Nonmonotonic Systems Research Articles

Improvement of least mean square adaptive algorithm for non-monotonic systems

This paper introduces the zlLMS algorithm, an improvement over the traditional least mean square (LMS) algorithm, addressing its limitations in handling nonlinear and non-monotonic transfer functions commonly encountered in engineering systems. The proposed method replaces the LMS algorithm’s error inputs with a function monotonically correlated with the controllable signal. Through mathematical derivations and simulations, the zlLMS algorithm demonstrates superior performance in nonlinear adaptive filtering scenarios, such as the raised-cosine transfer function of Mach-Zehnder modulators and the Lorentz transfer function of diode lasers. Simulation results reveal that the amplitude difference between the ideal and zlLMS-equalized signals is reduced to 1/1000th of the original input signal’s amplitude, underscoring its effectiveness in optics and terahertz technologies. These findings highlight the algorithm’s robustness and potential for broad applications in engineering, especially where nonlinear dynamics are involved.

Exploring the Definition of Non-Monotonicity – Logical and Psychological Considerations

When humans reason, they are able to revise their beliefs in light of new information and abandon obsolete conclusions. Logicians argued, that in some cases, such reasonings appear to be non-monotonic. Thus, many different, seemingly non-monotonic systems were created to formally model such cases. The purpose of this article is to re-examine the definition of non-monotonicity and its implementation in non-monotonic logics and in examples of everyday human reasoning. We will argue that many non-monotonic logics employ some weakened versions of the definitions of non-monotonicity, since in-between different steps of reasoning they either: a) allow previously accepted premises to be removed, or b) change the rules of inference. Of the two strategies, the second one seems downright absurd, since changing the rules of a given logic is a mere replacement of that logic with the rules of another. As a consequence we obtain two logics, whereas the definition of a non-monotonic logic is supposed to define one. The definition of non-monotonicity does not permit either of these cases, which means that such logics are monotonic.

Nonmonotonic inferences: Classical conclusions in an intuitionistic modal framework

Abstract We introduce a conceptual foundation for nonmonotonic reasoning which integrates an intuitionistic plurimodal logic with classical logic. The need for a multilogical system arises once we understand that to legitimately infer less than certain conclusions, we make two different kinds of assessments: an external one (what are the alternatives) and an internal one (which logic should govern the set of beliefs). We exploit this scheme to capture in novel ways basic nonmonotonic systems such as autoepistemic logic, default logic, one of the best known interpretations of negation-as-failure, as well as some conditional logics of normality. Among the aims of this paper is to explain why there are substantial overlaps between these systems and also to settle some problematic features that are of concern in logical investigations on common sense reasoning.

The successful branching algorithm comparison in probabilistic safety assessment

In this paper, the technical requirements of quantitative analysis in the PSA standards sought to carry out corresponding research. Different models are used to analyze and compare the commonly used successful branching algorithms, and the compliance of each algorithm with the PSA standards is listed. Algorithm analysis and testing show that the ability to propagate success branch logic is the key to ensuring reasonable results. On the other hand, before carrying out quantitative analysis, we should identify whether there are high probability events (>0.1), non-monotonic system, mutual exclusion logic structure, etc, in the model. Then select the appropriate algorithm, and the analysis results of cutest, importance measure and uncertainty can be reasonable expounded. This paper conducts further research on the technical requirements related to quantitative analysis in the PSA standard, and gives the compliance status of the commonly used successful branch algorithms with the PSA standard. After research, the algorithms that are not recommended include the Ignore Event Tree Success algorithm and the “success probability” processing method. The former is provided in some PSA software, but because it does not consider the logic of successful branching, it will produce erroneous cut sets, which is also a practice that the PSA standard clearly does not support. Even if the numerical results are very different from other algorithms in some cases, they should not be used. Although the latter is considered to be usable in the current PSA standard, manual processing will bring a great workload, and it is not feasible in actual engineering.

