Discrete-Event Systems: Generalizing Metric Spaces and Fixed Point Semenatics

Adam Cataldo, Edward Lee, Xiaojun Liu, Eleftherios Matsikoudis and Haiyang Zheng

UCB/ERL M05/12, draft version, April 8, 2005.




This paper studies the semantics of discrete-event 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 discrete-event 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 totally-ordered monoid rather than an element of the non-negative reals.) A straightforward generalization of the Banach fixed point theorem to tetric spaces supports the definition of a fixed-point semantics and generalizations of well-known sufficient conditions for avoidance of Zeno conditions.