Search

In this article, we study systems of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a private equation known to this node only, and the aim of this article is to develop distributed algorithms that allow all the nodes to obtain solutions to the network Boolean equations without exchanging their local Boolean equations. First, we observe that the Boolean equations can be locally lifted to a system of linear algebraic equations under a basis of Boolean vectors, which is distributedly solvable using existing distributed linear equation algorithms as a subroutine. Next, we construct a distributed Boolean equation solver by the nodes solving the lifted linear network equation for a number of randomly selected initial values, and then converting the algebraic solutions into solutions to the original Boolean equations by a novel Boolean vector search algorithm. We prove that for solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are independent identically distributed selected according to a uniform distribution in a high-dimensional cube, such an algorithm returns the exact solution set of the Boolean equations at each node with high probability. We also present an algorithm for distributed verification of the satisfiability of Boolean equations when the solvability is not known beforehand, and prove its correctness. Finally, we show that by utilizing linear equation solvers with differential privacy to replace the in-network computing routines, the distributed Boolean equation algorithms can be made differentially private

Vessel segmentation plays a crucial role in the diagnosis of many diseases, as well as assisting surgery. With the development of deep learning, many segmentation methods have been proposed, and the results have become more and more accurate. However, most of these methods are based on supervised learning, which require a large amount of labeled data as training data. To overcome this shortcoming, unsupervised and self-supervised methods have also received increasing attention. In this paper, we generate a synthetic training datasets through L-system, and utilize adversarial learning to narrow the distribution difference between the generated data and the real data to obtain the ultimate network. Our method achieves state-of-the-art (SOTA) results on X-ray angiography artery disease (XCAD) by a large margin of nearly 10.4%.

This paper designs a distributed algorithm to seek generalized Nash equilibria of a robust game with uncertain coupled constraints. Due to the uncertainty of parameters in set constraints, the authors aim to find a generalized Nash equilibrium in the worst case. However, it is challenging to obtain the exact equilibria directly because the parameters are from general convex sets, which may not have analytic expressions or are endowed with high-dimensional nonlinearities. To solve this problem, the authors first approximate parameter sets with inscribed polyhedrons, and transform the approximate problem in the worst case into an extended certain game with resource allocation constraints by robust optimization. Then the authors propose a distributed algorithm for this certain game and prove that an equilibrium obtained from the algorithm induces an ε-generalized Nash equilibrium of the original game, followed by convergence analysis. Moreover, resorting to the metric spaces and the analysis on nonlinear perturbed systems, the authors estimate the approximation accuracy related to εand point out the factors influencing the accuracy of ε.

Boolean networks, as logical dynamical systems whose system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. The representation and control of Boolean networks have attracted a lot of attention in recent years. On a parallel line of research, Koopman developed an operator view of nonlinear dynamical systems in early 1930s, which shows that using observable functions all nonlinear dynamics can be represented as (infinite dimensional) linear systems. In this paper, we present a framework for representing Boolean networks via Koopman approach. First of all, we construct addition and scalar multiplication operations over the set of logical functions over the binary field, defining a linear Boolean function space. Then associated with any Boolean mapping, the induced Koopman operator is defined as an operator over such Boolean function space. Next, we show that if there exists a set of logical functions that are invariant in the Koopman sense, a Boolean network can be represented by a finite-dimensional linear system. This Koopman perspective for Boolean networks is also extended to controlled Boolean networks.

Boolean networks, as logical dynamical systems where the system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. In this paper, we present a framework to evaluate whether and how the closed-loop dynamics of a controlled Boolean network can be shaped into any prescribed form by state-feedback control. We refer to this problem as to the feedback shaping for Boolean networks. First of all, based on the linear representation of Boolean networks, we establish a necessary and sufficient rank condition for a controlled Boolean network to be feedback shapable or not. Next, we design an algorithm for the synthesis of closed-loop dynamics for a feedback shapable Boolean network, such that for any given controlled Boolean network and a desired closed-loop dynamics, one can always find a feedback control law so that the closed-loop dynamics is precisely realized.