A generalized ternary decision diagram for reliability analysis on fault-tolerant systems in different coverage models

ABSTRACT For systems subject to imperfect fault coverage, ternary decision diagrams (TDDs) can be used for reliability analysis, as the component state becomes ternary, namely, operational, failed in a covered mode, and failed in an uncovered mode. There are different coverage models (CMs), including the traditional imperfect fault coverage model (IFCM) and its new extension irrelevance coverage model (ICM). The ICM extends the coverage to irrelevant components in addition to the fault components but is limited to monotonic systems so far. In addition, the TDDs for different CMs are developed based on different operation rules, and can only be used in specific CMs. To address these problems, this paper presents a generalized ternary decision diagram (GTDD). On one hand, the GTDD can handle non-monotonic systems in the ICM by extending the calculation of minimal irrelevance triggers to non-monotonic systems. On the other hand, the GTDD is designed as an all-in-one solution for the reliability analysis in various CMs, including perfect fault coverage model, element-level coverage model, fault-level coverage model, one-on-one-level coverage model, and ICM. Two case studies are presented to demonstrate the effectiveness of GTDD.

State of Power Prediction for Battery Systems With Parallel-Connected Units

To meet the ever-increasing demand for energy storage and power supply, battery systems are being vastly applied to, e.g., grid-level energy storage and automotive traction electrification. In pursuit of safe, efficient, and cost-effective operation, it is critical to predict the maximum acceptable battery power on the fly, commonly referred to as the battery system’s state of power (SoP). Compared to the SoP prediction at the battery cell level, predicting the SoP of a multi-battery system, especially including parallel-connected cells/modules/packs, is much more complicated and far less investigated. To solve this problem, a system-model-based SoP prediction method is first proposed in this work. Specifically, based on the formulated system model and generic state-space representation, the challenge of nonmonotonic system state evolution, arising from the dynamic parallel current distribution, is identified and systematically addressed by the proposed method. As demonstrated by tests on a battery system set up with experimentally verified parameter values, the proposed method outperforms the commonly applied cell-SoP-based methods for providing a more accurate and reliable prediction of the battery system SoP. Moreover, the proposed prediction framework presented in generic forms can be readily applied to other system structures.

Simple dynamics in non-monotone Kolmogorov systems

In this paper we analyse the global dynamical behaviour of some classical models in the plane. Informally speaking we prove that the folkloric criteria based on the relative positions of the nullclines for Lotka–Volterra systems are also valid in a wide class of discrete systems. The method of proof consists of dividing the plane into suitable positively invariant regions and applying the theory of translation arcs in a subtle manner. Our approach allows us to extend several results of the theory of monotone systems to nonmonotone systems. Applications in models with weak Allee effect, population models for pioneer-climax species, and predator–prey systems are given.

Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission.

In this paper, we focus on spreading speed of a reaction-diffusion SI epidemic model with vertical transmission, which is a non-monotone system. More specifically, we prove that the solution of the system converges to the disease-free equilibrium as $ t \rightarrow \infty $ if $ R_{0} \leqslant 1 $ and if $ R_0 > 1 $, there exists a critical speed $ c^\diamond > 0 $ such that if $ \|x\| = ct $ with $ c \in (0, c^\diamond) $, the disease is persistent and if $ \|x\| \geqslant ct $ with $ c > c^\diamond $, the infection dies out. Finally, we illustrate the asymptotic behaviour of the solution of the system via numerical simulations.

Traveling waves in a nonlocal dispersal SIR model with non-monotone incidence

It is well-known that the nonlocal dispersal operator has the advantage of capturing short-range as well as long-range factors for the dispersal of the spices by choosing the kernel function properly, and is also capable to include spatial dispersal strategies of the species beyond random (local) diffusion. This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with non-monotone incidence, which is a non-monotone system. The method of sub and super solutions combined with Schauder’s fixed-point theorem is applied to establish the existence of positive traveling waves as the wave speed is over critical speed. We further prove the existence of traveling waves with critical speed and the nonexistence of bounded positive traveling waves by the delicate analysis method. The main difficulty is to get the boundedness of traveling waves caused by the nonlocal dispersal operator.

