Set Membership Research Articles

Zero-knowledge proofs for set membership: efficient, succinct, modular

We consider the problem of proving in zero knowledge that an element of a public set satisfies a given property without disclosing the element, i.e., for some u, “u in S and P(u) holds”. This problem arises in many applications (anonymous cryptocurrencies, credentials or whitelists) where, for privacy or anonymity reasons, it is crucial to hide certain data while ensuring properties of such data. We design new modular and efficient constructions for this problem through new commit-and-prove zero-knowledge systems for set membership, i.e. schemes proving u in S for a value u that is in a public commitment c_u. We also extend our results to support non-membership proofs, i.e. proving u notin S. Being commit-and-prove, our solutions can act as plug-and-play modules in statements of the form “u in S and P(u) holds” by combining our set (non-)membership systems with any other commit-and-prove scheme for P(u). Also, they work with Pedersen commitments over prime order groups which makes them compatible with popular systems such as Bulletproofs or Groth16. We implemented our schemes as a software library, and tested experimentally their performance. Compared to previous work that achieves similar properties—the clever techniques combining zkSNARKs and Merkle Trees in Zcash—our solutions offer more flexibility, shorter public parameters and 3.7 times –30times faster proving time for a set of size 2^{64}.

Fuzzy Logic

Fuzzy logic is a mathematical framework for reasoning about ambiguous or inaccurate information. It is founded on the idea that truth can be stated as a degree of membership in a fuzzy set rather than as a binary value of true or untrue. Fuzzy logic is used in control systems, artificial intelligence, and decision-making. This paper defines fuzzy logic and discusses its key concepts, mathematical underpinnings, and applications. We look at the benefits and drawbacks.

Trajectory planning and control for autonomous vehicles: a “fast” data-aided NMPC approach

A huge research effort is being spent worldwide by automotive companies and academic institutions for developing vehicles with high levels of autonomy, ranging from advanced driving-assisted systems to fully automated vehicles. Nonlinear Model Predictive Control (NMPC) has the potential to become a key technology in this context, thanks to its capability to deal with linear and nonlinear systems, manage physical constraints and satisfy multi-objective performance criteria. However, NMPC is based on the on-line solution of a nonconvex optimization problem and this operation may require a high computational cost, compromising its real-time implementation. In this paper, a “fast” data-aided NMPC approach is developed, aimed at trajectory planning and control for autonomous vehicles. In particular, a Set Membership approximation method is used to derive from data tight bounds on the optimal NMPC control law. These bounds are used to restrict the search domain of the underlying NMPC optimization process, allowing a significant reduction of the computation time. The proposed NMPC trajectory planning and control approach is tested in simulation and compared with other state-of-the-art methods, considering different road scenarios.

Set membership filtering for discrete saturated time‐varying networked systems with incomplete measurements under round‐robin protocol

AbstractIn this paper, the set membership filtering problem is investigated for a class of discrete networked systems with time delay and state saturation under the round‐robin (RR) protocol, where the system noise is unknown but bounded. In order to reduce the communication burden, the RR protocol is used to schedule the measurement information of the system. In addition, the incompleteness of the measurement information is considered. A set membership filter is established based on incomplete measurements. The objective is to give a criterion that the filtering error is confined to the ellipsoidal set by using the recursive linear matrix inequality (RLMI) method. Moreover, a convex optimization method is proposed to minimize the obtained ellipsoids. Finally, a numerical simulation illustrates that the designed set membership filtering scheme is effective.

