A general automaton model for timing-based systems is presented and is used as the context for developing a variety of simulation proof techniques for such systems. These techniques include (1) refinements, (2) forward and backward simulations, (3) hybrid forward-backward and backward-forward simulations, and (4) history and prophecy relations. Relationships between the different types of simulations, as well as soundness and completeness results, are stated and proved. These results are (with one exception) analogous to the results for untimed systems in Part I of this paper. In fact, many of the results for the timed case are obtained as consequences of the analogous results for the untimed case.

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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...

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Abstract. We show that timed branching bisimilarity as defined by van der Zwaag [14] and Baeten & Middelburg [2] is not an equivalence relation, in case of a dense time domain. We propose an adaptation based on van der Zwaag’s definition, and prove that the resulting timed branching bisimilarity is an equivalence indeed. Furthermore, we prove that in case of a discrete time domain, van der Zwaag’s definition and our adaptation coincide. 1

Abstract. The cones and foci verification method from Groote and Springintveld [9] was extended to timed systems by van der Zwaag [17]. We present an extension of this cones and foci method for timed systems, which can cope with infinite τ-sequences. We prove soundness of our approach and give small verification examples. 1

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