State Space Explosion Problem Research Articles

Efficient Algorithm for Proportional Lumpability and Its Application to Selfish Mining in Public Blockchains

This paper explores the concept of proportional lumpability as an extension of the original definition of lumpability, addressing the challenges posed by the state space explosion problem in computing performance indices for large stochastic models. Lumpability traditionally relies on state aggregation techniques and is applicable to Markov chains demonstrating structural regularity. Proportional lumpability extends this idea, proposing that the transition rates of a Markov chain can be modified by certain factors, resulting in a lumpable new Markov chain. This concept facilitates the derivation of precise performance indices for the original process. This paper establishes the well-defined nature of the problem of computing the coarsest proportional lumpability that refines a given initial partition, ensuring a unique solution exists. Additionally, a polynomial time algorithm is introduced to solve this problem, offering valuable insights into both the concept of proportional lumpability and the broader realm of partition refinement techniques. The effectiveness of proportional lumpability is demonstrated through a case study that consists of designing a model to investigate selfish mining behaviors on public blockchains. This research contributes to a better understanding of efficient approaches for handling large stochastic models and highlights the practical applicability of proportional lumpability in deriving exact performance indices.

Narrowing and heuristic search for symbolic reachability analysis of concurrent object-oriented systems

A concurrent system specified as a rewrite theory can be analyzed symbolically using narrowing-based reachability analysis. Narrowing-based approaches have been applied to formally analyze cryptographic protocols and parameterized protocols. However, existing narrowing-based analysis methods, based on a breadth-first-search strategy, cannot deal with generic distributed systems with objects and messages due to the symbolic state-space explosion problem and implicit constraints imposed on object-oriented systems. This paper proposes a heuristic search approach for narrowing-based reachability analysis to guide the search for counterexamples with a small number of objects. As a result, our method can effectively find a counterexample if an error state is reachable. In addition, this paper also shows how to encode implicit object-oriented constraints using order-sorted signatures and equational constraints. We demonstrate the effectiveness of our technique using a nontrivial distributed consensus algorithm.

Temporal false data injection attack and detection on cyber‐physical power system based on deep reinforcement learning

AbstractFalse data injection (FDI) attacks are serious threats to a cyber‐physical power system (CPPS), which may be launched by a malicious software or virus accessing only the measurements from one substation. This study proposes a novel attack method named the temporal FDI (TFDI) attack. Namely, the virus makes decisions based on temporal observations of the CPPS, and the attack is driven by a deep Q network (DQN) algorithm. As DQN takes vectors of continuous variables as input states, the proposed method is free of the state space explosion problem, which helps the virus to learn the optimal attack strategy efficiently. Moreover, for adopting time‐series measurements as quasi‐dynamic observations, long short‐term memory cells are employed as a layer of the Q network. The TFDI attack enables the virus to discern trends of load variations and enhance the attack’s effectiveness. Meanwhile, a countermeasure is also presented to detect the proposed FDI attack. Binary classifiers are trained for each bus to detect suspicious local measurements according to their deviations from system‐state manifolds. When suspicious measurements are spotted frequently, the corresponding bus is believed to be under FDI attacks. Test cases validate the efficacy of the proposed FDI attack method as well as its countermeasure.

Reliability Analysis of Safety-Critical Systems using Optimized Petri Nets

Unified modeling language (UML) has emerged as a powerful tool for designing and modeling safety-critical systems. However, UML is not useful in illustrating the dynamic behavior of a critical system. UML does not consider the important critical aspects of the reliability of a safety-critical system, such as non-liveliness, deadlock, stability, and throughput. Therefore, we propose a framework based on UML and Petri Net (PN). In this framework, UML is used to capture all the safety-critical system requirements, whereas PN is used to deliver an in-depth analysis of the reliability aspects of a safety-critical system. Since the PN model suffers from the state space explosion problem, the converted PN may have a large number of redundant states, resulting in a high amount of time required for reliability analysis. Therefore, six reduction algorithms have been derived in this framework to overcome this limitation. The proposed framework is validated with 32 safety-critical system instances of the Nuclear Power Plant on the Reactor Core Isolation Cooling System.