In this article, we aim to design a distributed approximate algorithm for seeking Nash equilibria (NE) of an aggregative game. Due to the local set constraints of each player, projection-based algorithms have been widely employed for solving such problems actually. Since it may be quite hard to get the exact projection in practice, we utilize inscribed polyhedrons to approximate local set constraints, which yields a related approximate game model. We first prove that the NE of the approximate game is the $\epsilon $ -NE of the original game and then propose a distributed algorithm to seek the $\epsilon $ -NE, where the projection is then of a standard form in quadratic optimization with linear constraints. With the help of the existing developed methods for solving quadratic optimization, we show the convergence of the proposed algorithm and also discuss the computational cost issue related to the approximation. Furthermore, based on the exponential convergence of the algorithm, we estimate the approximation accuracy related to $\epsilon $ . In addition, we investigate the computational cost saved by approximation in numerical simulation.

In this article, we study the recursion corresponding to the measurement outcomes for open quantum networks under sequential measurements. Open quantum networks are networked quantum subsystems (e.g., qubits) with the state evolutions described by a continuous Lindblad master equation. When measurements are performed sequentially along such continuous dynamics, the quantum network states undergo probabilistic jumps and the corresponding measurement outcomes can be described by a vector of probabilistic Boolean variables. The induced recursion of the Boolean vectors forms a probabilistic Boolean network. First of all, we show that the state transition of the induced Boolean network can be explicitly represented through a real version of the master equation. Next, when the open quantum dynamics are relaxing in the sense that they possess a unique equilibrium as a global attractor, structural properties including absorbing states, reducibility, and periodicity for the induced Boolean network are direct consequences of this relaxing property. Particularly, we show that generically, relaxing quantum dynamics leads to irreducible and aperiodic chains for the measurement outcomes. Finally, we show that for quantum consensus networks, which are a type of nonrelaxing open quantum network dynamics, the communication classes of the measurement-induced Boolean networks are encoded in the quantum Laplacian of the underlying interaction graph

In the near future, the road traffic flow will consist of both human-driven vehicles (HDVs) and connected autonomous vehicles (CAVs). Since HDVs cannot communicate with CAVs and road side units (RSU), they are unobservable to CAVs if outside the range of sensing. In such case, the advantages of CAVs will be compromised and various high-level tasks in mixed autonomous traffic flow cannot be achieved. This study proposes a model to infer HDVs information using sensing data of CAVs. The rationale is that CAVs react to HDVs based on the car-following (CF) logic. Inversely, real-time locations of HDVs can be reconstructed using the data from CAVs. The Bayesian network is used to reflect the CF logic and develop a real-time vehicle location inference method for both single and multiple CAVs scenarios. Last, the method is tested using real-world dataset. Both time consumption and near-field estimation precision are validated.

A model-free approach is presented, based on the Monte Carlo tree search (MCTS) algorithm, for the control of mixed traffic flow of human-driven vehicles (HDV) and connected and autonomous vehicles (CAV), named MCTS-MTF, on a one-lane roadway with signalized intersection control. Previous research has often simplified the problem with certain assumptions to reduce computational burden, such as dividing a vehicle trajectory into several segments with constant speed or linear acceleration/deceleration, which was rather unrealistic. This study departs from the existing research in that minimum constraints on CAV trajectory control were required, as long as the basic rules such as safety considerations and vehicular performance limits were followed. Modeling efforts were made to improve the algorithm solution quality and the run time efficiency over the naïve MCTS algorithm. This was achieved by an exploration-exploitation balance calibration module, and a tree expansion determination module to expand the tree more effectively along the desired direction. Results of a case study found that the proposed algorithm was able to achieve a travel time saving of 3.5% and a fuel consumption saving of 6.5%. It was also demonstrated to run at eight times the speed of a naïve MCTS model, suggesting a promising potential for real-time or near real-time applications.

