Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment

In this article, a new decision‐making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval‐valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.     

Control of stochastic discrete event systems modeled byprobabilistic languages

In Garg et al. (1999) and Garg (1992) the formalism of probabilistic languages for modeling the stochastic qualitative behavior of discrete event systems (DESs) was introduced. In this paper, we study their supervisory control where the control is exercised by dynamically disabling certain controllable events thereby nulling the occurrence probabilities of disabled events, and increasing the occurrence probabilities of enabled events proportionately. This is a special case of “probabilistic supervision” introduced in Lawford and Wonham (1993). The control objective is to design a supervisor such that the controlled system never executes any illegal traces (their occurrence probability is zero), and legal traces occur with minimum prespecified occurrence probabilities. In other words, the probabilistic language of the controlled system lies within a prespecified range, where the upper bound is a “nonprobabilistic language” representing a legality constraint. We provide a condition for the existence of a supervisor. We also present an algorithm to test this existence condition when the probabilistic languages are regular (so that they admit probabilistic automata representation with finitely many states). Next, we give a technique to compute a maximally permissive supervisor online     

Calculation of probabilistic testability measures for digital circuits with Structurally Synthesized BDDs

A method is proposed for probabilistic testability analysis of digital circuits focusing on calculating the probabilistic controllability measures in terms of signal probabilities with the goal of assessment of pseudorandom test quality in digital circuits. The structure of the circuit is modeled as a macro-level network, where macros denote Fan-out-Free Regions (FFRs) of the circuit, which are represented as Structurally Synthesized BDDs (SSBDDs). SSBDD based representation allows signal probability calculation with higher speed and accuracy than using gate-level calculation approach. The proposed method is based on tracing true paths in SSBDDs, which avoids errors caused by signals' correlation and possible redundancy in the circuit, that is not possible in gate-by-gate probability calculation. A method is proposed for proving redundancy of faults, which is an extension of the same idea of SSBDD path tracing used for probability calculation. Experimental results show higher accuracy and considerable speed-up of probabilistic analysis using the proposed new macro-level approach, compared to gate-level calculation.     

统计分析在系统级故障诊断中的应用

本文探讨统计分析在系统级故障诊断中的应用,提出两个概率诊断模型,讨论了给出的两个模型的测试算法及有关测试数据的统计推断。     

Algebraic theory of probabilistic processes

In this paper we extend de Nicola and Hennessy’s testing theory to deal with probabilities. We say that two processes are testing equivalent if the probabilities with which they pass any test are equal. We present three alternative semantic views of our testing equivalence. First, we introduce adequate extensions of acceptance sets (inducing an operational characterization) and acceptance trees (inducing a denotational semantics). We also present a sound and complete axiomatization of our testing equivalence. So, this paper represents a complete study of the adaptation of the classical testing theory for probabilistic processes.     

一种新的缩短自测试序列长度的方法

本文建立在逻辑电路内部自测试的基础上，提出了一种新的缩短伪随机测试序列长度的方法。文中首先找到了最难测故障在电路中的分布，建立了对应于最难测故障的电路模型，然后用线性反馈移位寄存器对这些电路模型的输出信号进行压缩，通过分析压缩后的特征码，得出最难测故障的测试长度。最后利用电路的原始输入概率与测试长度之间的关系，提出了一种缩短测试序列长度的算法，求出了最短的测试长度与最佳的输入概率。     

Optimal adaptive bandwidth monitoring for QoS based retrieval

Network aware multimedia delivery applications are a class of applications that provide certain level of quality of service (QoS) guarantees to end users while not assuming underlying network resource reservations. These applications guarantee QoS parameters like media object transmission time limit by actively monitoring the available bandwidth of the network and adapting the object to a target size that can be transmitted within a given time limit. A critical problem is how to obtain an accurate enough estimation of available bandwidth while not wasting too much time in bandwidth testing. In this paper, we present an algorithm to determine optimal amount of bandwidth testing given a probabilistic confidence level for network-aware multimedia object retrieval applications. The model treats the bandwidth testing as sampling from an actual bandwidth population. It uses statistical estimation method to quantify the benefit of each new bandwidth-testing sample, which is used to determine the optimal amount of bandwidth testing by balancing the benefit with the cost of each sample. Our implementation and experiments shows the algorithm determines the optimal amount of bandwidth testing effectively with minimum computation overhead.     

Fault diagnosis of a logical circuit by use of a pseudorandom signal and a neural network

This paper describes a new method of pseudorandom testing of a digital circuit by use of a correlation method and a neural network. The authors have recently proposed a new method of fault diagnosis in a logical circuit by applying a pseudorandom M-sequence to the circuit under test, calculating the cross-correlation function between the input and the output, and comparing the cross-correlation functions with the references. This method, called the M-sequence correlation (MSEC) method, is further extended by using a neural network in order not only to detect the existence of faults, but also to find the place or location of the faults. The authors investigated the effects of using parts of the fault patterns to train the neural network to be able to detect faults. It is shown that more than 95% of faults can be detected even when only 60% of the possible training data are used.     

