Abstract. We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature. 1

This paper describes an experimental investigation of the application of dynamical systems theory to the verification of digital VLSI circuits. We analyze the behavior of a nine-transistor toggle element using a simple, SPICE-like model. We show how such properties as minimum and maximum clock frequency can be identified from topological features of solutions to the corresponding system of differential equations. This dynamical systems perspective also gives a clear, continuous-model interpretations of such phenomena as dynamic storage and timing hazards. Keywords: Dynamical systems, hardware verification, hybrid models, real time systems 1 Introduction Most verification of VLSI designs, synchronous and asynchronous, assumes discrete models for signal values and transition times. These discrete models lend themselves well to event-driven simulation [3], model checking [4], and theorem proving [17]. However, many important circuit phenomena cannot be modeled with discrete time and va...

As a follow-up to the volume Three Decades of Mathematical System Theory that the authors of this paper edited in 1989 at the occasion of the ...ftieth birthday of Jan Willems, we discuss some trends that have become more visible in the past decade. It is argued that system theory is ready to play a role in many dierent ...elds of science. Three examples of nontraditional application areas are discussed. 1 Introduction Ten years ago, the ...ftieth birthday of Jan Willems was marked by the appearance of the volume Three Decades of Mathematical System Theory. A Collection of Surveys at the Occasion of the 50th Birthday of Jan C. Willems [18]. We, as editors of the volume, started the project in 1988. We decided that Jan could be best honored by collecting contributions from his closest collaborators (and friends) which together should form a survey of what had been achieved in system theory. Confronted with the problem of ...nding a title for the volume we quickly arrived at the ...

As VLSI fabrication technology progresses to 65nm feature sizes and smaller, transistors no longer operate as ideal switches. This motivates verifying digital circuits using continuous models. This paper presents the verification of the high-speed, toggle flip-flop proposed by Yuan and Svensson [1]. Our approach builds on the projection based methods originally proposed by Greenstreet and Mitchell [2], [3]. While they were only able to demonstrate their approach with two- and threedimensional systems, we apply projection based analysis to a seven-dimensional model for the flip-flop. We believe that this is the largest verification to date of a digital circuit using non-linear circuit-level models. In this paper, we describe how we overcame problems of numerical errors and instability associated with the original projection based methods. In particular, we present a novel linear-program solver and new methods for constructing accurate linear approximations of non-linear dynamics. We use the toggle flip-flop as an example and consider how these methods could be extended to verify a standard cell library for digital design.

Machine (CHAM) extends these ideas with a focus on the expression of semantic of non deterministic processes [5]. The CHAM introduces a mechanism to isolate some parts of the chemical solution. This idea has been seriously taken into account in the notion of P systems. P systems [44, 45] are a recent distributed parallel computing model based on the notion of a membrane structure. A membrane structure is a nesting of cells represented, e.g, by a Venn diagram without intersection and with a unique superset: the skin. Objects are placed in the regions defined by the membranes and evolve following various transformations: an object can evolve into another object, can pass trough a membrane or dissolve its enclosing membrane. As for Gamma, the computation is finished when no object can further evolve. By using nested multisets, MGS is able to emulate more or less the notion of P systems. In addition, patterns like the iteration + go beyond what is possible to specify in the l.h.s. of a Gamma rule.

This paper presents a smooth dynamical system that implements a toggle flip-flop. The flip-flop is described as a system of smooth, non-linear ODE's. We identify a period-2, invariant set of this system, and show that this corresponds to the discrete state transitions of a discrete model. We show that this behaviour is robust for a large class of inputs and that these toggle elements can be composed to implement a binary counter of any number of bits. I.

In previous work I have concentrated on formalizing the notion of a hybrid system as switching among an indexed collection of dynamical systems. I have also studied in some depth the modeling, analysis, and control of such systems. Here, I give a quick overview of the area of hybrid systems. I also briefly review the formal definition and discuss the main approaches taken in the study of hybrid systems. Finally, I elucidate issues in each of the research areas in light of previous results. 1

We present the circuit-level verification of a common arbiter circuit. To perform this verification, we address three issues. First, we present a specification for the arbiter and show how this specification amounts to a set of topological constraints on trajectories of the continuous model. Second, we show that computing bounding sets for these trajectories is complicated by stiffness of the differential equation model and present novel techniques for handling stiff equations in a formal verification context. Finally, we note that while no arbiter can be guaranteed to always grant a pending request, we can show liveness in the presence of concurrent requests in an “almost surely ” sense. I.

Abstract. We present a novel implementation of the Simplex algorithm for solving linear programs that arise when manipulating “projectagons”. A projectagon is a compact representation of a high-dimensional geometric object by its projections onto two-dimensional subspaces. These two-dimensional projections lead to linear programs where each inequality constraint has one or two non-zero coefficients. In this paper, we breifly describe the use of projetagons in the verification of VLSI circuits. We then present an efficient and robust implementation of Simplex for the linear programs that arise from projectagons. We show how this solver can be used to implement projectagons and illustrate these ideas with a small example.

: One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. This paper addresses this problem for a class of piecewise linear discontinuous systems under the definition of solutions of Caratheodory. The concepts of jump solutions or a sliding mode are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and su#cient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multi-modal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which...