We describe the main features of SmArT, a software package providing a seamless environment for the logic and probabilistic analysis of complex systems. SmArT can combine dierent formalisms in the same modeling study. For the analysis of logical behavior, both explicit and symbolic state-space generation techniques, as well as symbolic CTL model-checking algorithms, are available. For the study of stochastic and timing behavior, both sparse-storage and Kronecker numerical solution approaches are available when the underlying process is a Markov chain. In addition,

Abstract. 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 their extensions. In this paper we focus on BDDs, multi-valued decision diagrams (MDDs), multi-terminal binary decision diagrams (MTBDDs) and matrix diagrams. 1

We present techniques for computing the solution of large Markov chain models whose generators can be represented in the form of a generalized tensor algebra, such as Stochastic Automata Networks (SAN). Many large systems include a number of replications of identical components. This paper exploits replication by aggregating similar components. This leads to a state space reduction, based on lumpability. We define SAN with replicas, and we show how such SAN models can be strongly aggregated, taking functional rates into account. A tensor representation of the matrix of the aggregated Markov chain is proposed, allowing to store this chain in a compact manner and to handle larger models with replicas more efficiently. Examples and numerical results are presented to illustrate the reduction in state space and, consequently, the memory and processing time gains.

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al collections of homogeneous objects indexed by set elements. a[3][0.2] aggregates : analogous to the Pascal \record". p:3 A type can be further modied by the following natures, which describe stochastic characteristics: const: (the default) a non-stochastic quantity. ph: a random variable with discrete or continuous phase-type distribution. rand: a random variable with arbitrary distribution. ctmc, dtmc, spn, . . . : stochastic formalisms dening a stochastic process indexed by time. 1.1 Function declarations Syntactically, objects dened in SMART are functions, possibly recursive, and can be overloaded: real pi := 3.14; /* a parameter-less function */ bool close(real a, real b) := abs(a-b) < 0.00001; /* a two-parameter function */ int pow(int base, int e

We review high-level specification formalisms for Markovian performability models, thereby emphasising the role of structuring concepts as realised par excellence by stochastic process algebras. Symbolic representations based on decision diagrams are presented, and it is shown that they quite ideally support compositional model construction and analysis.

Implicit techniques for representing and generating the reachability set of a high-level model have become quite efficient. However, such techniques are usually restricted to models whose events have equal priority. Models containing events with differing classes of priority or complex priority structure, in particular models with immediate events, have thus been required to use explicit reachability set generation techniques. In this paper, we present an efficient implicit technique, based on multi-valued decision diagram representations for sets of states and matrix diagram representations for next-state functions, that can handle models with complex priority structure. If the model contains immediate events, the vanishing states can be eliminated either during generation, by manipulating the matrix diagram, or after generation, by manipulating the multi-valued decision diagram. We apply both techniques to several models and give detailed results. 1.

Petri nets and stochastic Petri nets have been widely adopted as one of the best tools to model the logical and timing behavior of discrete-state systems. However, their practical applicability is limited by the state-space explosion problem. We survey some of the techniques that have been used to cope with large state spaces, starting from early explicit methods, which require data structures of size proportional to the number of states or state-to-state transitions, then moving to implicit methods, which borrow ideas from symbolic model checking (binary decision diagrams) and numerical linear algebra (Kronecker operators) to drastically reduce the computational requirements. Next, we describe the structural decomposition approach which has been the topic of our research in the last few years. This method only requires to specify a partition of the places in the net and, combining decision diagrams and Kronecker operators with the new concepts of event locality and node saturation, achieves fundamental gains in both memory and time efficiency. At the same, the approach is applicable to a wide range of models. We conclude by considering several research directions that could further push the range of solvable models, eventually leading to an even greater industrial acceptance of this simple yet powerful modeling formalism.

Abstract. This paper presents the advantages in extending Classical Tensor Algebra (CTA), also known as Kronecker Algebra, to allow the definition of functions, i.e., functional dependencies among its operands. Such extended tensor algebra have been called Generalized Tensor Algebra (GTA). Stochastic Automata Networks (SAN) and Superposed Generalized Stochastic Petri Nets (SGSPN) formalisms use such Kronecker representations. The advantages of GTA do not imply in a reduction or augmentation of application scope, since there is a representation equivalence between SAN, which uses GTA, and SGSPN, which uses only CTA. Two modeling examples are presented in order to draw comparisons between the memory needs and CPU time required for the generation and solution using both formalisms, showing the computational advantages in using GTA. 1

In this thesis, we present symbolic implementation techniques for model checking prob-abilistic timed automata as models for systems, for example, communication networks and randomised distributed algorithms. Given a system model as probabilistic timed au-tomata and a specification, such as, “a leader will be elected within 5 time units with probability 0.999 ” and “the message can be successfully delivered within 3 time units with probability 0.985”, a probabilistic real-time model checker can automatically verify if the model satisfies the specification. Motivated by the success of symbolic approaches to both non-probabilistic real-time model checking and untimed probabilistic model checking, we present a symbolic implementation of model checking for probabilistic timed automata, which is based on the data structure called Multi-terminal Binary Decision Diagrams (MTBDDs). Our MTBDD-based symbolic implementation is generic with respect to the support of the real-time information. Two data structures representing the real-time information are supported, Difference Bound Matrices (DBMs) and Difference Decision Diagrams (DDDs). We develop explicit and symbolic implementations of model check-