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A Constructive FixedPoint Theorem and the Feedback Semantics of Timed Systems
 in Workshop on Discrete Event Systems (WODES), Ann Arbor
, 2006
"... Abstract — Deterministic timed systems can be modeled as fixed point problems [15], [16], [4]. In particular, any connected network of timed systems can be modeled as a single system with feedback, and the system behavior is the fixed point of the corresponding system equation, when it exists. For d ..."
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Abstract — Deterministic timed systems can be modeled as fixed point problems [15], [16], [4]. In particular, any connected network of timed systems can be modeled as a single system with feedback, and the system behavior is the fixed point of the corresponding system equation, when it exists. For deltacausal systems, we can use the Cantor metric to measure the distance between signals and the Banach fixedpoint theorem to prove the existence and uniqueness of a system behavior. Moreover, the Banach fixedpoint theorem is constructive: it provides a method to construct the unique fixed point through iteration. In this paper, we extend this result to systems modeled with the superdense model of time [7], [8] used in hybrid systems. We call the systems we consider eventually deltacausal, a strict generalization of deltacausal in which multiple events may be generated on a signal in zero time. With this model of time, we can use a generalized ultrametric [14] instead of a metric to model the distance between signals. The existence and uniqueness of behaviors for such systems comes from the fixedpoint theorem of [13], but this theorem gives no constructive method to compute the fixed point. This leads us to define petrics, a generalization of metrics, which we use to generalize the Banach fixedpoint theorem to provide a constructive fixedpoint theorem. This new fixedpoint theorem allows us to construct the unique behavior of eventually deltacausal systems. I.
94720 DiscreteEvent Systems: Generalizing Metric Spaces and FixedPoint Semantics
, 2005
"... Abstract. This paper studies the semantics of discreteevent systems as a concurrent model of computation. The classical approach, which is based on metric spaces, does not handle well multiplicities of simultaneous events, yet such simultaneity is a common property of discreteevent models and mode ..."
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Abstract. This paper studies the semantics of discreteevent systems as a concurrent model of computation. The classical approach, which is based on metric spaces, does not handle well multiplicities of simultaneous events, yet such simultaneity is a common property of discreteevent models and modeling languages. (Consider, for example, delta time in VHDL.) In this paper, we develop a semantics using an extended notion of time. We give a generalization of metric spaces that we call tetric spaces. (A tetric functions like a metric, but its value is an element of a totallyordered monoid rather than an element of the nonnegative reals.) A straightforward generalization of the Banach fixed point theorem to tetric spaces supports the definition of a fixedpoint semantics and generalizations of wellknown sufficient conditions for avoidance of Zeno conditions. 1