Basic Concurrency Models Figure 1 : Buffer_v0 Model Figure 2 : Buffer_v0 Semantics

In this section we introduce shortly the two basic CM considered in this paper. The running example of a FIFO buffer is used here to illustrate their modelling capabilities. The buffer we consider is defined as a set of independent successive cells (cell1, ... , celln). Write (W) and read (R) operations are available. The Write operation, logically, always put a value in the empty cell 'cell1' of the buffer and Read, logically, removes a value from the full cell 'celln'. Values must be read in the order they have been written. This description is left in a abstract way and further constraints or details are given in the next paragraphs. The well known Petri net CM, introduced in 1962 by C.A. Petri, allows the specification of concurrent systems in terms of state variables set (places), events set (transitions) and a flow relation indicating for each event name the pre and post conditions over the state variables set. A place is a bag of resources (tokens) which are all independent inside a place and between several places. A event precondition indicates what resources must be consumed in each net place and the post-condition which resources must be produced. As resources are considered as independent and are shared by all net transition then concurrent behaviour can be modelled in an elegant and easy way. The order relation between events is deduced from the causality relation induced by the flow relation. More precisely, the execution order of transitions is dependent on the places which are considered in the pre or post conditions. For example, simultaneity of transitions is possible when there are no shared places 2 or when enough resources exist in the shared places with respect to the preconditions. Sequence is possible when simultaneity is possible and imposed when an event e' needs resources produced exclusively by an event e. The Petri net semantics is usually given using step sequences or partial languages which express both in an equivalent way the true concurrency of finite Petri nets as it is shown in [ 1] 3. In Figure 1 , we have a Petri net model of the buffer (using the conventionnal graphical representation [ 9] with double arrows for short) whom step semantics is given in Figure 2. The buffer is made of 3 ordered cells b1,b2 and b3 such that values arrive through the cell1 and are removed from the …