Saturation flow rate (SFR) denotes the maximum sustainable flow rate during the green signal. Calibration of SRF is not a problem that can be solved once and for all. Due to various reasons such as degrading infrastructure or changes in the surrounding environment, a well-calibrated SFR could become outdated and it is expensive to recalibrate following traditional methods. This manuscript proposes a model to calculate saturation flow rate in a real-time fashion from loop detector-data that is readily available. The problem is formulated as a Markov Chain model with the goal of identifying traffic headway states. A total of five states and the transitional behavior are defined. The distribution of headway given the underlying state is also presented. The SFR estimation is converted to the identification of stable headway. The proposed model is tested and validated, which shows the proposed model is able to generate satisfactory results.

This paper considers the remote state estimation with multiple sensors. Each sensor transmits its sensing data to a remote estimator over a shared channel, where simultaneous transmissions are allowed. Regrading the transmission of other sensors as interference signals, the system designer should coordinate the sensors appropriately in order to maximize the overall estimation performance. Motivated by microeconomics, we treat sensors as self-interested power buyers under different unit prices announced by the system designer. Accordingly, the strategic interactions among sensors are formulated in a non-cooperative game, upon which the existence and uniqueness of a pure equilibrium solution are proved. Even if the game admits a conflict of interests among sensors, under well-designed prices, the game outcome aligns with the global optimal solution. We also devise an algorithm to compute these prices with simple iterations, which is given in explicit forms for ease of implementation. Numerical examples are given to demonstrate the developed results.

In this paper, we study the induced probabilistic Boolean dynamics for dynamical quantum networks subject to sequential quantum measurements. In this part of the paper, we focus on closed networks of quits whose states evolutions are described by a continuous Lindblad master equation. When measurements are performed sequentially along such continuous dynamics, the quantum network states undergo random jumps and the corresponding measurement outcomes can be described by a probabilistic Boolean network. First of all, we show that the state transition of the induced Boolean networks can be explicitly represented through realification of the master equation. Next, when the open quantum dynamics is relaxing in the sense that it possesses a unique equilibrium as a global attractor, structural properties including absorbing states, reducibility, and periodicity for the induced Boolean network are direct consequences of the relaxing property. Finally, we show that for quantum consensus networks as a type of non-relaxing open quantum network dynamics, the communication classes of the measurement-induced Boolean networks are encoded in the quantum Laplacian of the underlying interaction graph.

ABSTRACTRoad network bottleneck would shift around according to the demand or supply, and therefore behaves differently from static bottlenecks such as on–off ramp and lane drop. The identification of dynamic bottlenecks is crucial for management and control. Traditional methods use direct measurements. Such methods may be biased and can only analyze the recurrent or non-recurrent bottleneck, not both. This research uses rank measurement based on speed to identify bottlenecks. The underlying idea is that bottleneck links or areas have relatively lower rank, and neighboring links covered by the same congestion event should behave similarly with respect to the rank distribution. The identification procedure is implemented by three steps: firstly, the rank distribution is fit using Gaussian Mixture Model (GMM), and then the Kullback–Leibler (KL) divergence is employed to measure the similarity between adjacent links; at last, the bottleneck areas are clustered. Using rank information would enhance the bottleneck identification performance.

ABSTRACTA typical arterial consists of two sections: an upstream section (U-section) and a channelized section (C-section). When vehicles queue beyond the interface of the two sections, channelized section spillovers (CSSs) occur. This research investigates the dynamic behaviours of CSSs under various traffic scenarios, by utilizing an enhanced lane queuing model. The model is able to explicitly capture the queue profiles at signalized intersections when CSSs occur, and also reflect the influences of road geometry and first-in-first-out behaviour on CSSs. The major findings include: (1) CSSs may be ‘created’ by exiting CSSs; (2) CSSs may be either ‘erased’ or delayed by earlier CSSs; (3) due to the interactions among different CSSs, queues generally need time to stabilize, even when demand and supply are fixed. These uncovered characteristics of CSSs necessitate the re-modelling of delay, travel time and capacity at signalized intersections, under a more dynamic framework.

