https://arxiv.org/pdf/2001.02511.pdf
V. CONCLUSIONS
We have developed a characterisation of deterministic
processes in the presence of CTCs for an arbitrary number of localised regions. Our proofs have demonstrated
that non-trivial time travel between multiple regions is
consistent with the absence of a logical paradox as long
as once the outputs of all but two regions are fixed, at
most one-way signalling is possible.
The most significant result of our work is our discovery of distinct non-trivial quadripartite process functions
which are compatible with the presence of CTCs. This
demonstrates that when multiple local regions communicate with each other in the presence of CTCs, there is a
broad range of communication scenarios which still allow
freedom of choice for observers in each region without the
development of a logical inconsistency such as a grand father paradox. The range of distinct communication scenarios which are consistent with the presence of CTCs
proves that the way CTCs allow multiple observers in distinct regions to communicate is not incredibly restricted
by a conflict with locality, free will and logical inconsistencies. As a result, we have demonstrated that there is
a range of scenarios in which multiple observers can communicate without causal order in a classical framework.