RectEuler: Visualizing Intersecting Sets using Rectangles

AbstractEuler diagrams are a popular technique to visualize set‐typed data. However, creating diagrams using simple shapes remains a challenging problem for many complex, real‐life datasets. To solve this, we propose RectEuler: a flexible, fully‐automatic method using rectangles to create Euler‐like diagrams. We use an efficient mixed‐integer optimization scheme to place set labels and element representatives (e.g., text or images) in conjunction with rectangles describing the sets. By defining appropriate constraints, we adhere to well‐formedness properties and aesthetic considerations. If a dataset cannot be created within a reasonable time or at all, we iteratively split the diagram into multiple components until a drawable solution is found. Redundant encoding of the set membership using dots and set lines improves the readability of the diagram. Our web tool lets users see how the layout changes throughout the optimization process and provides interactive explanations. For evaluation, we perform quantitative and qualitative analysis across different datasets and compare our method to state‐of‐the‐art Euler diagram generation methods.

A Case for Partitioned Bloom Filters

In a partitioned Bloom Filter the $m$ bit vector is split into $k$ disjoint $m/k$ sized parts, one per hash function. Contrary to hardware designs, where they prevail, software implementations mostly adopt standard Bloom filters, considering partitioned filters slightly worse, due to the slightly larger false positive rate (FPR). In this paper, by performing an in-depth analysis, first we show that the FPR advantage of standard Bloom filters is smaller than thought; more importantly, by studying the per-element FPR, we show that standard Bloom filters have weak spots in the domain: elements which will be tested as false positives much more frequently than expected. This is relevant in scenarios where an element is tested against many filters, e.g., in packet forwarding. Moreover, standard Bloom filters are prone to exhibit extremely weak spots if naive double hashing is used, something occurring in several, even mainstream, libraries. Partitioned Bloom filters exhibit a uniform distribution of the FPR over the domain and are robust to the naive use of double hashing, having no weak spots. Finally, by surveying several usages other than testing set membership, we point out the many advantages of having disjoint parts: they can be individually sampled, extracted, added or retired, leading to superior designs for, e.g., SIMD usage, size reduction, test of set disjointness, or duplicate detection in streams. Partitioned Bloom filters are better, and should replace the standard form, both in general purpose libraries and as the base for novel designs.

LinSets.zip: Compressing Linear Set Diagrams.

Linear diagrams are used to visualize set systems by depicting set memberships as horizontal line segments in a matrix, where each set is represented as a row and each element as a column. Each such line segment of a set is shown in a contiguous horizontal range of cells of the matrix indicating that the corresponding elements in the columns belong to the set. As each set occupies its own row in the matrix, the total height of the resulting visualization is as large as the number of sets in the instance. Such a linear diagram can be visually sparse and intersecting sets containing the same element might be represented by distant rows. To alleviate such undesirable effects, we present LinSets.zip, a new approach that achieves a more space-efficient representation of linear diagrams. First, we minimize the total number of gaps in the horizontal segments by reordering columns, a criterion that has been shown to increase readability in linear diagrams. The main difference of LinSets.zip to linear diagrams is that multiple non-intersecting sets can be positioned in the same row of the matrix. Furthermore, we present several different rendering variations for a matrix-based representation that utilize the proposed row compression. We implemented the different steps of our approach in a visualization pipeline using integer-linear programming, and suitable heuristics aiming at sufficiently fast computations in practice. We conducted both a quantitative evaluation and a small-scale user experiment to compare the effects of compressing linear diagrams.

Zero-Knowledge Arguments for Lattice-Based Accumulators: Logarithmic-Size Ring Signatures and Group Signatures Without Trapdoors

An accumulator is a function that hashes a set of inputs into a short, constant-size string while preserving the ability to efficiently prove the inclusion of a specific input element in the hashed set. It has proved useful in the design of numerous privacy-enhancing protocols, in order to handle revocation or simply prove set membership. In the lattice setting, currently known instantiations of the primitive are based on Merkle trees, which do not interact well with zero-knowledge proofs. In order to efficiently prove the membership of some element in a zero-knowledge manner, the prover has to demonstrate knowledge of a hash chain without revealing it, which is not known to be efficiently possible under well-studied hardness assumptions. In this paper, we provide an efficient method of proving such statements using involved extensions of Stern’s protocol. Under the Small Integer Solution assumption, we provide zero-knowledge arguments showing possession of a hash chain. As an application, we describe new lattice-based group and ring signatures in the random oracle model. In particular, we obtain: (i) the first lattice-based ring signatures with logarithmic size in the cardinality of the ring and (ii) the first lattice-based group signature that does not require any GPV trapdoor and thus allows for a much more efficient choice of parameters.

