Timed Acceptances: A Model of Time Dependent Processes

Timed Acceptances is a process model for describing and reasoning about processes with explicit t iming constraints (real-time processes). A process is an enti ty tha t interacts with other processes and makes internal decisions tha t affect its subsequent behavior. An interaction corresponds to synchronization or communication among processes. An internal or unobservable decision is not affected by any other process. In a real-t ime process, the actions tha t can occur are constrained both logically and temporally. A sequence of i~teractions and internal decisions occurring at par t icular t imes consti tutes an execution of a realt ime process. To model execution, we use a temporal extension of Hennessy's Acceptance Trees and characterize a real-t ime process by the set of all its possible executions. The interactions of a process are represented by events. An event is an instantaneous visible action in which a process engages during its execution. The s tate of a process controls the possible interactions of the process and the times at which the interactions can occur. The internal decisions represent the nondeterministic behavior of a process. They affect the subsequent behavior of the process by changing the internal state of the process. Thus, the possibili ty of an internal decision is represented by the set of states which a process may be in at a given time. We now give the formal description of an execution, which consists of a timed trace and the possible states the process may be in after executing the trace. We assume tha t the choice of s ta te is made immediately upon executing the last event in the trace. A t imed trace is a finite sequence ( (a l , n l ) , ( a2 ,n2 ) , . . . , ( am ,nm) ) e (~ × N')* where N is the nonnegative integers. Each (ai, hi) represents the occurrence of the i th event in the execution of the process. The t ime nl is the t ime between 0 and the occurrence of a l . For i > 1, ni represents the relative t ime between events ai-1 and ai. A state is represented by a set of event-time pairs {(An,n1), . . . ,(Am,n,,~)} C 5P(~o(~) x N t3 {(0, c¢))) where ~ is the set of events, ~O(A) is the set of all finite subsets of A and Ai is either {al) or 0. Each ({ai) , n~) represents the possibili ty of event al occurring ni t ime units after a process enters the state. The pair ($, hi) represents the possibili ty of the process stopping at t ime hi. If ni = c¢, the state represents the process diverging. A divergent process is one tha t continues to execute, but does not engage in any further useful computat ion. The set of all executions of a process is i ts acceptance set, A(P) . Formally, a process is represented by a pair (~P, .A(P)) satisfying the following six constraints. Firs t , the empty trace is a t race of P since i t represents the behavior of P the moment it is switched on, but before i t engages in any action. Second, the trace set of P is prefix dosed because if P has executed a trace s, it must have executed all prefixes of s first. Third, all events in a nonempty