This paper discusses a distributed design for clustering based on the K-means algorithm in a switching multi-agent network, for the case when data are decentralized stored and unavailable to all agents. The authors propose a consensus-based algorithm in distributed case, that is, the double-clock consensus-based K-means algorithm (DCKA). With mild connectivity conditions, the authors show convergence of DCKA to guarantee a distributed solution to the clustering problem, even though the network topology is time-varying. Moreover, the authors provide experimental results on various clustering datasets to illustrate the effectiveness of the fully distributed algorithm DCKA, whose performance may be better than that of the centralized K-means algorithm.

Hot spotting is a thermal localisation phenomenon in which multiple hot regions form on a brake disc surface during high-energy and/or high-speed braking events. As an undesired problem, hot spots can result in high-order brake judder, an audible droning noise and thermal cracking. This paper presents a finite element model for hot-spot modelling which introduces the classical axisymmetric assumptions to the brake pad in three dimensions by scaling the material properties combined with a subroutine to simulate the heat generation instead of modelling the rotation of the brake pad. The results from the initial feasibility models showed significant improvement in the computing efficiency with acceptable accuracy when compared with a traditional finite element model without such simplifications. This method was then applied to three-dimensional simulations of hot spotting on a realistic ventilated brake disc–pad pair, and the results showed good correlation with the experiments. In order to improve the understanding of the hot-spotting mechanism, parametric studies were performed including the effects of a solid-disc geometry and a ventilated-disc geometry, the rotational speed and energy, the pins, the disc run-out and the brake pad length. Based on the analysis of the results, it was identified that the vents and the pins affected the hot-spot distribution. The speed was shown to be more important in the hot-spot generation time and the hot-spot distribution than either the pressure or the total energy input was. The brake disc run-out was shown to affect the magnitude of both the hot-spot temperature and the hot-spot height because of the non-linear relationship between the local deformation, the contact pressure and the heat generation. Finally, increasing the brake pad length generated fewer hot spots, but the temperature of each hot spot increased.

Given a static logical network, partition-based solutions are investigated. Easily verifiable necessary and sufficient conditions are obtained, and the corresponding formulas are presented to provide all types of the partition-based solutions. Then, the results are extended to mix-valued logical networks. Finally, two applications are presented: 1) an implicit function (IF) theorem of logical equations, which provides necessary and sufficient condition for the existence of IF and 2) converting the difference-algebraic network into a standard difference network. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive survey on semi-tensor product (STP) of matrices and its applications to different disciplines. First of all, the STP and its basic properties are introduced. Meanwhile, its inside physical meaning is explained. Second, its application to conventional dynamic systems is presented. As an example, the region of attraction of stable equilibriums is discussed. Third, its application to logical systems is presented. Particularly, the algebraic state space representation of logical systems and the important role it plays in analysis and control of logical systems are emphasized. Fourth, its application to finite games is discussed. The most interesting problems include potential game, evolutionary game, and game theoretic control. Finally, the mathematical essence of STP is briefly introduced.

There are important information security problems for system identification, which have not been paid attention to and well studied. In this paper, we discuss the security problem of input information according to a class of system identification problems, and present security protocols in the sense of cryptography. It is ensured that the input information is not leaked during solving the identification problem, which can deal with passive attacks effectively. In addition, a quantitative relationship between the input information and the number of participants is given in this paper. The simulation results show that, in the range of allowed errors, the identification algorithm with the security protocol achieves the estimation accuracy. Moreover, the corresponding time complexity increases comparing with the original identification algorithm.

ABSTRACTIn the near future, road traffic is expected to be a mixture of CAVs (connected and autonomous vehicles) and HDVs (human-driven vehicles). CAVs can be monitored by the system. By contrast, the information of HDVs is not readily available. Since full traffic flow topology information is valuable for applications such as data routing, this study proposes a method to infer the information of HDVs using CAV trajectories in partial VANET (Vehicular Ad Hoc Network). Specifically, two deep networks, i.e. VexSen and VelSen, are constructed to infer the existence and location of HDVs. Then, the networks are combined and fused into one sensing procedure. The proposed method was validated by conducting numerous experiments. The results show that HDV existence inferrence accuracy can reach 80%, whereas for the case of multiple CAVs at least the information of 20% HDVs can be inferred given the CAVs penetration rate of 20%.

