Modeling dependencies and simultaneity in membrane system computations

Membrane system computations proceed in a synchronous fashion: at each step all the applicable rules are actually applied. Hence each step depends on the previous one. This coarse view can be refined by looking at the dependencies among rule occurrences, by recording, for an object in a membrane and at a certain stage of the computation, which were the rules that produced it and subsequently (in a later step), which was the rule that consumed it. By recording the rules that have contributed to produce an object we can keep track of its history. Recording more than one rule for each object could be used to consider as equivalent some of the past histories, from the point of view of future use of this object. The other main ingredient in membrane system computations, namely the simultaneity in the rule applications, is usually enforced at the level of observation. We propose a way to look at it in a structural way. This is achieved using zero-safe nets that allows to synchronize transitions, i.e., rule occurrences. Zero-safe nets can be unfolded into causal nets in a classical way, and this unfolding can be manipulated to obtain a merged net where some histories could be considered as equivalent. To these notions of unfoldings, suitable event structures can be associated. The capability of capturing simultaneity of zero-safe nets is then transferred to the level of event structures by adding a way to express which events occur simultaneously.