discovering dynamic integrity rules.

Let $V$ be a strongly rational vertex operator algebra, and let $g_1, g_2, g_3$ be three commuting finitely ordered automorphisms of $V$ such that $g_1g_2=g_3$ and $g_i^T=1$ for $i=1, 2, 3$ and $T\in \N$. Suppose $M^1$ is a $g_1$-twisted module. For any $n, m\in \frac{1}{T}\N$, we construct an $A_{g_3, n}(V)$-$A_{g_2, m}(V)$-bimodule $\mathcal{A}_{g_3, g_2, n, m}(M^1)$ associated to the quadruple $(M^1, g_1, g_2, g_3)$. Given an $A_{g_2, m}(V)$-module $U$, an admissible $g_3$-twisted module $\mathcal{M}(M^1, U)$ is constructed. For the quadruple $(V, 1, g, g)$ with some finitely ordered $g\in \text{Aut}(V)$, $\mathcal{A}_{g, g, n, m}(V)$ coincides with the $A_{g, n}(V)$-$A_{g, m}(V)$-bimodules $A_{g, n, m}(V)$ constructed by Dong-Jiang, and $\mathcal{M}(V, U)$ is the generalized Verma type admissible $g$-twisted module generated by $U$. When $U=M^2(m)$ is the $m$-th component of a $g_2$-twisted module $M^2$ for some $m\in\frac{1}{T}\N$, we show that the submodule of $\M(M^1, M^2(m))$ generated by the $m$-th component satisfies the universal property of the tensor product of $M^1$ and $M^2$. Using this result, we obtain a twisted version of Frenkel-Zhu-Li's fusion rules theorem.

We present the Feynman rules for leading-twist gauge-invariant quark and gluon operators with an arbitrary number of total derivatives and applicable to any order in perturbation theory. This generalizes previous results and constitutes a necessary ingredient in the computation of the matrix elements of the corresponding operators. The results are written in a form appropriate for implementation in a computer algebra system. To illustrate the latter we provide implementations in Mathematica and FORM, which are made available at https://github.com/vtsam/NKLO.

A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by the optimal rate of convergence for the function space. The focus is on algebraic rates of convergence $O(N^{-α+ε})$ for $α\ge 1$ and any $ε> 0$, where $α$ is the decay of a series representation of the integrand function. The dependence of the implied constant on the dimension can be controlled by weights which determine the influence of the different dimensions. Different types of weights are discussed. The construction of good lattice rules, and polynomial lattice rules, can be done using the same method for all $1 < p \le \infty$; but the case $p=1$ is special from the construction point of view. For $1 < p \le \infty$ the component-by-component construction and its fast algorithm for different weighted function spaces is then discussed.

Neural learning rules for principal component / subspace analysis (PCA / PSA) can be derived by maximizing an objective function (summed variance of the projection on the subspace axes) under an orthonormality constraint. For a subspace with a single axis, the optimization produces the principal eigenvector of the data covariance matrix. Hierarchical learning rules with deflation procedures can then be used to extract multiple eigenvectors. However, for a subspace with multiple axes, the optimization leads to PSA learning rules which only converge to axes spanning the principal subspace but not to the principal eigenvectors. A modified objective function with distinct weight factors had to be introduced produce PCA learning rules. Optimization of the objective function for multiple axes leads to symmetric learning rules which do not require deflation procedures. For the PCA case, the estimated principal eigenvectors are ordered (w.r.t. the corresponding eigenvalues) depending on the order of the weight factors. Here we introduce an alternative objective function where it is not necessary to introduce fixed weight factors; instead, the alternative objective function uses squared sum

In functional logic programs, rules are applicable independently of textual order, i.e., any rule can potentially be used to evaluate an expression. This is similar to logic languages and contrary to functional languages, e.g., Haskell enforces a strict sequential interpretation of rules. However, in some situations it is convenient to express alternatives by means of compact default rules. Although default rules are often used in functional programs, the non-deterministic nature of functional logic programs does not allow to directly transfer this concept from functional to functional logic languages in a meaningful way. In this paper we propose a new concept of default rules for Curry that supports a programming style similar to functional programming while preserving the core properties of functional logic programming, i.e., completeness, non-determinism, and logic-oriented use of functions. We discuss the basic concept and propose an implementation which exploits advanced features of functional logic languages.

While pre-trained language models (PLMs) are the go-to solution to tackle many natural language processing problems, they are still very limited in their ability to capture and to use common-sense knowledge. In fact, even if information is available in the form of approximate (soft) logical rules, it is not clear how to transfer it to a PLM in order to improve its performance for deductive reasoning tasks. Here, we aim to bridge this gap by teaching PLMs how to reason with soft Horn rules. We introduce a classification task where, given facts and soft rules, the PLM should return a prediction with a probability for a given hypothesis. We release the first dataset for this task, and we propose a revised loss function that enables the PLM to learn how to predict precise probabilities for the task. Our evaluation results show that the resulting fine-tuned models achieve very high performance, even on logical rules that were unseen at training. Moreover, we demonstrate that logical notions expressed by the rules are transferred to the fine-tuned model, yielding state-of-the-art results on external datasets.