# 20.6 Branching Immediate Observation nets

Branching Immediate Observation (BIO) nets are a subclass of Petri nets defined by their transitions, which are of the form $$|{}^\bullet t - t^{\bullet}| \le 1$$. This multiset subtraction means that all input places of $$t$$ with arc weight $$k$$ are also output places of $$t$$ with arc weight $$k$$, except for one which may have input arc weight $$k$$ and output arc weight $$k-1$$. BIO nets generalize Basic Parallel Processes (BPP) nets and Immediate Observation (IO) nets - a token can “branch” into any number of tokens as in BPP, but may need to “observe” that another token is in a certain place to proceed with its transition as in IO.

BIO nets have a non-semilinear reachability relation but a reachability problem which is PSPACE-complete. This makes BIO nets the first natural net class with non-semilinear reachability relation for which the reachability problem is provably simpler than for general Petri nets. Additionally, although the reachability relation is not flat (in the sense of [Leroux, Sutre, 05]), it is locally flat (specifically it is $$pre^*$$ flat) allowing the use of efficient existing model checking tools with acceleration techniques like FAST [Bardin et al., 03].

We think this is an interesting class, and we are looking both for application domains for BIO nets (e.g. in the direction of chemical reaction networks) and for consequences of this result in other domains that have problems connected to the Petri net reachability problem (e.g. data nets, formal languages, process calculi…).