Non-Commutativity, Incompatibility, Emergent Behavior and Decision Support Systems

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.

Asset Prices and Google's Search Data

Ensuring the correct behavior of cyber physical systems at run time is of critical importance for their safe deployment. Any malfunctioning of such systems should be detected in a timely manner for further actions. This paper addresses the issue of how quickly a monitor raises an alarm after the occurrence of a failure in cyber physical systems. Towards this end, it introduces a class of systems called exponentially converging monitorable systems. The paper shows that failures in these systems can be detected fast by employing the traditional threshold monitors. It shows that the expected failure detection time for exponentially converging monitorable systems has logarithmic relationship with the inverse of the chosen threshold value. The paper identifies well defined natural classes of these systems. Experimental results are presented that confirm the theoretical results on the relationship between the failure detection time and the chosen threshold values.

Experimental Digital Twins (EDT) provide a comprehensive view on the behavior of Cyber-Physical Systems (CPS). This is achieved by simulation technology and the application of virtual environments to develop and test desired functionality, including models, data structures, services and communication capabilities. This paper outlines a meta-information driven method of describing services provided by a CPS for consumption by peers via any means of communication. In addition, a set of application rules is presented, which handle cases such as service composition and the interaction of CPS components in distributed systems. We present the fundamental meta-information system that facilitates the specification process by providing the necessary type information for the dynamic object construction and automated (de-)serialization of data passed between provider and consumer.

What is the proper attitude toward what is expressed by a vague sentence in the face of borderline evidence? Some call this attitude “ambivalence” and distinguish it from uncertainty. It has been argued that Classical Epistemicism conjoined with classical probability theory fails to characterize this attitude, and that we must therefore abandon classical logic or classical probabilities in the presence of vagueness. In this paper, I give a characterization of ambivalence assuming a supervaluationist semantics for vague terms that does not revise either. The theory, which I call the theory of Superprobabilities, identifies the proper attitude toward a vague sentence, in the presence of exact borderline evidence, as the set of classical probabilities of the evidence on each member of the set of all precisifications of a vague sentence. I defend the use of sets of probabilities against objections by generalizing the theory of Superprobabilities to a decision theory called Superrationality. I then compare the merits of the theory of Superprobabilities to Classical Epistemicism and nonclassical probabilities theories with respect to the problem of ambivalence.

In this chapter we present a brief introductions to Kolmogorov’s measure-theoretic and von Mises’ frequency probability theories. These are classical probability theories. A rather common opinion is that classical probability can not describe the probabilistic structure of quantum mechanics. Such an opinion is based on belief that classical probability can not reproduce the main distinguishing probabilistic features of quantum formalism, e.g., interference of probabilities and Born’s rule. It is also believed that the violation of Bell’s inequality can not be explained in the framework of classical probability theory. The aim of this book is to show that such a viewpoint to the relation between classical and quantum probabilities is not justified. We shall see that the mathematical formalisms of Kolmogorov’s measure-theoretic [Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933; Shiryaev in Ann. Probab. 17:866–944, 1989; Shiryaev, Probability, 1984] and Mises’ frequency [Von Mises in Math. Z. 5:52–99, 1919; Von Mises, Probability, Statistics and Truth, 1957; Von Mises, The Mathematical Theory of Probability and Statistics, 1964] models can be represented in the QL way. However, these models should be interpreted in the contextual framework.

An operator-valued T1 theory for symmetric CZOs

In systems development, it is necessary to think thoroughly about any emergent behaviors that might occur and create strategies to prevent the undesirable emergent behavior from occurring or at least mitigate them. Often dealing with that at later development stages is costly. Emergent behaviors are the main concern, particularly in the development of complex systems. However, identifying and predicting them is challenging in complex systems as it involves lots of components and interactions. Prior to making wise design decisions, developers need to elucidate and understand the emergent behaviors. For this reason, a conceptual model that highlights the presence of emergent behavior at an early stage is essential in the systems development. The conceptual model is useful in achieving higher coverage as it is correlated with the probability of identifying more emergent behaviors. Therefore, this paper goes through the existing work focusing on effort in modeling emergent behavior and presents perspectives on developing a conceptual model that is able to inform the presence of emergent behavior.

This manuscript critically examines existing research on cyber battle management systems (CBMS) and underscores the importance of advancing complex structure thinking, cybernetics, wicked problem-solving, and emerging behavior analysis. It advocates for a systems-thinking approach to solving complex problems by identifying and understanding associated systems, predicting their behavior, and managing changes. The manuscript explores the integration of cybernetics meta-methodology and the viable system model with metasystems reductionism to address negative emergent behavior in complex systems. The study highlights the roles of individual systems, systems of systems, and metasystems, emphasizing the deterministic nature of single systems and the stochastic characteristics of systems of systems. By integrating cybernetics, viable system models, and meta-metasystems, the manuscript explores key parameters for building intelligent systems, revealing that meta-metasystems offer superior capabilities for coordinating and integrating multiple systems. The research results demonstrate the successful development of a meta-metasystem tailored for CBMS, providing a strategic framework for the future of cyber battle management.

The paper considers the idea of detecting security incidents and anomalies in physical access control systems (PACS) based on combined use of different types of user identification. As an example, an implementation of PACS is considered that performs identification by RFID cards and face recognition. The architecture of such system is introduced. A BPMN diagram for RFID identification of employees by entry and exit to the organization premises is built. A basic data model is proposed for storing information about users, devices, and events. Situations that can be interpreted as anomalous in the considered system are described and classified. The theoretical contribution of the work consists in enhancing the approaches to modeling user behavior in cyber-physical systems, especially in case when this modeling is performed on the basis of heterogeneous data collected from sources of different nature.