An Explicit Dual Control Approach for Constrained Reference Tracking of Uncertain Linear Systems

A finite horizon optimal tracking problem is considered for linear dynamical systems subject to parametric uncertainties in the state-space matrices and exogenous disturbances. A suboptimal solution is proposed using a model predictive control (MPC) based explicit dual control approach which enables active uncertainty learning. A novel algorithm for the design of robustly invariant online terminal sets and terminal controllers is presented. Set membership identification is used to update the parameter uncertainty online. A predicted worst-case cost is used in the MPC optimization problem to model the dual effect of the control input. The cost-to-go is estimated using contractivity of the proposed terminal set and the remaining time horizon, so that the optimizer can estimate future benefits of exploration. The proposed dual control algorithm ensures robust constraint satisfaction and recursive feasibility, and navigates the exploration-exploitation trade-off using a robust performance metric.

A Neutrosophic Set Approach on Chest X-rays for Automatic Lung Infection Detection

COVID cases and its variants is noted enormously in the past three years. In many medical cases, lung infections such as viral pneumonia, bacterial pneumonia have been initially interpreted as COVID-19. Hence, the proposed work is concentrating on differentiating these lung infection types. This work focuses on using neutrosophic approach of classifying into True (T), False (F) and Indeterminacy (I) set membership to reduce the fuzziness and retain more significant information for feature extraction of the opacity to differentiate the types of lung infections. Initially, the images are preprocessed by alpha-mean and beta-enhancement operation to reduce the indeterminacy and enhancing the image components as the range of lung opacity levels to determine the types. Then, these neutrosophic set enhanced images are fed to various deep learning models like ResNet-50, VGG-16 and XGBoost for classification. Experiments are conducted on ActualMed COVID-19 Chest X-ray and COVID-19 radiography dataset and a comparative analysis on several domain set of images such as the original image, neutrosophic domain (T, I, F) and enhanced neutrosophic domain (alpha, beta) are trained and tested through transfer learning by tuning the various validation parameters. On experimental analysis, an enhanced neutrosophic image achieves a better accuracy of 97.33% among the other domain sets.

Автоматизація підбору кадрового складу за принципами нечіткої логіки

This article discusses the development of an expert system for personnel selection using fuzzy logic. The system uses a fuzzy set structure and membership functions for each criterion to calculate how well a candidate matches the criteria. The algorithm was implemented in Python and the results show that the system can improve automated recruitment for vacant positions and be developed further as a software product.

Robust particle filter for state estimation in presence of bounded but uncertain parameters based on ellipsoidal set membership approach

Accurate estimation of state variables plays an important role in control, monitoring and optimization. Considering that the uncertainty of parameter may back-propagate to the particle and then influence the state estimation, this paper proposes a robust particle filtering method in the presence of bounded uncertain parameters based on the ellipsoidal set membership filtering (ESMF) approach. This proposed method employs ellipsoidal sets to enclose the parameter uncertainty, and each prior particle and posterior particle are determined according to the ellipsoidal calculation based on the ESMF algorithm: The prior particle is characterized by an ellipsoid derived by the ellipsoidal summation of linear transformation ellipsoid, linearization error ellipsoid and system noise ellipsoid; Each posterior particle is obtained by updating the prior particle through the ellipsoidal intersection of the prior particle ellipsoid and the measurement ellipsoid. As a result, the uncertainty of parameter may be incorporated into the state estimation, and the advantages of both ESMF and particle filter are taken by the proposed method. The efficacy of the proposed method is shown by three simulation examples.

