We present real time and discrete time versions of ACP with absolute timing and relative timing. The starting-point isanewrealtimeversion with absolute timing, called ACPsat, featuring urgent actions and a delay operator. The discrete time versions are conservative extensions of the discrete time versions of ACP being known as ACP dat and ACP drt. The principal version is an extension of ACP sat with integration and initial abstraction to allow for choices over an interval of time and relative timing to be expressed. Its main virtue is that it generalizes ACP without timing and most other versions of ACP with timing in a smooth and natural way. This is shown for the real time version with relative timing and the discrete time version with absolute timing.

ion The previous section treated BPA ffir with recursion modulo timed strong bisimulation. In this section the alphabet is extended with a special constant ø , to obtain BPA ffiø r with recursion, and process terms are considered modulo rooted timed branching bisimulation. In the sequel, a and ff will represent elements from A [ føg and A [ fffi; øg, respectively. 3.1 Time Shift In order to define timed branching bisimulation, the syntax is extended with the time shift operator (r)p, which takes a rational number r and a process term p. The process term (r)p denotes the behaviour of p that is shifted r units in time. Its ultimate delay is defined by U((r)p) = maxfU(p) + r; 0g The transition rules and axioms for the time shift are given in Table 4. Using axioms TS1-4, this operator can be eliminated from all process terms. 3.2 Timed Branching Bisimulation The operational semantics consists of the transition rules in Table 1 and Table 2 and Table 4. The definition of timed strong...

Timed and hybrid automata are models designed for describing real-time or hybrid systems.

This paper deals with prefix integration, and integration is parametrized by conditions, which are inequalities between linear expressions of variables. We present an axiomatization for process terms, and propose a strategy to decide bisimulation equivalence between process terms, by means of this axiomatization. 1 Introduction