We introduce a geometrical setting which seems promising for the study
of computation in multiset rewriting systems, but could also be applied to register machines and other models of computation. This approach will be applied here to membrane
systems (also known as P systems) without dynamical membrane creation. We discuss
the role of maximum parallelism and further simplify our model by considering only one
membrane and sequential application of rules, thereby arriving at asynchronous multiset
rewriting systems (AMR systems). Considering only one membrane is no restriction, as
each static membrane system has an equivalent AMR system. It is further shown that
AMR systems without a priority relation on the rules are equivalent to Petri Nets. For
these systems we introduce the notion of asymptotically exact computation, which allows
for stochastic appearance checking in a priori bounded (for some complexity measure)
computations. The geometrical analogy in the lattice Nd0
; d 2 N, is developed, in which a
computation corresponds to a trajectory of a random walk on the directed graph induced
by the possible rule applications. Eventually this leads to symbolic dynamics on the partition generated by shifted positive cones C+
p , p 2 Nd0
, which are associated with the
rewriting rules, and their intersections. Complexity measures are introduced and we consider non-halting, loop-free computations and the conditions imposed on the rewriting
rules. Eventually, two models of information processing, control by demand and control by
availability are discussed and we end with a discussion of possible future developments