Nonlinear set membership filter with state estimation constraints via consensus-ADMM

This paper considers the state estimation problem for nonlinear dynamic systems with unknown but bounded noises. Set membership filter (SMF) is a popular algorithm to solve this problem. In the set membership setting, we investigate the filter problem where the state estimation requires to be constrained by a linear or nonlinear equality. We propose a consensus alternating direction method of multipliers (ADMM) based SMF algorithm for nonlinear dynamic systems. To deal with the difficulty of nonlinearity, instead of linearizing the nonlinear system, a semi-infinite programming (SIP) approach is used to transform the nonlinear system into a linear one, which allows us to obtain a more accurate estimation ellipsoid. For the solution of the SIP, an ADMM algorithm is proposed to handle the state estimation constraints, and each iteration of the algorithm can be solved efficiently. Finally, the proposed filter is applied to typical numerical examples to demonstrate its effectiveness.

Zonotopic recursive least‐squares parameter estimation: Application to fault detection

SummaryThe uncertain parameters estimation problem for linear discrete‐time systems modeled in regression form and assuming an unknown but bounded description of the noise, is the main subject of this work. Particularly, zonotopic sets for bounding the parametric uncertainty are considered. Then, a zonotopic recursive least‐squares (ZRLS) estimator is proposed and compared with its stochastic counterpart (RLS), as well as with the set‐membership (SM) approach. Likewise, the parameter estimation problem is addressed from the Bayesian general framework to define the relationship between stochastic and deterministic approaches. Moreover, both ZRLS and SM applications applied to fault detection are assessed taking as a reference the minimum detectable fault in the worst case. Finally, a well‐known quadruple‐tank process is used to illustrate the estimation and fault detection results.

A Set Membership approach to black-box optimization for time-varying problems

A novel method to tackle black-box optimization for time-varying problems is proposed. Using a Set Membership (SM) framework, the approach directly adjusts the uncertainty associated with old data points as new samples are introduced. Uninformative old samples are discarded, and the adjusted model guides the exploitation and exploration routines as characteristic of black-box optimization. With the proposed method, there is no need to estimate the time-related rate of change of the hidden function, as required in previous literature. We provide results of a benchmark test, comparing the performance of the proposed method to other approaches to time-varying black-box optimization, with promising results.

Loneliness and intersectionality: A progressive conditional approach

Both loneliness and intersectionality have become well established areas of academic research since the 1970s and 1980s. Nevertheless, only very recently some meaningful connections were made between the two, although researchers have paid attention to the interactive effects of two or more socio-demographic attributes on loneliness. For intersectionality, much of academic research is invested in establishing it as a theoretical approach in tackling social injustice, whilst how it should be studied empirically remains a major challenge. In contrast, research on loneliness has been predominantly empirical, and the small number of studies on loneliness from the intersectional perspective have adopted different research methodologies. This paper proposes and illustrates an approach loyal to the fundamental principles of intersectionality and simple to conduct in empirical investigations at the same time. First, it focuses exclusively on intersectional cross-classifications rather than both the main and the interactional effects; second, it demands a rationale of starting from one attribute and then moving on to include an additional attribute at a time; third, it examines the intersectional cross-classifications and their relationships with the interested outcome systematically without transforming the data in set memberships. The approach is illustrated with analyses of the data collected in Great Britain in the seventh round (2014/15) of European Social Survey. Young people (under 30) of ethnic minority and born inside Great Britain suffered from the highest percentage of frequent loneliness (15%), whilst their counterparts born outside the country enjoyed the lowest rate. Among the middle-aged, ethnicity determined how vulnerable they were to frequent loneliness. For older people (60+) born outside Great Britain, regardless of ethnicity, the percentage of frequent loneliness was 10%.