Hybrid nonlinear observer for battery state‐of‐charge estimation using nonmonotonic force measurements

This article focuses on state‐of‐charge (SOC) estimation in a lithium‐ion battery, using measurements of terminal voltage and bulk force. A nonlinear observer designed using Lyapunov analysis relying on lower and upper bounds of the Jacobian of the nonlinear output function is utilized. Rigorous analysis shows that the proposed observer has feasible design solutions only in each piecewise monotonic region of the output functions and has no constant stabilizing observer gain when the entire SOC range is considered. The nonmonotonicity challenge is then addressed by designing a hybrid nonlinear observer that switches between several constant observer gains. The global stability of the switched system is guaranteed by ensuring overlap between regions and an adequate dwell time between switches. The performance of the observer is evaluated first through simulations using a high‐fidelity battery model and then through experiments. The performance of the nonlinear observer is compared with that of an extended Kalman Filter. Simulation results show that with no model uncertainty the nonlinear observer provides estimates with an RMS error of 1.1%, while the EKF performs better, providing an RMS error less than 1%. However, when model error is introduced into this nonmonotonic system, the EKF becomes unstable for even very small model errors in the output curves. The nonlinear observer, on the other hand, continues to perform very well, providing accurate estimates and never becoming unstable. The experimental results verify the observations from the simulation and the experimental EKF is found to become unstable due to model errors, while the hybrid nonlinear observer continues to work reliably.

Join Idle Queue with Service Elasticity: Large-Scale Asymptotics of a Nonmonotone System

We consider the model of a token-based joint autoscaling and load-balancing strategy, proposed in a recent paper by Mukherjee et al. [Mukherjee D, Dhara S, Borst SC, Van Leeuwaarden JSH (2017) Optimal service elasticity in large-scale distributed systems. Proc. ACM Measurement Anal. Comput. Systems 1(1):25:1–25:28.], which offers an efficient scalable implementation and yet achieves asymptotically optimal steady-state delay performance and energy consumption as the number of servers N → ∞. In the aforementioned work, the asymptotic results are obtained under the assumption that the queues have fixed-size finite buffers, and therefore, the fundamental question of stability of the proposed scheme with infinite buffers was left open. In this paper, we address this fundamental stability question. The system stability under the usual subcritical load assumption is not automatic. Moreover, the stability may not even hold for all N. The key challenge stems from the fact that the process lacks monotonicity, which has been the powerful primary tool for establishing stability in load-balancing models. We develop a novel method to prove that the subcritically loaded system is stable for large enough N and establish convergence of steady-state distributions to the optimal one as N → ∞. The method goes beyond the state-of-the-art techniques; it uses an induction-based idea and a “weak monotonicity” property of the model. This technique is of independent interest and may have broader applicability.

Why Machines Don’t (yet) Reason Like People

AI has never come to grips with how human beings reason in daily life. Many automated theorem-proving technologies exist, but they cannot serve as a foundation for automated reasoning systems. In this paper, we trace their limitations back to two historical developments in AI: the motivation to establish automated theorem-provers for systems of mathematical logic, and the formulation of nonmonotonic systems of reasoning. We then describe why human reasoning cannot be simulated by current machine reasoning or deep learning methodologies. People can generate inferences on their own instead of just evaluating them. They use strategies and fallible shortcuts when they reason. The discovery of an inconsistency does not result in an explosion of inferences—instead, it often prompts reasoners to abandon a premise. And the connectives they use in natural language have different meanings than those in classical logic. Only recently have cognitive scientists begun to implement automated reasoning systems that reflect these human patterns of reasoning. A key constraint of these recent implementations is that they compute, not proofs or truth values, but possibilities.

Traveling Waves in Epidemic Models: Non-monotone Diffusive Systems with Non-monotone Incidence Rates

We study the existence and nonexistence of traveling waves of diffusive epidemic models with general incidence rates. The model systems are non-monotone because of the intrinsic predator–prey interaction between the susceptible and infective compartments in epidemic systems. Moreover, the incidence rate may not be monotone in the infected population because social behaviors and collective activities may change in response to the prevalence of disease. To find positive traveling solutions of the non-monotone system with a non-monotone incidence function, we will construct a suitable convex set in a weighted function space, and then apply Schauder fixed point theorem. It turns out that the basic reproduction number of the corresponding ordinary differential equations plays an important role in the existence theory of traveling waves. Moreover, the critical wave speed can be explicitly obtained in terms of the diffusion coefficient, recovery rate and death rate for the infected group, and partial derivative of incidence function at the disease-free equilibrium. Finally, we prove that the positive traveling wave solution does not exist if the basic reproduction number is no more than one, or the wave speed is less than the critical value.