Metascheduling Using Discrete Particle Swarm Optimization for Fault Tolerance in Time-Triggered IoT-WSN

In time-triggered systems, adaptation is performed by using metascheduling approaches to ensure temporal predictability. Wireless sensor networks (WSNs) are growing to be used as a reliable, an energy-efficient, and a scalable network infrastructure for numerous Internet of Things (IoT) applications. Because of run-time changes because of failure events, a metascheduling system is required to address the changes by precomputing schedules for several context events at design time. Therefore, this paper proposes a metascheduler that solves a new scheduling problem for each adaptation scenario in IoT-WSN using our offline algorithm named discrete particle swarm optimization for reliable task allocation (DPSO-TA), resulting in a multi-schedule graph that combines the repeated schedules. In this work, a time-triggered IoT-WSN metascheduler is proposed that takes into considerations the node or link failures that are common in WSN. Single and two failure events are considered, where a graph for all feasible schedules is formed at the network design time. Such a large number of schedules causes a state space explosion problem, which is managed using a re-convergence method. Different application model sizes and network topologies are used to assess the proposed metascheduler under various failure event scenarios. The results show a reduction in the size of the multi-schedule graph by applying inputs with different deadline values and different number of tasks. Furthermore, the validity suggested by the proposed metascheduler, which is the ratio of valid to invalid schedules, is improved when compared to algorithms that schedule tasks to hosts with the shortest completion time or algorithms that select hosts for a set of sorted tasks based on energy consumed and task arrival time.

Reliability Assessment of Safety-Critical Systems of Nuclear Power Plant using Ordinary Differential Equations and Reachability Graph

This study proposes a novel and integrated approach incorporating reachability graph (RG) and ordinary differential equations (ODE) to assess the reliability of safety–critical systems (SCS). The suggested framework is divided into nine steps. The system is first modeled using the Petri net (PN) to generate the ODE and RG. The ODE solution can be used to compute the marking probabilities of PN model. The marking probability values can be utilized to measure the state probabilities of RG. The resulting RG contains various states of the system. The failure states of RG are identified to assess the system's reliability. The suggested approach is implemented on the Shutdown System of Nuclear Power Plant, and is validated using Brown & Lipow model. The obtained accuracy of 99.53905% in the evaluation of reliability validates the technique. The state-space explosion is one of the major problems associated with conventional stochastic approaches like Markov chains, which use assumed transition probabilities to quantify reliability between different states. The proposed strategy is capable of avoiding the problem of state-space explosion.

Compositional verification of embedded real-time systems

In an embedded real-time system (ERTS), real-time tasks (software) are typically executed on a multicore shared-memory platform (hardware). The number of cores is usually small, contrasted with a larger number of complex tasks that share data to collaborate. Since most ERTSs are safety-critical, it is crucial to rigorously verify their software against various real-time requirements under the actual hardware constraints (concurrent access to data, number of cores). Both the real-time systems and the formal methods communities provide elegant techniques to realize such verification, which nevertheless face major challenges. For instance, model checking (formal methods) suffers from the state-space explosion problem, whereas schedulability analysis (real-time systems) is pessimistic and restricted to simple task models and schedulability properties. In this paper, we propose a scalable and generic approach to formally verify ERTSs. The core contribution is enabling, through joining the forces of both communities, compositional verification to tame the state-space size. To that end, we formalize a realistic ERTS model where tasks are complex with an arbitrary number of jobs and job segments, then show that compositional verification of such model is possible, using a hybrid approach (from both communities), under the state-of-the-art partitioned fixed-priority (P-FP) with limited preemption scheduling algorithm. The approach consists of the following steps, given the above ERTS model and scheduling algorithm. First, we compute fine-grained data sharing overheads for each job segment that reads or writes some data from the shared memory. Second, we generalize an algorithm that, aware of the data sharing overheads, computes an affinity (task-core allocation) guaranteeing the schedulability of hard-real-time (HRT) tasks. Third, we devise a timed automata (TA) model of the ERTS, that takes into account the affinity, the data sharing overheads and the scheduling algorithm, on which we demonstrate that various properties can be verified compositionally, i.e., on a subset of cores instead of the whole ERTS, therefore reducing the state-space size. In particular, we enable the scalable computation of tight worst-case response times (WCRTs) and other tight bounds separating events on different cores, thus overcoming the pessimism of schedulability analysis techniques. We fully automate our approach and show its benefits on three real-world complex ERTSs, namely two autonomous robots and an automotive case study from the WATERS 2017 industrial challenge.

