Let a directed graph $G = (V, E)$ be given, plus a constraint map $c: E \rightarrow V$ and a set $T \subseteq V$ of initial token locations. A valid move consists of sliding a token from $v$ to $w$ if:
- $v \in T$ — to slide a token, it must be there.
- $w \not \in T$ — to move a token somewhere, the new place must be empty.
- $c(v, w) \in T$ — (the unusual bit) to use an edge, some token must occupy the "activator" vertex. Note that the trivial constraint $c(v, w) = v$ is allowed.
(Then $T \Rightarrow^1 (T \setminus \{v\}) \cup \{w\}$. An interesting decision problem might be this: given $v$, is there a $T'$ s.t. $T \Rightarrow^* T'$ and $v \in T'$; there are plenty others.)
I think this can be reformulated as a Petri net, with transitions $\{v, c(v, w)\} \rightarrow \{w, c(v, w)\}$ when $c(v, w) \neq v$ and $\{v\} \rightarrow \{w\}$ when $c(v, w) = v$. However, I am very unfamiliar with Petri nets. Does this have some correspondence to Petri nets at all? Is the subtype of Petri nets where all transitions are on the above form well-studied? What are the major results, especially concerning the computational complexity of the most interesting decision and function problems?
