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Extra resources for Advances in Computers, Vol. 45: Emphasizing Parallel Programming Techniques
12 is as shown in Table IV, in which fy is the name of the start transition of y, p o is the name of the place between the final transition of or2 and t,, and p c is the name of the place between the final transition of c and ty. Now, we cannot derive a threading for Fig. 6(a). Consider the partial labeling shown in Table llI. Node Y has two labels that must be equivalent. The first label (the one going to translation t9) can be simplified to p, because all transitions in the label have only one predecessor.
4 Another method to prove simple control is by computing a labeling and showing that it is (or it is not) a balanced threading. An algorithm to compute a threading is derivable from the definition. The idea is to assign labels to the outputs and compute labels of predecessors. If ~ they are no1 equivalent, then there is no 4The rank theorem  shows that the well formedness property of free-choice nets can be checked in polynomial time. This property is intimately related to simple control, hence it can be proven that simple control is also polynomial in free choice nets [22,25,29].
3 Partial Order Behavior A sequence of enabled transitions in a system describes executions of the net. These sequences, however, do not determine causal dependencies between transition occurrences. To describe causal dependencies, an alternative representation of executions is needed, in which an execution is a partial order of firings (as opposed to a sequence, which is a total ordering). This partial order is also described by a net called a causal net. A causal net has no choice nor cycles.