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A survey on continuous time computations
 New Computational Paradigms
"... 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 resu ..."
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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
How Fast Will the Flip Flop?
 In Proceedings of the 1994 International Symposium on Advanced Research in Asynchronous Circuits and Systems
, 1994
"... 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 ninetransistor toggle element using a simple, SPICElike model. We show how such properties as minimum and maximum clock frequ ..."
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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 ninetransistor toggle element using a simple, SPICElike 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, continuousmodel 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 eventdriven simulation [3], model checking [4], and theorem proving [17]. However, many important circuit phenomena cannot be modeled with discrete time and va...
Four Decades of Mathematical System Theory
 Univ. of Groningen
, 1999
"... As a followup 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 p ..."
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As a followup 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 ...
A Smooth Dynamical System that Counts in Binary
, 1997
"... This paper presents a smooth dynamical system that implements a toggle flipflop. The flipflop is described as a system of smooth, nonlinear ODE's. We identify a period2, invariant set of this system, and show that this corresponds to the discrete state transitions of a discrete model. We sh ..."
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This paper presents a smooth dynamical system that implements a toggle flipflop. The flipflop is described as a system of smooth, nonlinear ODE's. We identify a period2, 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.
Formal Verification of an Arbiter
"... We present the circuitlevel 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, ..."
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We present the circuitlevel 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.
Hybrid dynamical systems, or hds: The ultimate switching experience
 In: Preprints of the Block Island Workshop on Control Using LogicBased Switching
, 1995
"... 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 ..."
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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
ProjectagonBased Reachability Analysis for CircuitLevel Formal Verification
, 2011
"... This dissertation presents a novel verification technique for analog and mixed signal circuits. Analog circuits are widely used in many applications include consumer electronics, telecommunications, medical electronics. Furthermore, in deep submicron design, physical effects might undermine common ..."
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This dissertation presents a novel verification technique for analog and mixed signal circuits. Analog circuits are widely used in many applications include consumer electronics, telecommunications, medical electronics. Furthermore, in deep submicron design, physical effects might undermine common digital abstractions of circuit behavior. Therefore, it is necessary to develop systematic methodologies to formally verify hardware design using circuitlevel models. We present a formal method for circuitlevel verification. Our approach is based on translating verification problems to reachability analysis problems. It applies nonlinear ODEs to model circuit dynamics using modified nodal analysis. Forward reachable regions are computed from given initial states to explore all possible circuit behaviors. Analog properties are checked on all circuit states to ensure full correctness or find a design flaw. Our specification language extends LTL logic with continuous time and values and applies Brockett’s annuli to specify analog signals. We also introduced probability into the specification to support practical analog properties such as metastability behavior.
Formal Verification of Celement Circuits
"... It is well known that the correct behavior of asynchronous circuits is not guaranteed when the inputs switch too slowly. The erroneous behavior is generally difficult to be spotted by simulation based methods. We applied formal methods to study the analog switching behavior of a fullbuffer circuit ..."
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It is well known that the correct behavior of asynchronous circuits is not guaranteed when the inputs switch too slowly. The erroneous behavior is generally difficult to be spotted by simulation based methods. We applied formal methods to study the analog switching behavior of a fullbuffer circuit composed of Celements. We used our reachability analysis tool COHO to compute all reachable states of two Celement designs and verified several analog properties. Based on these properties, we identified a sufficient condition under which the fullbuffer circuit always supports the designed handshaking protocol. We also improved the COHO tool to automate the verification process, reduce error and improve performance. I.