  The theory of general relativity predicts the existence of closed time-like
curves (CTCs), which theoretically would allow an observer to travel back in
time and interact with their past self. This raises the question of whether
this could create a grandfather paradox, in which the observer interacts in
such a way to prevent their own time travel. Previous research has proposed a
framework for deterministic, reversible, dynamics compatible with non-trivial
time travel, where observers in distinct regions of spacetime can perform
arbitrary local operations with no contradiction arising. However, only
scenarios with up to three regions have been fully characterised, revealing
only one type of process where the observers can verify to both be in the past
and future of each other. Here we extend this characterisation to an arbitrary
number of regions and find that there exist several inequivalent processes that
can only arise due to non-trivial time travel. This supports the view that
complex dynamics is possible in the presence of CTCs, compatible with free
choice of local operations and free of inconsistencies.