Modular neural networks for MAP classification of time series andthe partition algorithm

We apply the partition algorithm to the problem of time-series classification. We assume that the source that generates the time series belongs to a finite set of candidate sources. Classification is based on the computation of posterior probabilities. Prediction error is used to adaptively update the posterior probability of each source. The algorithm is implemented by a hierarchical, modular, recurrent network. The bottom (partition) level of the network consists of neural modules, each one trained to predict the output of one candidate source. The top (decision) level consists of a decision module, which computes posterior probabilities and classifies the time series to the source of maximum posterior probability. The classifier network is formed from the composition of the partition and decision levels. This method applies to deterministic as well as probabilistic time series. Source switching can also be accommodated. We give some examples of application to problems of signal detection, phoneme, and enzyme classification. In conclusion, the algorithm presented here gives a systematic method for the design of modular classification networks. The method can be extended by various choices of the partition and decision components.     

Robust sampled-data H ∞ output tracking control for a class of nonlinear networked systems with stochastic sampling

In this article, the problem of robust sampled-data H _∞ output tracking control is investigated for a class of nonlinear networked systems with stochastic sampling and time-varying norm-bounded uncertainties. For the sake of technical simplicity, only two different sampling periods are considered, their occurrence probabilities are given constants and satisfy Bernoulli distribution, and can be extended to the case with multiple stochastic sampling periods. By the way of an input delay, the probabilistic system is transformed into a stochastic continuous time-delay system. A new linear matrix inequality-based procedure is proposed for designing state-feedback controllers, which would guarantee that the closed-loop networked system with stochastic sampling tracks the output of a given reference model well in the sense of H _∞. Conservatism is reduced by taking the probability into account. Both network-induced delays and packet dropouts have been considered. Finally, an illustrative example is given to show the usefulness and effectiveness of the proposed H _∞ output tracking design.     

Observer-based residual generation for fault diagnosis for non-affine non-linear polynomial systems

This paper proposes an observer-based residual generator (OBRG) for diagnosing faults in a continuous non-affine system with polynomial non-linearities up to any finite degree where the fault and an unknown input affect both the system and part of the output. Firstly, given certain assumptions and the use of defined extended vectors, a parameterized polynomial system is considerred for which a compact set of sufficient conditions is given for the existence of a candidate OBRG. Conditions for error stability (by a Lyapunov method) and detectability are given. The calculation steps in the design of the OBRG are shown to involve the solution of three linear equations (with parameterizations) and the calculation of a set of constant matrices (for detectability of faults). A result is then given establishing that the design holds for a much wider class of systems. The residual design is applied to a real three-tank system.     

Software testability measurement for intelligent assertion placement

Voas defines software testability as the degree to which software reveals faults during testing. This software characteristic is important when determining how to best apply verification techniques and build quality assurance plans. When testability is low, testers often want advice on how to increase it. In this paper, we describe the use of testability measures (using VoasÕs definition) for intelligent assertion placement. Software assertions are one relatively simple trick for improving testability. VoasÕs perspective on what software testability is has been implemented via three algorithms that together comprise a technique termed Ôsensitivity analysisÕ. Sensitivity analysis analyses how likely a test scheme is to (1) propagate data state errors to the output space, (2) cause internal states to become corrupted when faults are exercised, and (3) exercise the code. By knowing where faults appear likely to hide from a particular test scheme, we have insight into where internal tests (assertions) are particularly beneficial. This paper explores using the one sensitivity analysis algorithm that measures propagation as a heuristic for where and how to inject software assertions.     

Formally transforming user-model testing problems into implementer-model testing problems and viceversa

There are several methods to assess the capability of a test suite to detect faults in a potentially wrong system. We explore two methods based on considering some probabilistic information. In the first one, we assume that we are provided with a probabilistic user model. This is a model denoting the probability that the entity interacting with the system takes each available choice. In the second one, we suppose that we have a probabilistic implementer model, that is, a model denoting the probability that the implementer makes each possible fault while constructing the system. We show that both testing scenarios are strongly related. In particular, we prove that any user can be translated into an implementer model in such a way that the optimality of tests is preserved, that is, a test suite is optimal for the user if and only if it is optimal for the resulting implementer. Another translation, working in the opposite direction, fulfills the reciprocal property. Thus, we conclude that any test selection criterium designed for one of these testing problems can be used for the other one, once the model has been properly translated. Besides, the applicability of user models to other kinds of testing approaches is considered.     