Guarantees for data-driven control of nonlinear systems using semidefinite programming: A survey

This survey presents recent research on determining control-theoretic properties and designing controllers with rigorous guarantees using semidefinite programming and for nonlinear systems for which no mathematical models but measured trajectories are available. Data-driven control techniques have been developed to circumvent a time-consuming modelling by first principles and because of the increasing availability of data. Recently, this research field has gained increased attention by the application of Willems’ fundamental lemma, which provides a fertile ground for the development of data-driven control schemes with guarantees for linear time-invariant systems. While the fundamental lemma can be generalized to further system classes, there does not exist a comparable data-based system representation for nonlinear systems. At the same time, nonlinear systems constitute the majority of practical systems. Moreover, they include additional challenges such as data-based surrogate models that prevent system analysis and controller design by convex optimization. Therefore, a variety of data-driven control approaches has been developed with different required prior insights into the system to ensure a guaranteed inference. In this survey, we will discuss developments in the context of data-driven control for nonlinear systems. In particular, we will focus on methods based on system representations providing guarantees from finite data, while the analysis and the controller design boil down to convex optimization problems given as semidefinite programming. Thus, these approaches achieve reasonable advances compared to the state-of-the-art system analysis and controller design by models from system identification. Specifically, the paper covers system representations based on extensions of Willems’ fundamental lemma, set membership, kernel techniques, the Koopman operator, and feedback linearization.

A New Adaptive Suboptimal Controller for a Class of Linear Discrete-Time Systems with Unknown Bounded Disturbances

This paper deals with adaptive suboptimal control of linear discrete-time, time-invariant, minimum phase, scalar systems in the presence of arbitrary bounded unmeasurable disturbances whose upper and lower bounds are a priori unknown. The distinguishing feature of the systems to be controlled is that there is no information concerning a priori knowledge of a bounded set to which the unknown parameters of these systems belong. Novelty of this paper is that, instead of unknown a priori membership set of parameters, the peculiar a posteriori membership sets are designed via the use of the measured signals. The adaptive suboptimal control algorithm utilizing this approach is proposed. The properties of the adaptive controller including the convergence of the adaptation procedure and also the ultimate boundedness of system's signals are established. To demonstrate an efficiency of this controller and support the theoretical study, simulation results are presented.

Nonlinear Model Predictive Control: an Optimal Search Domain Reduction

Nonlinear Model Predictive Control (NMPC) is a powerful control method, used in many industrial contexts. NMPC is based on the online solution of a suitable Optimal Control Problem (OCP) but this operation may require high computational costs, which may compromise its implementation in “fast” real-time applications. In this paper, we propose a novel NMPC approach, aiming to improve the numerical efficiency of the underlying optimization process. In particular, a Set Membership approximation method is applied to derive from data tight bounds on the optimal NMPC control law. These bounds are used to restrict the search domain of the OCP, allowing a significant reduction of the computation time. The effectiveness of the proposed NMPC strategy is demonstrated in simulation, considering an overtaking maneuver in a realistic autonomous vehicle scenario.

Comparative analysis of Fuzzy Tsukamoto's membership functions for determining irrigated rice field feasibility status

The representation of the fuzzy set membership curve consisting of trapezoidal, triangular, and linear shapes, has an important role in the fuzzy logic system. The selection of the curve's shapes determines the useable membership function and affects the fuzzy output value. Previous studies generally used curves that had been employed in predecessors or other studies that did not explain the reason for choosing a fuzzy member curve. This condition became problem because there was not a guide in selecting the appropriate membership function model for the parameters used in the fuzzy process so that most researchers only use membership functions that are commonly used in previous studies or in the same case as their research. The purpose of this study was to determine the effect of selecting trapezoidal and triangular curves on the performance of Tsukamoto's fuzzy logic for determining the rice-fields suitability status. The research methodology comprised 3 main stages. The first stage was data collecting, to collect soil pH values, soil moisture, and air temperature in rice fields. The second stage was the implementation of the Tsukamoto fuzzy. At this stage, two membership function curves were used. The third stage was a comparative analysis of Tsukamoto's fuzzy's output of trapezoidal and triangular curves. The results obtained indicate that there is no significant performance difference between the two different membership functions. The results of the research with the trapezoidal membership function have a better accuracy rate of 93% while the triangular membership function has an accuracy rate of 90%.