Microscopic behavior modeling plays center role in traffic flow analyais, simulation, and autonomous vehicles algorithm development. Numerous efforts are devoted to the development of it in both longitudinal dimension and lateral dimension. Empirical observations reveal that the jerk (differential of acceleration) greatly influences traffic safety and a speed-dependent jerk profile exists regarding longitudinal and lateral movement. Replication of the speed-dependent jerk profile is crucial when the microscopic models are employed to the analysis of traffic safety. However, this research shows that current stochastic microscopic models cannot describe speed-dependent jerks, and thus cannot be directly used to describe driving behavior with considerable jerk profiles. This research firstly derives the jerk distribution for a general stochastic car following model, and then shows that several car following models together with lateral movement model cannot generate the realistic jerk distribution. A compound Poisson formulation is proposed to remedy the drawback of these models. The model consists of a diffusion part and a jump part. The former describes normal driving stochasticity while the latter describes driving involving high jerk. The numerical studies show that the proposed model can replicate the speed-dependent jerk phenomenon. The propagation of the behavior in the traffic flow is also investigated.

This note provides the detailed description of the decomposed subspaces of finite games. First, the basis of potential games and the basis of non-strategic games are revealed. Then the bases of pure potential and pure harmonic subspaces are also obtained. These bases provide an explicit formula for the decomposition, and are convenient for investigating the properties of the corresponding subspaces. As an application, we consider the dynamics of networked evolutionary games (NEGs). Three problems are considered: 1) the dynamic equivalence of evolutionary games; 2) the dynamics of near potential games; and 3) the decomposition of NEGs. [ABSTRACT FROM PUBLISHER]

A vector space structure is proposed for the set of finite games with fixed numbers of players and strategies for each players. Two statical equivalences are used to reduce the dimension of finite games. Under the vector space structure the subspaces of exact and weighted potential games are investigated. Formulas are provided to calculate them. Then the finite evolutionary games (EGs) are considered. Strategy profile dynamics is obtained using different strategy updating rules (SURs). Certain SURs, which assure the convergence of trajectories to pure Nash equilibriums, are investigated. Using the vector space structure, the projection of finite games to the subspace of exact (or weighted) potential games is considered, and a simple formula is given to calculate the projection. The convergence of near potential games to an e-equilibrium is studied. Further more, the Lyapunov function of EGs is defined and its application to the convergence of EGs is presented. Finally, the near potential function for an EG is defined, and it is proved that if the near potential function of an EG is a Lyapunov function, the EG will converge to a pure Nash equilibrium. Some examples are presented to illustrate the results.

Consider a networked evolutionary game (NEG). According to its strategy updating rule, a fundamental evolutionary equation (FEE) for each node is proposed, which is based on local information. Using FEEs, the network strategy profile dynamics (SPD) is expressed as a $k$-valued (deterministic or probabilistic) logical dynamic system. The SPD is then used to analyze the network dynamic behaviors, such as the fixed points, the cycles, and the basins of attractions, etc. Particularly, when the homogeneous networked games are considered, a necessary and sufficient condition is presented to verify when a stationary stable profile exists. Then the equivalence of two NEGs is investigated. Finally, after a rigorous definition of controlled NEGs, some control problems, including controllability, stabilization, and network consensus, are considered, and some verifiable conditions are presented. Examples with various games are presented to illustrate the theoretical results. The basic tool for this approach is the semi-tensor product (STP) of matrices, which is a generalization of the conventional matrix product. [ABSTRACT FROM AUTHOR]

