Classifying invariant structures of step traces

A Precise Characterisation of Step Traces and Their Concurrent Histories

“…First of all, there is simultaneity indicating that two actions may occur together in a step; secondly, serialisability specifies possible execution orders for potentially simultaneous actions; thirdly, interleaving declares for actions that cannot occur simultaneously that no specific ordering is required. The latter two relations can also be captured in terms of a single sequentialisability relation, see [4,7]. This then leads to a notion of a step alphabet consisting of a finite set of symbols (action names) and two binary relations, simultaneity and sequentialisability.…”

“…On the other hand, the definition of step alphabets almost automatically leads to a hierarchy of step alphabets, depending on which combinations of simultaneity and sequentialisability are allowed. In [7], eight subclasses of step alphabets are distinguished. These include a class corresponding to Mazurkiewicz traces; step alphabets that combine the independence relation of Mazurkiewicz traces with step sequences (and simultaneity is the same as serialisability), and a class of step alphabets that leads to comtraces [9,11].…”

“…In [7], subfamilies of step traces have been investigated based on a classification of step alphabets, defined by assuming that one or more of the relations sim \ seq, seq \ sim, and sim ∩ seq are empty. This led to eight classes of step alphabets.…”

“…The alphabets in Θ wdp∪ssi∪con∪inl and Θ wdp∪con∪inl are of little interest. Those in Θ, Θ inl , Θ wdp∪ssi , Θ wdp∪con , Θ wdp∪ssi∪con and Θ wdp∪ssi∪inl are the ones that were studied in [7]. The eight others are as new subclasses, our subject of investigation in the next section.…”

“…Invariant structures are order structures (see [4,7]). Actually, every order structure can be 'closed' by adding pairs to their mutex and weak causality relations to obtain the unique invariant structure that has the same set of saturated extensions.…”