### Symbolic Representations and Analysis of Large Probabilistic Systems [chapter]

Andrew Miner, David Parker
2004 Lecture Notes in Computer Science
This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their efficiency by exploiting high-level structure and regularity in the models to which they are applied, increasing the size of the state spaces which can be tackled. In general, this is done by using data structures which provide compact storage but which are still efficient to manipulate, usually based on binary decision diagrams (BDDs) or
more » ... eir extensions. In this paper we focus on BDDs, multi-valued decision diagrams (MDDs), multi-terminal binary decision diagrams (MTB-DDs) and matrix diagrams. concurrently. This phenomenon is often referred to as 'the state space explosion problem', 'largeness' or 'the curse of dimensionality'. A great deal of work has been put into developing space and time efficient techniques for the storage and analysis of probabilistic models. Many of the recent approaches that have achieved notable success are symbolic techniques, by which we mean those using data structures based on binary decision diagrams (BDDs). Also known as implicit or structured methods, these approaches focus on generating compact model representations by exploiting structure and regularity, usually derived from the high-level description of the system. This is possible in practice because systems are typically modeled using structured, high-level specification formalisms such as Petri nets and process algebras. In contrast, explicit or enumerative techniques are those where the entire model is stored and manipulated explicitly. In the context of probabilistic models, sparse matrices are the most obvious and popular explicit storage method. The tasks required to perform analysis and verification of probabilistic models can be broken down into a number of areas, and we cover each one separately in this paper. In Section 2, we consider the storage of sets of states, such as the reachable state space of a model. In Section 3, we look at the storage of the probabilistic model itself, usually represented by a real-valued transition matrix. We also discuss the generation of each of these two entities. In Section 4, we describe how the processes of analysis, which usually reduce to numerical computation, can be performed in this context. Finally, in Section 5, we discuss the relative strengths and weaknesses of the various symbolic approaches, and suggest some areas for future work.