The evolutionarily stable strategy (ESS) of networked evolutionary games (NEGs) is studied. Analyzing the ESS of infinite popular evolutionary games and comparing it with networked games, a new verifiable definition of ESS for NEGs is proposed. Then, the fundamental evolutionary equation (FEE) is investigated and used to construct the strategy profile dynamics (SPDs) of homogeneous NEGs. Two ways for verifying the ESS are proposed: 1) using the SPDs to verify it directly. The SPDs provides complete information about the NEGs, and then necessary and sufficient conditions are revealed. It can be used for NEGs with small size and 2) some sufficient conditions are proposed to verify the ESS of NEGs via their FEEs. This method is particularly suitable for large scale networks. Some illustrative examples are included to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper a comprehensive introduction for modeling and control of networked evolutionary games (NEGs) via semi-tensor product (STP) approach is presented. First, we review the mathematical model of an NEG, which consists of three ingredients: network graph, fundamental network game, and strategy updating rule. Three kinds of network graphs are considered, which are i) undirected graph for symmetric games; ii) directed graph for asymmetric games, and iii) d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games (FEGs) are discussed, which are i) two strategies and symmetric (S-2); ii) two strategies and asymmetric (A-2); and iii) three strategies and symmetric (S-3). Three strategy updating rules (SUR) are introduced, which are i) Unconditional Imitation (UI); ii) Fermi Rule(FR); iii) Myopic Best Response Adjustment Rule (MBRA). First, we review the fundamental evolutionary equation (FEE) and use it to construct network profile dynamics (NPD)of NEGs. To show how the dynamics of an NEG can be modeled as a discrete time dynamics within an algebraic state space, the fundamental evolutionary equation (FEE) of each player is discussed. Using FEEs, the network strategy profile dynamics (NSPD) is built by providing efficient algorithms. Finally, we consider three more complicated NEGs: i) NEG with different length historical information, ii) NEG with multi-species, and iii) NEG with time-varying payoffs. In all the cases, formulas are provided to construct the corresponding NSPDs. Using these NSPDs, certain properties are explored. Examples are presented to demonstrate the model constructing method, analysis and control design technique, and to reveal certain dynamic behaviors of NEGs.

This paper presents a comprehensive modeling technique for networked evolutionary games (NEG). Three kinds of network graphs are considered, which are (i) undirected graph for symmetric games; (ii) directed graph for asymmetric games, and (iii) d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games (FEGs) are discussed, which are (i) two strategies and symmetric (S-2);(ii) two strategies and asymmetric (A-2);and (iii) three strategies and symmetric (S-3).Three strategy updating rules (SUR) are introduced, which are (i) Unconditional Imitation (UI); (ii) Fermi Rule (FR); (iii) Myopic Best Response Adjustment Rule (MBRA). Then we review the fundamental evolutionary equation (FEE), and give the detailed formulation for different models. Finally, the network profile dynamics (NPD) of NEGs are investigated via their FEE.

Networked noncooperative games are investigated, where each player (or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors' current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed. The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems.

Boolean network is a powerful tool in describing the genetic regulatory networks. Accompanying the development of systems biology, the analysis and control of Boolean networks have attracted much attention from biologists, physicists, and systems scientists. This paper surveys a series of recent results on Boolean (control) networks, obtained by the algebraic approach proposed by the authors. First, a brief introduction to the semi-tensor product of matrices and the matrix expression of logic is presented. Then the topological structure of Boolean networks is investigated. Finally, the basic control problems of Boolean control networks are explored, which include the controllability, observability, realization, stability and stabilization, disturbance decoupling, identification and optimization.

A new matrix product, called semi-tensor product of matrices, is reviewed. Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping. Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system. Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor. The corresponding algorithms are developed and used to some examples. [ABSTRACT FROM PUBLISHER]

Theory of completeness is essential for multi-valued logical functions. Using semi-tensor product (STP) of matrices, the algebraic form of k -valued logical functions is presented. Using algebraic form, a method is proposed to construct an adequate set of connectives (ASC), consisting of unary operators with conjunction/disjunction for k -valued logical functions, which can be used to express any k -valued logical functions. Based on it, two normal forms of k -valued logical functions are presented, which are extensions of the disjunctive normal form and conjunctive normal form of Boolean functions respectively. The ASC is then simplified to a condensed set. Finally, the normal forms are further extended to mix-valued logical functions. [ABSTRACT FROM AUTHOR]