Mutation testing in UTP

This paper presents a theory of testing that integrates into Hoare and He’s Unifying Theory of Programming (UTP). We give test cases a denotational semantics by viewing them as specification predicates. This reformulation of test cases allows for relating test cases via refinement to specifications and programs. Having such a refinement order that integrates test cases, we develop a testing theory for fault-based testing. Fault-based testing uses test data designed to demonstrate the absence of a set of pre-specified faults. A well-known fault-based technique is mutation testing. In mutation testing, first, faults are injected into a program by altering (mutating) its source code. Then, test cases that can detect these errors are designed. The assumption is that other faults will be caught, too. In this paper, we apply the mutation technique to both, specifications and programs. Using our theory of testing, two new test case generation laws for detecting injected (anticipated) faults are presented: one is based on the semantic level of UTP design predicates, the other on the algebraic properties of a small programming language.     

State estimation and detectability of probabilistic discrete event systems

A probabilistic discrete event system (PDES) is a nondeterministic discrete event system where the probabilities of nondeterministic transitions are specified. State estimation problems of PDES are more difficult than those of non-probabilistic discrete event systems. In our previous papers, we investigated state estimation problems for non-probabilistic discrete event systems. We defined four types of detectabilities and derived necessary and sufficient conditions for checking these detectabilities. In this paper, we extend our study to state estimation problems for PDES by considering the probabilities. The first step in our approach is to convert a given PDES into a nondeterministic discrete event system and find sufficient conditions for checking probabilistic detectabilities. Next, to find necessary and sufficient conditions for checking probabilistic detectabilities, we investigate the “convergence” of event sequences in PDES. An event sequence is convergent if along this sequence, it is more and more certain that the system is in a particular state. We derive conditions for convergence and hence for detectabilities. We focus on systems with complete event observation and no state observation. For better presentation, the theoretical development is illustrated by a simplified example of nephritis diagnosis.     

Probabilistic approaches to fault detection in networked discrete event systems

In this paper, we consider distributed systems that can be modeled as finite state machines with known behavior under fault-free conditions, and we study the detection of a general class of faults that manifest themselves as permanent changes in the next-state transition functionality of the system. This scenario could arise in a variety of situations encountered in communication networks, including faults occurred due to design or implementation errors during the execution of communication protocols. In our approach, fault diagnosis is performed by an external observer/diagnoser that functions as a finite state machine and which has access to the input sequence applied to the system but has only limited access to the system state or output. In particular, we assume that the observer/diagnoser is only able to obtain partial information regarding the state of the given system at intermittent time intervals that are determined by certain synchronizing conditions between the system and the observer/diagnoser. By adopting a probabilistic framework, we analyze ways to optimally choose these synchronizing conditions and develop adaptive strategies that achieve a low probability of aliasing, i.e., a low probability that the external observer/diagnoser incorrectly declares the system as fault-free. An application of these ideas in the context of protocol testing/classification is provided as an example.     

Robust probabilistic planning with ilao

In probabilistic planning problems which are usually modeled as Markov Decision Processes (MDPs), it is often difficult, or impossible, to obtain an accurate estimate of the state transition probabilities. This limitation can be overcome by modeling these problems as Markov Decision Processes with imprecise probabilities (MDP-IPs). Robust LAO* and Robust LRTDP are efficient algorithms for solving a special class of MDP-IPs where the probabilities lie in a given interval, known as Bounded-Parameter Stochastic-Shortest Path MDP (BSSP-MDP). However, they do not make clear what assumptions must be made to find a robust solution (the best policy under the worst model). In this paper, we propose a new efficient algorithm for BSSP-MDPs, called Robust ILAO* which has a better performance than Robust LAO* and Robust LRTDP, considered the-state-of-the art of robust probabilistic planning. We also define the assumptions required to ensure a robust solution and prove that Robust ILAO* algorithm converges to optimal values if the initial value of all states is admissible.     

A constructive method to static output stabilisation of Markov jump systems

This paper is concerned with the static output feedback stabilisation of Markov jump systems. Transition probabilities are assumed to be incomplete, namely, they may be known, uncertain with known lower and upper bounds and unknown. To linearise the nonlinearities induced by unknown transition probabilities, a linearisation is developed. To solve the static output feedback problem by means of linear matrix inequalities, a constructive method is proposed to separate the controller gain matrices from Lyapunov variables. Based on the linearisation and the separation, sufficient conditions are established to guarantee that the closed-loop system is stochastically stable by the designed static output feedback controller. The obtained results are further extended to deal with norm-bounded or polytopic uncertainties on system matrices. Numerical examples are given to demonstrate the effectiveness of the proposed method.     

Fault Detection Schemes for Continuous-Time Stochastic Dynamical Systems

In this paper the fault detection (FD) task in stochastic continuous-time dynamical systems is addressed. A new family of FD approaches is presented, which is based on the application of hypothesis testing on continuous-time estimators. The given FD schemes are widely analyzed in the framework of their characteristics, such as fault detectability, false alarms and missed detection. A collection of sufficient detectability conditions are given for a class of faults (referred here as generic), characterizing the faults which can be detected with certain formalized guarantee by the given FD schemes, and providing also an upper bound for the detection time in a probabilistic sense. The application and comparative performance of these FD approaches is illustrated for different faults in a simulation example.