Cromwell’s rule (also known as the zero priors paradox) refers to the constraint of classical probability theory that if one assigns a prior probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes’ rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell’s rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell’s rule. Quantum probability theory, the rules for how to assign probabilities from the mathematical formalism of quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of quantum probability theory is that it is not subject to Cromwell’s rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.

The classical entropies were formulated by means of the classical probability theory. The essence of the classical probability theory corresponds to the use of c‐number with commutative property. On the base of the classical probability theory, the classical information theory can be discussed by using the classical entropies. On the other hand, the principle of the quantum phenomenon seems to be having non‐commutative behavior. It leads to the use of the q‐operator with non‐commutative property to introduce the quantum entropies based on the quantum probability theory. In order to discuss the efficiency of the information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Shannon's entropy and the mutual entropy represent these measures of complexities in classical communication system, respectively. Von Neumann entropy and Ohya mutual entropy relate to these measures of complexities in quantum communication system, respectively. Recently, several mutual entropy type measures (Lindblad—Nielsen entropy and Coherent entropy) were proposed by using an entropy exchange with respect to a channel and an initial state. In this paper, we will show the most suitable one for discuss the efficiency of information transmission for quantum processes. In classical information theory, one of the most important tools to analyze communication processes are the classical coding theorems, which was investigated through the classical dynamical entropy. The classical dynamical entropy was discussed in [1] with respect to a measure preserving transformation defined on a message space for finite partitions of the measure space. In quantum information theory, Emch [2] and Conne, Stormer [3] first studied a complexity of quantum dynamical processes. After that, Ohya introduced a S‐ mixing entropy in general quantum systems, which was natural extension of the von Neumann entropy. Using the S—mixing entropy, Ohya defined a mean entropy and a mean mutual entropy for the quantum dynamical systems. Conne, Narnhoffer and Thirring introduced a dynamical entropy (CNT entropy) [4] and several researchers discussed the CNT entropy [5,6,7,8]. Alicki and Fannes defined a dynamical entropy (AF entropy) [9] using a finite operational partitions of unity. In 1995, Voiculescu proposed a dynamical entropy (dynamical approximation entropy) [10] in C* and W* algebras. Through the quantum Markov process, Accardi, Ohya and Watanabe defined a dynamical entropy (AOW entropy) [11,12] in 1997. Base on transition expectations and liftings in the sense of Accardi and Ohya [13], Kossakowski, Ohya and Watanabe introduced a dynamical entropy (KOW entropy) [14] with respect to completely positive maps in 1999. In this paper, we will discuss the complexity of the quantum dynamical process to calculate the generalized AOW entropy given by KOW entropy.

Time plays a major role in the specification of Cyber-physical Systems (CPS) behavior with concurrency, timeliness, asynchrony, and resource limits as their main characteristics. In addition to timeliness , the specification of CPS needs to assess and unambiguously define its behavior with respect to the other Quality-of-Information (QoI) properties: (1) Correctness, (2) Completeness, (3) Consistency, and (4) Accuracy. Very often, CPS need to handle these QoI properties, and any combination thereof, multiple times when performing computation and communication processes. However, a model-driven and systematic approach to specify CPS behavior that jointly considers combined QoI aspects is possible but missing in existing methodologies. As the first contribution of this work, we provide an extension to an established model of computation (MoC) based on “Functions driven by Finite State Machine” (FunState) to enable a model-driven composition mechanism to create CPS behavior specifications from reusable components. Second, we present a novel set of design patterns to illustrate the modeling of QoI-aware CPS specifications that can be applied in several state-of-the-art Electronic System Level (ESL) methodologies. The time semantics of the MoC are formalized using the tagged-signal-model, and the presented model-driven approach enables the composition of multiple design patterns. The main benefits of the presented model-driven approach and design patterns to create CPS specifications are as follows: (a) reduce modeling effort, errors, and time through the reuse of known recipes to re-incurring tasks and allow to automatically generate repetitive control flows based on extended Finite State Machines; (b) increase system robustness and facilitate the creation of holistic QoI management allowing to unambiguously define system behavior for scenarios with single/multiple QoI requirement violations in different models of computation; (c) dynamically validate timing behavior of system implementations to enable a multi-objective optimization of nonfunctional properties that influence CPS timing. We demonstrate the aforementioned benefits through the modeling and evaluation of an infrastructure-assisted automated driving case study using Infrastructure-to-Vehicle (I2V) communications to distribute QoI critical road environment information.

The evaluation of the emergent behaviour in complex systems requires an analytical framework which allows the observation of different phenomena that take place at different levels. In order to observe the dynamics of complex systems, it is necessary to perform simulations so that both local and the emergent behaviour can be observed. To this end, the way in which complex system simulators are built must be examined so that it will be feasible to model large scale scenarios. In this paper, the use of Model Driven Engineering methodology is proposed to deal with this issue. Among other benefits, it is shown that this methodology allows the representation and simulation of a complex system providing support for the analysis. This analysis is supported by a metamodel which describes the system components that are under study. The application of this methodology to the development of large scale simulators is explored through a case study. This case study analyses a complex socio-technical system: a power grid.