Building models of real-time systems from application software

s away from real time by mapping into the same logical instant independent events or events close enough in the real-time scale. The logical ordering takes into account causal relations between events. Thus, can be considered as an abstract specification of an input event handler (see Section IV-B) If is an input of the real-time system, then defines a time base, that is, an increasing sequence of instants in the following manner. As is a transducer, under some technical continuity conditions we have that, for any natural number , there exists a minimal real number , such that , where is the restriction of in the interval (0, ]. In other words, the logical input at instant depends only on the input values in the interval (0, ]. In practice, the time base can be defined by imposing separability constraints which guarantee that significant changes of the environment are not mapped into the same logical instant. These constraints can be any combination of: 1) constraints restricting environment state changes within two consecutive instants of the time base, e.g., values of a variable or the integral of some variable remain within some bounds; and 2) time constraints maintaining the distance between two successive instants of the time base close enough to separate significant input changes, e.g., sampling.The notion of a separator event is useful to define untiming 104 PROCEEDINGS OF THE IEEE, VOL. 91, NO. 1, JANUARY 2003 functions. An event is a separator if distinct occurrences of this event are mapped into distinct logical instants. For and , some examples of untiming input functions are the following. 1) A very simple case comprises sampling functions where a periodic event is a separator; that is, for any and , , where is the sampling period. This is a typical situation in time-triggered systems [23]. 2) Another simple case corresponds to the situation where all the input events are separators. This means that the application receives the input events one by one in the order they arrive; that is, for any and , , where are the arrival times of all the events. Synchronous languages often assume that the logical input at instant depends only on the inputs in the interval ( , ], i.e.,