Bounded State Space Research Articles

Convergence Rates of Attractive-Repulsive MCMC Algorithms

We consider MCMC algorithms for certain particle systems which include both attractive and repulsive forces, making their convergence analysis challenging. We prove that a version of these algorithms on a bounded state space is uniformly ergodic with explicit quantitative convergence rate. We also prove that a version on an unbounded state space is still geometrically ergodic, and then use the method of shift-coupling to obtain an explicit quantitative bound on its convergence rate.

Card-Based Cryptography Meets Formal Verification

Card-based cryptography provides simple and practicable protocols for performing secure multi-party computation with just a deck of cards. For the sake of simplicity, this is often done using cards with only two symbols, e.g., clubsuit and heartsuit . Within this paper, we also target the setting where all cards carry distinct symbols, catering for use-cases with commonly available standard decks and a weaker indistinguishability assumption. As of yet, the literature provides for only three protocols and no proofs for non-trivial lower bounds on the number of cards. As such complex proofs (handling very large combinatorial state spaces) tend to be involved and error-prone, we propose using formal verification for finding protocols and proving lower bounds. In this paper, we employ the technique of software bounded model checking (SBMC), which reduces the problem to a bounded state space, which is automatically searched exhaustively using a SAT solver as a backend. Our contribution is threefold: (a) we identify two protocols for converting between different bit encodings with overlapping bases, and then show them to be card-minimal. This completes the picture of tight lower bounds on the number of cards with respect to runtime behavior and shuffle properties of conversion protocols. For computing AND, we show that there is no protocol with finite runtime using four cards with distinguishable symbols and fixed output encoding, and give a four-card protocol with an expected finite runtime using only random cuts. (b) We provide a general translation of proofs for lower bounds to a bounded model checking framework for automatically finding card- and run-minimal (i.e., the protocol has a run of minimal length) protocols and to give additional confidence in lower bounds. We apply this to validate our method and, as an example, confirm our new AND protocol to have its shortest run for protocols using this number of cards. (c) We extend our method to also handle the case of decks on symbols clubsuit and heartsuit , where we show run-minimality for two AND protocols from the literature.

Аналіз та оптимізація систем з повторними викликами і чергою зі змінною інтенсивністю вхідного потоку

The paper is focused on in-depth study of the promising area of the stochastic systems theory related with scrutiny of queuing systems with repeated calls. We research Markov`s models of retrial systems with queue and variable rate of input flow controlled by threshold strategy with no restriction on the capacity of the orbit. We defined stationary regime existence conditions and investigated probability characteristics of process for two-dimension Markov process with continuous time which we took as a main model of the specified system. In stationary regime for probability characteristics of the service process were found explicit formulas. Research methods which we used are based on the initial process approximation by the process with bounded state space. Results of the research allow us to evaluate convergence rates of stationary distribution of finite systems with repeated calls to stationary distribution of infinite systems. Method of probability flow equating is used for obtain explicit expressions for stationary system probabilities through the closed path which are defined in a special way. For threshold control strategies the optimization problem of the total income of the system was stated and solved.

First-hitting times under drift

For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected first-hitting time of a random process by bounding expected local changes of the process – the drift. This is usually far easier than bounding the expected first-hitting time directly.Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be lower-bounded, that is, we remove unnecessary restrictions like a finite, discrete, or bounded state space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift. By bounding the step size of the process, we derive new lower-bounding multiplicative and variable drift theorems. Last, we also state theorems that are applicable when the process has a drift of 0, by using a drift on the variance of the process.

