Boolean networks is a well-established formalism for modelling biological systems. A vital challenge for analysing a Boolean network is to identify all the attractors. This becomes more challenging for large asynchronous Boolean networks, due to the asynchronous updating scheme. Existing methods are prohibited due to the well-known state-space explosion problem in large Boolean networks. In this paper, we tackle this challenge by proposing a SCC-based decomposition method. We prove the correctness of our proposed method and demonstrate its efficiency with two real-life biological networks.

Abstract. We present ASSA-PBN, a tool for approximate steady-state analysis of large probabilistic Boolean networks (PBNs). ASSA-PBN contains a constructor, a simulator, and an analyser which can approximately compute the steadystate probabilities of PBNs. For large PBNs, such approximate analysis is the only viable way to study their long-run behaviours. Experiments show that ASSA-PBN can handle large PBNs with a few thousands of nodes.

As a well-established computational framework, probabilistic Boolean networks (PBNs) are widely used for modelling, simulation, and analysis of biological systems. To analyze the steady-state dynamics of PBNs is of crucial importance to explore the characteristics of biological systems. However, the analysis of large PBNs, which often arise in systems biology, is prone to the infamous state-space explosion problem. Therefore, the employment of statistical methods often remains the only feasible solution. We present ${\mathsf{ASSA-PBN}}$ , a software toolbox for modelling, simulation, and analysis of PBNs. ${\mathsf{ASSA-PBN}}$ provides efficient statistical methods with three parallel techniques to speed up the computation of steady-state probabilities. Moreover, particle swarm optimisation (PSO) and differential evolution (DE) are implemented for the estimation of PBN parameters. Additionally, we implement in-depth analyses of PBNs, including long-run influence analysis, long-run sensitivity analysis, computation of one-parameter profile likelihoods, and the visualization of one-parameter profile likelihoods. A PBN model of apoptosis is used as a case study to illustrate the main functionalities of ${\mathsf{ASSA-PBN}}$ and to demonstrate the capabilities of ${\mathsf{ASSA-PBN}}$ to effectively analyse biological systems modelled as PBNs.

Abstract-Probabilistic Boolean networks (PBNs) is a well-established computational framework for modelling biological systems. The steady-state dynamics of PBNs is of crucial importance in the study of such systems. However, for large PBNs, which often arise in systems biology, obtaining the steady-state distribution poses a significant challenge. In this paper, we revive the two-state Markov chain approach to solve this problem. This paper contributes in three aspects. First, we identify a problem of generating biased results with the approach and we propose a few heuristics to avoid such a pitfall. Secondly, we conduct an extensive experimental comparison of the extended two-state Markov chain approach and another approach based on the Skart method. We analyse the results with machine learning techniques and we show that statistically the two-state Markov chain approach has a better performance. Finally, we demonstrate the potential of the extended two-state Markov chain approach on a case study of a large PBN model of apoptosis in hepatocytes.

Many automated system analysis techniques (e.g., model checking, model-based testing) rely on first obtaining a model of the system under analysis. System modeling is often done manually, which is often considered as a hindrance to adopt model-based system analysis and development techniques. To overcome this problem, researchers have proposed to automatically "learn" models based on sample system executions and shown that the learned models can be useful sometimes. There are however many questions to be answered. For instance, how much shall we generalize from the observed samples and how fast would learning converge? Or, would the analysis result based on the learned model be more accurate than the estimation we could have obtained by sampling many system executions within the same amount of time? In this work, we investigate existing algorithms for learning probabilistic models for model checking, propose an evolution-based approach for better controlling the degree of generalization and conduct an empirical study in order to answer the questions. One of our findings is that the effectiveness of learning may sometimes be limited.

Boolean networks are an important formalism for modelling biological systems and have attracted much attention in recent years. An important direction in Boolean networks is to exhaustively find attractors, which represent steady states when a biological network evolves for a long term. In this paper, we propose a new approach to improve the efficiency of BDD-based attractor detection. Our approach includes a monolithic algorithm for small networks, an enumerative strategy to deal with large networks, and two heuristics on ordering BDD variables. We demonstrate the performance of our approach on a number of examples, and compare it with one existing technique in the literature.

Boolean networks is a well-established formalism for modelling biological systems. An important challenge for analysing a Boolean network is to identify all its attractors. This becomes challenging for large Boolean networks due to the well-known state-space explosion problem. In this paper, we propose a new SCC-based decomposition method for attractor detection in large synchronous Boolean networks. Experimental results show that our proposed method is significantly better in terms of performance when compared to existing methods in the literature.

Many automated system analysis techniques (e.g., model checking, model-based testing) rely on first obtaining a model of the system under analysis. System modeling is often done manually, which is often considered as a hindrance to adopt modelbased system analysis and development techniques. To overcome this problem, researchers have proposed to automatically "learn" models based on sample system executions and shown that the learned models can be useful sometimes. There are however many questions to be answered. For instance, how much shall we generalize from the observed samples and how fast would learning converge? Or, would the analysis result based on the learned model be more accurate than the estimation we could have obtained by sampling many system executions within the same amount of time? Moreover, how well does learning scale to real-world applications? If the answer is negative, what are the potential methods to improve the efficiency of learning? In this work, we first investigate existing algorithms for learning probabilistic models for model checking and propose an evolution-based approach for better controlling the degree of generalization. Then, we present existing approaches to learn abstract models to improve the efficiency of learning for scalability reasons. Lastly, we conduct an empirical study in order to answer the above questions. Our findings include that the effectiveness of learning may sometimes be limited and it is worth investigating how abstraction should be done properly in order to learn abstract models.