Integral and probabilistic representations for systems of elliptic equations

We present probabilistic representations for some systems of elliptic equations constructed as expectations of functionals of some specific Markov chains, in particular, the walk on spheres processes. These representations are deduced from converse mean value theorems that we prove for the equations under analysis, especially the Lamé equation of elasticity theory. We construct Monte Carlo algorithms, and estimate the variances and the cost.     

Approximate representation of probabilistic data in expert systems

We propose the use of a third-order approximation for the representation of probabilistic data in expert systems and compare it to tree-structured representations. The differences are illustrated using the example of a reliability problem. We show that using the third-order representation results in significantly reduced losses as compared to tree structures, with a small increase in computational complexity. We present heuristic and exact techniques to determine the optimal third-order representation and propose a decomposition technique that allows the exact algorithm to be efficiently used for solving large problem instances.     

Basic probabilistic and statistical concepts for maintenance of parts and systems

Mathematical models are presented which are useful for determiningwhen replacement or maintenance is needed. In addition, techniquesfor assessing the efficacy of maintenance and/or overhaul arediscussed. Since the underlying concepts and techniques fornonrepairable items are relatively well known, attention isfocused on repairable items. Moreover, great emphasis is placedon the major differences between the concepts, probabilisticmodels, and statistical analysis techniques appropriate fornonrepairable and repairable items respectively. Such emphasisis still required because the superficial similarities betweennonrepairable and repairable items have contributed to the widespreaduse of poor terminology and notation which, in turn, make thesimilarities appear to be substantive, rather than just superficial.This vicious circle—-which is still evident in most currentreliability texts and standards—-must be broken, and thispaper is intended to contribute to this campaign. It is alsostressed that, even to the very limited extent that repairablesystems concepts and techniques are discussed in the literature,excessive emphasis is placed on reliability growth or improvement.This has resulted in even less understanding of basic notionsof repairable-systems deterioration, i.e. of basic conceptsassociated with systems maintenance. This paper focuses on conceptsconnected with systems maintenance to help rectify this imbalance.Nonetheless, it is also stressed that the same models (withdifferent parameters) can often be used for both situations.     

Cost and probabilistic analysis of series systems with mixed standby components

This paper deals with the reliability and availability characteristics of four different series system configurations with mixed standby (include cold standby and warm standby) components. The failure times of the primary and warm standby components are assumed to be exponentially distributed with parameters λ and , respectively. The repair time distribution of each server is also exponentially distributed with parameter μ. We derive the mean time-to-failure, MTTF, and the steady-state availability, A_T(∞), for four configurations and perform comparisons. For all four configurations, comparisons are done for specific values of distribution parameters and of the cost of the components. Finally, the configurations are ranked based on: MTTF, A_T(∞), and cost/benefit where benefit is either MTTF or A_T(∞).     

A framework for the probabilistic analysis of hierarchical planning systems

As we have argued in previous papers, multi-level decision problems can often be modeled as multi-stage stochastic programs, and hierarchical planning systems designed for their solution, when viewed as stochastic programming heuristics, can be subjected to analytical performance evaluation. The present paper gives a general formulation of such stochastic programs and provides a framework for the design and analysis of heuristics for their solution. The various ways to measure the performance of such heuristics are reviewed, and some relations between these measures are derived. Our concepts are illustrated on a simple two-level planning problem of a general nature and on a more complicated two-level scheduling problem.     

Constraint-based analysis of concurrent probabilistic hybrid systems: An application to networked automation systems

In previous publications, the authors have introduced the notion of stochastic satisfiability modulo theories (SSMT) and the corresponding SiSAT solving algorithm, which provide a symbolic method for the reachability analysis of probabilistic hybrid systems. SSMT extends satisfiability modulo theories (SMT) with randomized (or stochastic), existential, and universal quantification, as known from stochastic propositional satisfiability. In this paper, we extend the SSMT-based procedures to the symbolic analysis of concurrent probabilistic hybrid systems. After formally introducing the computational model, we provide a mechanized translation scheme to encode probabilistic bounded reachability problems of concurrent probabilistic hybrid automata as linearly sized SSMT formulae, which in turn can be solved by the SiSAT tool. We furthermore propose an algorithmic enhancement which tailors SiSAT to probabilistic bounded reachability problems by caching and reusing solutions obtained on bounded reachability problems of smaller depth. An essential part of this article is devoted to a case study from the networked automation systems domain. We explain in detail the formal model in terms of concurrent probabilistic automata, its encoding into the SiSAT modeling language, and finally the automated quantitative analysis.     

(2, 1)-Separating systems beyond the probabilistic bound

Building on previous results of Xing, we give new lower bounds on the rate of linear intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry (although nothing beyond the basic theory of linear systems on curves). Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years.     

Temporal logic control for stochastic linear systems using abstraction refinement of probabilistic games

We consider the problem of computing the set of initial states of a dynamical system such that there exists a control strategy to ensure that the trajectories satisfy a temporal logic specification with probability 1 (almost-surely). We focus on discrete-time, stochastic linear dynamics and specifications given as formulas of the Generalized Reactivity(1) fragment of Linear Temporal Logic over linear predicates in the states of the system. We propose a solution based on iterative abstraction-refinement, and turn-based 2-player probabilistic games. While the theoretical guarantee of our algorithm after any finite number of iterations is only a partial solution, we show that if our algorithm terminates, then the result is the set of all satisfying initial states. Moreover, for any (partial) solution our algorithm synthesizes witness control strategies to ensure almost-sure satisfaction of the temporal logic specification. While the proposed algorithm guarantees progress and soundness in every iteration, it is computationally demanding. We offer an alternative, more efficient solution for the reachability properties that decomposes the problem into a series of smaller problems of the same type. All algorithms are demonstrated on an illustrative case study.     

