Step traces

Goal-Oriented Reduction of Automata Networks

“…Automata networks as presented can be considered as a class of 1-safe Petri Nets [3] (at most one token per place) having groups of mutually exclusive places, acting as the automata, and where each transition has one and only one incoming and out-going arc and any number of read arcs. The semantics considered in this paper where transitions within different automata can be applied simultaneously echoes with Petri net step-semantics and concurrent/maximally concurrent semantics [20,30,19]. In the Boolean network community, such a semantics is referred to as the asynchronous generalized update schedule [2].…”

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.…”

“…For step traces, obviously, more general dependence and causal structures are needed to describe the invariant relationships between action occurrences. The relational structures studied in [2,3] and used in [4] to describe the causality in step traces, have -instead of a single strict partial order (causality) relation -two relations: a 'not later than' relation to represent weak causality (i.e., before or in the same step) and a 'mutual exclusion' mutex relation for pure interleaving (not allowed in the same step but not necessarily causally ordered). Order structures are (labelled)…”

“…Invariant structures are order structures which can be considered as generalised partial orders in the sense that they satisfy a generalised Szpilrajn's property: each invariant structure is the intersection of all its saturations. In [4], it is moreover demonstrated how to capture the intrinsic dependencies in a step sequence (over a given step alphabet). Furthermore, equivalent step sequences generate the same dependence structure; every step sequence corresponds to a saturated version of its dependence structure; and, finally, all step sequences obtained by saturating a dependence structure are equivalent.…”

“…Step traces [4] are an extension of the classical Mazurkiewicz traces, a basic and well-established model to represent concurrent behaviour [1,12,13,14].…”