Constrained BSDEs Driven by a Non-Quasi-Left-Continuous Random Measure and Optimal Control of PDMPs on Bounded Domains

We consider an optimal control problem for piecewise deterministic Markov processes (PDMPs) on a bounded state space. A pair of controls acts continuously on the deterministic flow and on the two transition measures (in the interior and from the boundary of the domain) describing the jump dynamics of the process. For this class of control problems, the value function can be characterized as the unique viscosity solution to the corresponding fully nonlinear Hamilton--Jacobi--Bellman equation with a nonlocal type boundary condition. By means of the recent control randomization method, we are able to provide a probabilistic representation for the value function in terms of a constrained backward stochastic differential equation (BSDE), known as the nonlinear Feynman--Kac formula. This result considerably extends the existing literature, where only the case with no jumps from the boundary is considered. The additional boundary jump mechanism is described in terms of a non-quasi-left-continuous random measure and induces predictable jumps in the PDMP's dynamics. The existence and uniqueness results for BSDEs driven by such a random measure are nontrivial, even in the unconstrained case, as emphasized in the recent work [E. Bandini, Electron. Commun. Probab., 20 (2015), pp. 1--13].

Швидкість збіжності стаціонарного розподілу системи з повторними викликами і чергою

This paper describes a steady state behavior of the retrial system in the case of one server, one place in the queue and an infinity orbit. We research Markov`s models of retrial systems and variable rate of input flow controlled by threshold strategy. We defined stationary regime existence conditions and investigated probability characteristics of process for two-dimension Markov process with continuous time which we took as a main model of the specified system. In stationary regime for probability characteristics of the service process were found explicit formulas. Research methods which we used are based on the initial process approximation by the process with bounded state space. Results of the research allow us to evaluate convergence rates of stationary distribution of finite systems with repeated calls to stationary distribution of infinite systems. Method of probability flow equating is used for obtain explicit expressions for stationary system probabilities through the closed path which are defined in a special way. We considered model for one service devices and one place in the queue, which are controlled by threshold strategies.

Bounded Brownian Motion

Diffusions are widely used in finance due to their tractability. Driftless diffusions are needed to describe ratios of asset prices under a martingale measure. We provide a simple example of a tractable driftless diffusion which also has a bounded state space.

Suppressing escape events in maps of the unit interval with demographic noise.

We explore the properties of discrete-time stochastic processes with a bounded state space, whose deterministic limit is given by a map of the unit interval. We find that, in the mesoscopic description of the system, the large jumps between successive iterates of the process can result in probability leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating these escape events, and we compare the results with those obtained for the metastable behavior of the microscopic model, where escape events are not possible. We find that, rather than truncating the noise distribution, censoring this distribution to prevent escape events leads to results which are more consistent with the microscopic model. The addition of higher moments to the noise distribution does not increase the accuracy of the final results, and it can be replaced by the simpler Gaussian noise.

Goal-oriented state control of a cognitive linear model with a bounded state space

Representing the basic concept of this paper, cognitive map is used to construct a cognitive linear dynamic model with a bounded state space. We consider the problem of transferring this model from an arbitrary initial state to some asymptotically stable state belonging to a neighborhood of a given state. We suggest two classes of controls and introduce the transfer "quality" as the proximity of the resulting steady state to the desired state. And finally, an illustrative example is provided.

Bounded Auditable Restoration of Distributed Systems

We focus on protocols for auditable restoration of distributed systems. The need for such protocols arises due to conflicting requirements (e.g., access to the system should be restricted but emergency access should be provided). One can design such systems with a tamper detection approach (based on the intuition of In-case-of-emergency-break-glass ). However, in a distributed system, such tampering, which are denoted as auditable events, is visible only for a single node. This is unacceptable since the actions they take in these situations can be different than those in the normal mode. Moreover, eventually, the auditable event needs to be cleared so that system resumes the normal operation. With this motivation, in this paper, we present two protocols for auditable restoration, where any process can potentially identify an auditable event. The first protocol has an unbounded state space while the second protocol uses bounded state space that does not increase with the length of the computation. In both protocols, whenever a new auditable event occurs, the system must reach an auditable state where every process is aware of the auditable event. Only after the system reaches an auditable state, it can begin the operation of restoration. Although any process can observe an auditable event, we require that only authorized processes can begin the task of restoration. Moreover, these processes can begin the restoration only when the system is in an auditable state. Our protocols are self-stabilizing and can effectively handle the case where faults or auditable events occur during the restoration protocol. Moreover, they can be used to provide auditable restoration to other distributed protocols.

