Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudo-metrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our approximation framework is compositional for a synchronous composition operator. Algorithms are developed for computing the proposed pseudo-metrics, both exactly and approximately. The exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations, or dually, the solution of a static game whose cost is the so-called branching distance between the systems. Approximations for the pseudo-metrics can be obtained by considering Lyapunov-like functions called simulation and bisimulation functions. We illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems.

Abstract. In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisimulation allows the observation to be different provided they are and remain arbitrarily close. Approximate bisimulation relations are conveniently defined as level sets of a function called bisimulation function. For the class of linear systems with constrained initial states and constrained inputs, we develop effective characterizations for bisimulation functions that can be interpreted in terms of linear matrix inequalities, set inclusion and games. We derive a computationally effective algorithm to evaluate the precision of the approximate bisimulation between a constrained linear system and its projection. This algorithm has been implemented in a MATLAB toolbox: MATISSE. Two examples of use of the toolbox in the context of safety verification are shown. 1.

A Probabilistic I/O Automaton (PIOA) is a countable-state automaton model that allows nondeterministic and probabilistic choices in state transitions. A task-PIOA adds a task structure on the locally controlled actions of a PIOA as a means for restricting the nondeterminism in the model. The task-PIOA framework defines exact implementation relations based on inclusion of sets of trace distributions. In this paper we develop the theory of approximate implementations and equivalences for task-PIOAs. We propose a new kind of approximate simulation between task-PIOAs and prove that it is sound with respect to approximate implementations. Our notion of similarity of traces is based on a metric on trace distributions and therefore, we do not require the state spaces nor the space of external actions (output alphabet) of the underlying automata to be metric spaces. We discuss applications of approximate implementations to probabilistic safety verification.

Abstract. Statistical zero-knowledge (SZK) properties play an important role in designing cryptographic protocols that enforce honest behavior while maintaining privacy. This paper presents a novel approach for verifying SZK properties, using recently developed techniques based on approximate simulation relations. We formulate statistical indistinguishability as an implementation relation in the Task-PIOA framework, which allows us to express computational restrictions. The implementation relation is then proven using approximate simulation relations. This technique separates proof obligations into two categories: those requiring probabilistic reasoning, as well as those that do not. The latter is a good candidate for mechanization. We illustrate the general method by verifying the SZK property of the well-known identification protocol proposed by Girault, Poupard and Stern.

This paper presents a fundamental study of similarity and bisimilarity for labelled Markov processes: a particular class of probabilistic labelled transition systems. The main results characterize similarity as a testing preorder and bisimilarity as a testing equivalence.

Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006)

Fifty years ago, control and computing were part of a broader system science. After a long period of separate development within each discipline, embedded and hybrid systems have challenged us to re-unite the, now sophisticated theories of continuous control and discrete computing on a broader system theoretic basis. In this paper, we present a framework of system approximation that applies to both discrete and continuous systems. We define a hierarchy of approximation metrics between two systems that quantify the quality of the approximation, and capture the established notions in computer science as zero sections. The central notions in this framework are that of approximate simulation and bisimulation relations and their functional characterizations called simulation and bisimulation functions and defined by Lyapunov-type inequalities. In particular, these functions can provide computable upper-bounds on the approximation metrics by solving a static game. Our approximation framework will be illustrated by showing some of its applications in various problems such as reachability analysis of continuous systems and hybrid systems, approximation of continuous and hybrid systems by discrete systems, hierarchical control design, and simulation-based approaches to verification of continuous and hybrid systems.