On Quantitative Properties Preservation in Reconfigurable Generalized Stochastic Petri Nets

Generalized stochastic Petri nets (GSPNs) have been extended to several dynamic-structure formalisms providing suitable tools for the modeling and verification of reconfigurable discrete-event systems (R-DESs). However, analyzing the performance of large-complex R-DESs remains a big challenging issue. Indeed, dynamic-structure GSPNs still rely on old-fashioned techniques often causing the state-space explosion problem. In this article, we present a new technique for the quantitative analysis of a dynamic-structure formalism called reconfigurable GSPNs without computing the whole state space. This work describes new reconfiguration forms used to preserve desired quantitative properties of parts of interest after each reconfiguration. Therefore, it is only required to verify the examined properties at an initial configuration. The proposed technique is proven to effectively reduce the state space and shorten the computation time in such cases. Finally, some experimental results are provided to illustrate that, from a computational perspective, the developed approach outperforms the existing tools.

Robustness evaluation of trust and reputation systems using a deep reinforcement learning approach

One of the biggest challenges facing trust and reputation systems (TRSs) is evaluating their ability to withstand various attacks, as these systems are vulnerable to multiple types of attacks. While simulation methods have been used to evaluate TRSs, they are limited because they cannot detect new attacks and do not ensure the system’s overall robustness. To address this limitation, verification methods have been proposed that can examine the entire state space and detect all possible attacks. However, these methods are not always practical for large models and real environments because they suffer from the state space explosion problem. To tackle this issue, we propose a deep reinforcement learning approach for evaluating the robustness of TRSs. In this approach, an agent can learn how to attack a system and find the best attack plan without prior knowledge. Additionally, our method avoids the state space explosion problem because it uses a deep Q-network instead of storing and examining the entire state space. We tested our proposed method on five well-known reputation models, assessing various attack goals such as selfishness, maliciousness, competition, and slandering. The results showed that our method was successful in identifying the best attack plan and executing it successfully in the system, demonstrating its effectiveness in evaluating the robustness of TRSs.

Property Preservation of Object-Oriented Petri Reduction Net Based Representation for Embedded Systems

Embedded systems are widely used in automotive electronics, smart home, smart medical, aerospace and other fields. Aiming at the problem of formal modeling and verification analysis of embedded systems, a solution is proposed using extended Petri net reduction operations. Petri net based representation for embedded system (PRES+) and the object-oriented technology are combined to obtain the object-oriented PRES+ (OOPRES+). Two kinds of subnet reduction rules of OOPRES+ are presented. The preservation of boundedness and liveness of the reduction net system has been investigated to alleviate the problem of state space explosion of OOPRES+. The modeling and analysis of the embedded control system of a smart restaurant is used as an example to verify the effectiveness of the subnet reduction rules. Results obtained can provide an effective way to examine the reduction property of Petri net systems, and present a powerful means to model and verify the large-scale complex embedded systems.

Checking Missing-Data Errors in Cyber-Physical Systems Based on the Merged Process of Petri Nets

Missing-data errors easily occur in cyber-physical systems (CPSs). Although many business process modeling notation (BPMN)-based methods are proposed to model CPSs and detect errors, it is hard to automatically verify their correctness, especially in the data flows, due to their lack of formal specifications. By comparison, Petri nets, as a formal method, are widely used to detect data-flow errors. However, these methods easily suffer from the state-space explosion problem. This is mainly because their reachability graphs or state transition graphs are based on the interleaving semantics. As an unfolding technique of Petri net, a merged process can characterize concurrency relations and alleviate this problem. Thus, we utilize the merged process of Petri net with data (PD-net) to check the missing-data errors of the CPS. We first transform a BPMN of the CPS into a PD-net and generate its merged process. Meanwhile, we analyze its structural behaviors and data-adjacent events. Furthermore, we propose an algorithm for checking missing-data errors. In addition, a case study and some experiments are done to show the practicality and effectiveness of our method.