Bounded saturation-based CTL model checking

Formal verification is becoming a fundamental step of safety-critical and model-based software development. As part of the verification process, model checking is one of the current advanced techniques to analyze the behavior of a system. Symbolic model checking is an efficient approach to handling even complex models with huge state spaces. Saturation is a symbolic algorithm with a special iteration strategy, which is efficient for asynchronous models. Recent advances have resulted in many new kinds of saturation-based algorithms for state space generation and bounded state space generation and also for structural model checking. In this paper, we examine how the combination of two advanced model checking algorithms – bounded saturation and saturation-based structural model checking – can be used to verify systems. Our work is the first attempt to combine these approaches, and this way we are able to handle and examine complex or even infinite state systems. Our measurements show that we can exploit the efficiency of saturation in bounded model checking.

Adaptive Regularized Particle Filter for Synchronization of Chaotic Colpitts Circuits in an AWGN Channel

For chaotic trajectories, when the system parameters are fixed, they are generally confined in a bounded state space. In this paper, we propose an adaptive regularized particle filter (RPF), which makes the best of this inherent characteristic, for identical synchronization of chaotic Colpitts circuits combating additive white Gaussian noise (AWGN) channel distortion. This proposed filter incorporates RPF that resamples from a continuous approximation of the posterior density to avoid sample impoverishment and then utilizes the revised Kullback–Leibler distance (KLD) sampling to adaptively select the number of particles used. Compared with the existing particle filters (PFs) with fixed large number of particles, this proposed adaptive RPF propagates less number of particles with similar performance and thus provides a much more efficient solution for this problem.

HPobSAM for modeling and analyzing IT Ecosystems – Through a case study

The next generation of software systems includes systems composed of a large number of distributed, decentralized, autonomous, interacting, cooperating, organically grown, heterogeneous, and continually evolving subsystems, which we call IT Ecosystems. Clearly, we need novel models and approaches to design and develop such systems which can tackle the long-term evolution and complexity problems. In this paper, our framework to model IT Ecosystems is a combination of centralized control (top-down) and self-organizing (bottom-up) approach. We use a flexible formal model, HPobSAM, that supports both behavioral and structural adaptation/evolution. We use a detailed, close to real-life, case study of a smart airport to show how we can use HPobSAM in modeling, analyzing and developing an IT Ecosystem. We provide an executable formal specification of the model in Maude, and use LTL model checking and bounded state space search provided by Maude to analyze the model. We develop a prototype of our case study designed by HPobSAM using Java and Ponder2. Due to the complexity of the model, we cannot check all properties at design time using Maude. We propose a new approach for run-time verification of our case study, and check different types of properties which we could not verify using model checking. As our model uses dynamic policies to control the behavior of systems which can be modified at runtime, it provides us a suitable capability to react to the property violation by modification of policies.

Admission control for a multi-server queue with abandonment

In a M/M/N+M queue, when there are many customers waiting, it may be preferable to reject a new arrival rather than risk that arrival later abandoning without receiving service. On the other hand, rejecting new arrivals increases the percentage of time servers are idle, which also may not be desirable. We address these trade-offs by considering an admission control problem for a M/M/N+M queue when there are costs associated with customer abandonment, server idleness, and turning away customers. First, we formulate the relevant Markov decision process (MDP), show that the optimal policy is of threshold form, and provide a simple and efficient iterative algorithm that does not presuppose a bounded state space to compute the minimum infinite horizon expected average cost and associated threshold level. Under certain conditions we can guarantee that the algorithm provides an exact optimal solution when it stops; otherwise, the algorithm stops when a provided bound on the optimality gap is reached. Next, we solve the approximating diffusion control problem (DCP) that arises in the Halfin---Whitt many-server limit regime. This allows us to establish that the parameter space has a sharp division. Specifically, there is an optimal solution with a finite threshold level when the cost of an abandonment exceeds the cost of rejecting a customer; otherwise, there is an optimal solution that exercises no control. This analysis also yields a convenient analytic expression for the infinite horizon expected average cost as a function of the threshold level. Finally, we propose a policy for the original system that is based on the DCP solution, and show that this policy is asymptotically optimal. Our extensive numerical study shows that the control that arises from solving the DCP achieves a very similar cost to the control that arises from solving the MDP, even when the number of servers is small.

Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics

AbstractWe study equilibrium states in relativistic galactic dynamics which are described by stationary solutions of the Einstein–Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses.

State-constrained agile missile control with adaptive-critic-based neural networks

In this study, we develop an adaptive-critic-based controller to steer an agile missile that has a constraint on the minimum flight Mach number from various initial Mach numbers to a given final Mach number in minimum time while completely reversing its flightpath angle. This class of bounded state space, free final time problems is very difficult to solve due to discontinuities in costates at the constraint boundaries. We use a two-neural-network structure called in this study to carry out the optimization process. This structure obtains an optimal controller through solving optimal control-related equations resulting from a Hamiltonian formulation. Detailed derivations of equations and conditions on the constraint boundary are provided. For numerical experiments, we consider vertical plane scenarios. Flight Mach number and the flightpath angle are the states and the aerodynamic angle of attack is treated as the control. Numerical results bring out some attractive features of the adaptive critic approach and show that this formulation works very well in guiding the missile to its final conditions for this state constrained optimization problem from an envelope of initial conditions.

Controlling the Lorenz system: combining global and local schemes

The Lorenz equations are one of the best-known and analyzed systems exhibiting chaotic behavior. In this paper, a new control scheme for the Lorenz system combining local and global techniques is introduced. This scheme is based on a feedback law which is only applied in a bounded state space region of control (SSRC). The SSRC is determined by the enclosure of the Lorenz attractor.

Some physical models with Minkowski spacetime structure and Lorentz group symmetry

Projecting the linear equation X ̇ = AX , A ∈ so(n, 1) , yields a primitive model. It is a prototype of several physical models, namely perfect elastoplasticity (on phase), spacetime of special relativity, special relativistic mechanics and so on, all of which may endow a Minkowskian spacetime and the Lorentz group left acts on it. The mathematical structure of the perfect elastoplastic equations is then compared with those of the spacetime of special relativity and of the special relativistic equations of motion of a massive charged particle; their similarities in group properties and subtle differences in phase spaces are discussed. It is remarkable that the evolutions from elastic constitutive equations to elastoplastic constitutive equations and from the Newtonian equations to the special relativistic equations are very similar in several facets, notably (a) from a linear theory to a non-linear theory, (b) the state space being enlarged from the usual Euclidean space to Minkowski spacetime, (c) from an n-space to a cone of ( n+1)-space, (d) from a non-bounded state space to a bounded state space, (e) from a non-causal relation of states to a causal relation of augmented states.

ON OPTIMALITY OF LOCAL CONTROL OF CHAOS

Local control stabilizes chaotic motions by applying a state feedback only within a possible strongly bounded state space region of control (SSRC). In this paper, we consider the question of when a local control is optimal. Therefore, a cost function is employed that accounts for two components: time and control force needed until stabilization occurs. We present analytical and numerical results on the dependency of the cost on the size of the SSRC and a weighting constant between the two components of the cost function.

Fractal encoding of context‐free grammars in connectionist networks

Connectionist network learning of context‐free languages has so far been applied only to very simple cases and has often made use of an external stack. Learning complex context‐free languages with a homogeneous neural mechanism looks like a much harder problem. The current paper takes a step toward solving this problem by analyzing context‐free grammar computation (without addressing learning) in a class of analog computers called dynamical automata, which are naturally implemented in connectionist networks. The result is a widely applicable method of using fractal sets to organize infinite‐state computations in a bounded state space. An appealing consequence is the development of parameter‐space maps, which locate various complex computers in spatial relationships to one another. An example suggests that such a global perspective on the organization of the parameter space may be helpful for solving the hard problem of getting connectionist networks to learn complex grammars from examples.