A Coderivative Approach to the Robust Stability of Composite Parametric Variational Systems: Applications in Nonsmooth Mechanics

The main concern of this paper is to investigate the Lipschitzian-like stability property (namely Aubin property) of the solution map of possibly nonmonotone variational systems with composite superpotentials. Using Mordukhovich coderivative criterion and a second-order subdifferential analysis, we provide simple and verifiable characterizations of this property in terms of the data involved in the problem. Applications are given in nonsmooth mechanics.

Optimal control formulation of pulse-based control using Koopman operator

In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty

In applications, such as biomedicine and systems/synthetic biology, technical limitations in actuation complicate implementation of time-varying control signals. In order to alleviate some of these limitations, it may be desirable to derive simple control policies, such as step functions with fixed magnitude and length (or temporal pulses). In this technical note, we further develop a recently proposed pulse-based solution to the convergence problem, i.e., minimizing the convergence time to the target exponentially stable equilibrium, for monotone systems. In particular, we extend this solution to monotone systems with parametric uncertainty. Our solutions also provide worst case estimates on convergence times. Furthermore, we indicate how our tools can be used for a class of nonmonotone systems, and more importantly how these tools can be extended to other control problems. We illustrate our approach on switching under parametric uncertainty and regulation around a saddle point problems in a genetic toggle switch system.

FDE Circumscription

In his article "Reassurance via Translation" Marcel Crabbe proposed a formalism to obtain reassurance and classical recapture in the setting of minimal FDE. His formalism proved to be general enough to be extended in order to formalize other forms of non-monotonic systems based on preference relations. It is the aim of this article to show how his result can be extended in a natural way by combining two different reasoning systems, namely minimal FDE and circumscription, in order to get a paraconsistent and paracomplete version of circumscription, which we will call paracomplistent circumscription, which has the advantages of FDE and circumscription but is neither explosive nor lacks modus ponens in consistent contexts. Furthermore, we will complete a proof Crabbe left unfinished.

Formal Nonmonotonic Theories and Properties of Human Defeasible Reasoning

The knowledge representation and reasoning of both humans and artificial systems often involves conditionals. A conditional connects a consequence which holds given a precondition. It can be easily recognized in natural languages with certain key words, like "if" in English. A vast amount of literature in both fields, both artificial intelligence and psychology, deals with the questions of how such conditionals can be best represented and how these conditionals can model human reasoning. On the other hand, findings in the psychology of reasoning, such as those in the Suppression Task, have led to a paradigm shift from the monotonicity assumptions in human inferences towards nonmonotonic reasoning. Nonmonotonic reasoning is sensitive for information change, that is, inferences are drawn cautiously such that retraction of previous information is not required with the addition of new information. While many formalisms of nonmonotonic reasoning have been proposed in the field of Artificial Intelligence, their capability to model properties of human reasoning has not yet been extensively investigated. In this paper, we analyzed systematically from both a formal and an empirical perspective the power of formal nonmonotonic systems to model (i) possible explicit defeaters, as in the Suppression Task, and (ii) more implicit conditional rules that trigger nonmonotonic reasoning by the keywords in such rules. The results indicated that the classical evaluation for the correctness of inferences has to be extended in the three major aspects (i) regarding the inference system, (ii) the knowledge base, and (iii) possible assumed exceptions for the rule.

On the stability of a class of non-monotonic systems of parallel queues

We investigate, under general stationary e`rgodic assumptions, the stability of systems of S parallel queues in which any incoming customer joins the queue of the server having the p + 1-th shortest workload (p

Existence and qualitative properties of travelling waves for an epidemiological model with mutations

In this article, we are interested in a non-monotonic system of logistic reaction–diffusion equations. This system of equations models an epidemic where two types of pathogens are competing, and a mutation can change one type into the other with a certain rate. We show the existence of travelling waves with minimal speed, which are usually non-monotonic. Then we provide a description of the shape of those constructed travelling waves, and relate them to some Fisher-KPP fronts with non-minimal speed.