Hierarchical multilevel optimization for reliability target allocation in probabilistic design of decomposed systems

Analytical target cascading (ATC) is a methodology for translating system-level design targets to subsystem and component design specifications in hierarchically decomposed optimal system design problems. In previous work we extended the ATC formulation to account for uncertainties, where the bounds of the probabilistic design constraints were chosen arbitrarily and held fixed during the ATC process. In this work, we extend the probabilistic ATC formulation to include reliability targets in the vector of cascaded quantities. In this manner, we quantify the optimality-reliability tradeoffs for each element of the decomposed system and compute the probabilistic constraint bounds required to satisfy the overall system reliability target. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deterministic and probabilistic discrepancies

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the d-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension d under consideration.     

Robust sampled‐data control of uncertain switched neutral systems with probabilistic input delay

This article focuses on the robust sampled‐data control for a class of uncertain switched neutral systems based on the average dwell‐time approach. In particular, the system is considered with probabilistic input delay using sampled state vectors, which are described by the stochastic variables with a Bernoulli distributed white sequence and time‐varying norm‐bounded uncertainties. By constructing a novel Lyapunov–Krasovskii functional which involves the lower and upper bounds of the delay, a new set of sufficient conditions are derived in terms of linear matrix inequalities for ensuring the robust exponential stability of the uncertain switched neutral system about its equilibrium point. Moreover, based on the stability criteria, a state feedback sampled‐data control law is designed for the considered system. Finally, a numerical example based on the water‐quality dynamic model for the Nile River is given to illustrate the effectiveness of the proposed design technique. © 2015 Wiley Periodicals, Inc. Complexity 21: 308–318, 2016     

Monotone and probabilistic wavelet approximation

Continuous functions are approximated by wavelet operators. These preserve mone tonicity and transform continuous probability distribution functions into probability distribution functions. The degree of this approximation is estimated by establishing some Jackson type inequalities     

Some properties for probabilistic and multinomial (probabilistic) values on cooperative games

We investigate the conditions for the coefficients of probabilistic and multinomial values of cooperative games necessary and/or sufficient in order to satisfy some properties, including marginal contributions, balanced contributions, desirability relation and null player exclusion property. Moreover, a similar analysis is conducted for transfer property of probabilistic power indices on the domain of simple games.     

Structured probabilistic inference

Probabilistic inference is among the main topics with reasoning in uncertainty in AI. For this purpose, Bayesian Networks (BNs) is one of the most successful and efficient Probabilistic Graphical Model (PGM) so far. Since the mid-90s, a growing number of BNs extensions have been proposed. Object-oriented, entity-relationship and first-order logic are the main representation paradigms used to extend BNs. While entity-relationship and first-order models have been successfully used for machine learning in defining lifted probabilistic inference, object-oriented models have been mostly underused. Structured inference, which exploits the structural knowledge encoded in an object-oriented PGM, is a surprisingly unstudied technique. In this paper we propose a full object-oriented framework for PRM and propose two extensions of the state-of-the-art structured inference algorithm: SPI which removes the major flaws of existing algorithms and SPISBB which largely enhances SPI by using d-separation.     

Exponential probabilistic inequalities

We extend the mapping properties of martingale transforms, decoupling inequalities, differential subordination, and the Stein inequalities to exponential Orlicz spaces.     

Interior gradient and Hessian estimates for the Dirichlet problem of semi-linear degenerate elliptic systems: A probabilistic approach

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.     

Robust consensus of nonlinear multi‐agent systems via reliable control with probabilistic time delay

In this paper, the consensus problem of uncertain nonlinear multi‐agent systems is investigated via reliable control in the presence of probabilistic time‐varying delay. First, the communication topology among the agents is assumed to be directed and fixed. Second, by introducing a stochastic variable which satisfies Bernoulli distribution, the information of probabilistic time‐varying delay is equivalently transformed into the deterministic time‐varying delay with stochastic parameters. Third, by using Laplacian matrix properties, the consensus problem is converted into the conventional stability problem of the closed‐loop system. The main objective of this paper is to design a state feedback reliable controller such that for all admissible uncertainties as well as actuator failure cases, the resulting closed‐loop system is robustly stable in the sense of mean‐square. For this purpose, through construction of a suitable Lyapunov–Krasovskii functional containing four integral terms and utilization of Kronecker product properties along with the matrix inequality techniques, a new set of delay‐dependent consensus stabilizability conditions for the closed‐loop system is obtained. Based on these conditions, the desired reliable controller is designed in terms of linear matrix inequalities which can be easily solved by using any of the effective optimization algorithms. Moreover, a numerical example and its simulations are included to demonstrate the feasibility and effectiveness of the proposed control design scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 138–150, 2016     

Logic and Religion

We first start by describing the happening of the 1st World Congress on Logic and Religion. We then explain the motivation for developing the interaction between logic and religion. In a third part we discuss some papers presented at this event published in the present special issue.