Resilience modeling for multi-state systems based on Markov processes

Modern systems are increasingly under the threaten of disruptive events like earthquakes, floods and storms. Under real life scenarios, multi-state models are often used to describe the behaviors of the system exposed to disruptive events. This article develops a comprehensive resilience modeling and quantifying framework for a multi-state system in which the evolution of the performance level over time is described by a time-homogeneous Markov process. In order to characterize the different dimensions of the system resilience, four types of resilience metrics are proposed to describe the resistant, absorption, recovery, and overall resilience, where each type is divided into an inherent resilience metric and an acquired resilience metric. The theory of aggregated stochastic processes is applied to derive explicit formulas for the four types of resilience metrics. Meanwhile, simulation-based algorithms are proposed to verify the correctness of analytical formulas. They are first exploited to the resilience analysis of a nuclear power plant under the threat of earthquakes, and then used in a numerical example to illustrate the applicability of the proposed method in dealing with the state space explosion problem. The results show that the developed resilience modeling and quantifying framework is able to comprehensively describe the resilience of multi-state systems threatened by disruptive events, and further, some practical suggestions are given to the designers and operators of the system based on the results.

Compositional Verification of Railway Interlocking Systems

Model checking techniques have often been applied to the verification of railway interlocking systems, responsible for guiding trains safely through a given railway network. However, these techniques fail to scale to the interlocking systems controlling large stations, composed of hundreds and even thousands of controlled entities, due to the state space explosion problem. Indeed, interlocking systems exhibit a certain degree of locality that allows some reasoning only on the mere set of entities that regard the train movements, but safe routing through a complex station layout requires a global reservation policy, which can require global state conditions to be taken into account. In this article, we present a compositional approach aimed at chopping the verification of a large interlocking system into that of smaller fragments, exploiting in each fragment a proper abstraction of the global information on routing state. A proof is given of the thesis that verifying the safety of the smaller fragments is sufficient to verify the safety of the whole network. Experiments using this compositional approach have shown important gains in performance of the verification, as well as in the size of affordable station layouts.

Possibilistic Cost Computation Tree Logic and Related Equivalence, Abstraction Technique

Recently, probabilistic Kripke structures have been used to represent uncertain systems; nevertheless, important transition costs were ignored in earlier studies, making it impossible to model some uncertain systems with costs. To compensate for the major shortcoming, this research addresses the transition costs suggested in uncertain systems. First of all, we offer possibilistic Kripke cost structures and associated possibilistic costs such as cumulative cost and possibilistic expected cost. As the verification model, furthermore, we investigated the syntax and semantics of possibilistic cost computation tree logic and utilized them to develop verification characteristics of possibilistic Kripke cost structures. Finally, as solutions to the state space explosion problem of possibilistic cost computation tree logic model checking, abstraction and equivalence are presented as methods to reduce the state numbers and frameworks of possibilistic Kripke cost structures.

Performance measurement of safety-critical systems based on ordinary differential equations and Petri nets: A case study of nuclear power plant

This article proposes a novel approach to measure the performance of Safety-Critical Systems (SCS). Such systems contain multiple processing nodes that communicate with each other is modeled by a Petri nets (PN). The paper uses the PN for the performance evaluation of SCS. A set of ordinary differential equations (ODEs) is derived from the Petri net model that represent the state of the system, and the solutions can be used to measure the system's performance. The proposed method can avoid the state space explosion problem and also introduces new metrics of performance, along with their measurement: deadlock, liveness, stability, boundedness, and steady state. The proposed technique is applied to Shutdown System (SDS) of Nuclear Power Plant (NPP). We obtained 99.887% accuracy of performance measurement, which proves the effectiveness of our approach.

Partial-order reduction for parity games and parameterised Boolean equation systems

In model checking, reduction techniques can be helpful tools to fight the state-space explosion problem. Partial-order reduction (POR) is a well-known example, and many POR variants have been developed over the years. However, none of these can be used in the context of model checking stutter-sensitive temporal properties. We propose POR techniques for parity games, a well-established formalism for solving a variety of decision problems, including model checking. As a result, we obtain the first POR method that is sound for the full modal upmu -calculus. We show how our technique can be applied to the fixed point logic called parameterised Boolean equation systems, which provides a high-level representation of parity games. Experiments with our implementation indicate that substantial reductions can be achieved.

