Creating Evolution Scenarios for Hybrid Systems Victor W. Marek Department of Computer Science University of Kentucky Lexington, KY 40506-0046 marek@cs.engr.uky.edu Walton Sumner II Department of Internal Medicine Washington University St. Louis, MO sumner@oster.wustl.edu Miroslaw Truszczy´ nski Department of Computer Science University of Kentucky Lexington, KY 40506-0046 mirek@cs.engr.uky.edu Abstract We discuss the evolution scenarios for hybrid systems described by families of parameters changing in time. The systems are in discrete states and the transi- tions between states are probabilistic. The need for such scenarios arises in applications. In particular, in medicine such scenarions can be viewed as patient histories and can be used in medical training and test- ing. 1. Introduction In this paper we investigate the issue of generating evolution scenarios for hybrid systems. The systems we have in mind are defined by continuously chang- ing parameters and discretely defined states. States are determined by ranges of parameter values. Each system has its age (time from inception) and at any time moment it is in one of the states. Consider a state s and age t. The problem of interest is to de- velop a technique to create a hybrid system which at age t is in state s. This technique should endow the system with a history consistent with state s and explaining how the system evolved into it. The his- tory is specified by a family of functions describing the parameters of the system over time until moment t. Examples of such systems abound. Living organ- isms (including humans) can be viewed as systems in medical conditions (diseases as well as normal con- ditions [SC1, SC2]). Values of parameters such as height, weight, blood pressure, pulse, levels of hor- mone, etc define health states. Humans also exist as economic entities described by such parameters as wealth, investment preferences, level of education, levels of spending, savings, etc. Other examples in- clude physical systems, such as industrial plants, with continuously changing parameters: engine power, fuel consumption, speed, direction, external conditions, etc.) or complex computer and telephone networks (with throughput capacity, and several traffic charac- teristics). In all these examples, states are determined by combinations of parameters, and systems change over time from one state to another. An additional important characteristic of these sys- tems is that the changes from state to state can be affected by actions of external “supervisors”: physi- cians, financial advisors, plant and computer network managers. Over time, systems evolve from state to state. The goal of the supervisor is to avoid unde- sirable states, that is, to maintain the system in its current state if it is desirable, or move it to another, more desirable one. For instance, if the current eco- nomic condition of an individual is inconsistent with projected economic needs, then the financial advisor must recognize this fact and propose a strategy to correct this state of affairs. Thus, in general, a supervisor should recognize the state of the system and design corrective actions. The first of these two tasks is the problem of diagnosis, re- quiring probing the system for current and past values of its parameters. Some of these values may not be available or may be very difficult or costly to obtain. Knowledge of past values is often helpful. In medicine complete or partial knowledge of the history of an in- dividual is often critical for the correct diagnosis and efficient management. A similar problem occurs in a class of games in which