On Optimization of Automation Systems: Integrating Modular Learning and Optimization

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Compositional Optimization</i> (CompOpt) was recently proposed for optimization of discrete-event systems of systems. A modular optimization model allows CompOpt to divide the optimization into separate sub-problems, mitigating the state space explosion problem. This paper presents the Modular Optimization Learner (MOL), a method that interacts with a simulation of a system to automatically learn these modular optimization models. MOL uses a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">modular learning</i> that takes as input a hypothesis structure of the system and uses the provided structural information to split the acquired learning into a set of modules, and to prune parts of the search space. Experiments show that modular learning reduces the state space by many orders of magnitude compared to a monolithic learning, which enables learning of much larger systems. Furthermore, an integrated greedy search heuristic allows MOL to remove many sub-optimal paths in the individual modules, speeding up the subsequent optimization. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Automation systems are becoming increasingly large and complex and the automation of more and more advanced tasks often requires the coordination of multiple subsystems. Optimization can have a great impact on the efficiency of these systems in terms of cost and operation speed. Finding optimal solutions is, however, a difficult task. As the number of tasks and subsystems of the automation system increases, the search space of the optimization problems tends to grow exponentially. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Compositional Optimization</i> (CompOpt) is a method specifically designed for the optimization of large-scale automation systems. A challenge with the application of CompOpt is that it takes as input a specific type of optimization model that divides the system into subsystems; like machines, vehicles, etc. Formulating these models requires a high level of expertise and system knowledge. This paper addresses this challenge with an algorithm that learns these models from a simulation of the system. The simulation can be implemented in any software as long as a suitable interface exists or can be constructed. To divide the learning into subsystems, the algorithm uses an initial <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">plant structure hypothesis</i> (PSH). This can be viewed as a meta-model that includes known structural information, such as the number of subsystems and which actions that affect each subsystem. The more structural information that is added to PSH, the more efficient the learning and subsequent optimization will be. The purpose of this is to reduce the level of expertise and system knowledge needed in the application of optimization, to simplify the transition to Industry 4.0.

Automated Synthesis of Generalized Invariant Strategies via Counterexample-Guided Strategy Refinement

Strategy synthesis for multi-agent systems has proved to be a hard task, even when limited to two-player games with safety objectives. Generalized strategy synthesis, an extension of generalized planning which aims to produce a single solution for multiple (possibly infinitely many) planning instances, is a promising direction to deal with the state-space explosion problem. In this paper, we formalize the problem of generalized strategy synthesis in the situation calculus. The synthesis task involves second-order theorem proving generally. Thus we consider strategies aiming to maintain invariants; such strategies can be verified with first-order theorem proving. We propose a sound but incomplete approach to synthesize invariant strategies by adapting the framework of counterexample-guided refinement. The key idea for refinement is to generate a strategy using a model checker for a game constructed from the counterexample, and use it to refine the candidate general strategy. We implemented our method and did experiments with a number of game problems. Our system can successfully synthesize solutions for most of the domains within a reasonable amount of time.

Predicting reliability of software in industrial systems using a Petri net based approach: A case study on a safety system used in nuclear power plant

Software reliability prediction in the early stages of development can be propitious in many ways. The combinatorial models used to predict reliability using architectures such as fault trees, binary decision diagrams, etc. have limitations in modeling complex system behavior. On the other hand, state-based models such as Markov chains suffer from the state-space explosion problem, and they need transition probability among different system states to measure reliability. These probabilities are usually assumed or are obtained from the operational profile for which the system should be used in the field.

Data-Driven Application-Oriented Reliability Model of a High-Performance Computing System

Reliability analysis and performance evaluation are complementary methods to quantify nonfunctional aspects of a system. However, a range of factors such as concurrency and heterogeneity quickly exacerbate the state-space explosion problem when attempting detailed system-level modeling and simulation of high-performance computing (HPC) systems. To overcome these impediments to modeling and analysis, this article develops a hierarchical model of an application that implements checkpointing running in an HPC environment subject to application, network, and system-wide outages. The modeling approach ensures that the number of states is linear in the number of checkpoints and possesses a low constant factor for the number of recovery states most relevant to the external influences contributing to degraded application performance. We illustrate the types of analysis enabled by the model through a series of examples with parameters determined empirically from data logs of the Blue Waters supercomputer located at the University of Illinois at Urbana–Champaign. A comprehensive comparative analysis of the model parameters indicates that lowering the failure rate of network nodes would most significantly reduce application downtime. We also discuss how the modeling approach can be used to objectively assess both current and hypothetical future systems to identify competitive designs and enhancements.