Undecidability in Tensor Network States

Recent work has examined how undecidable problems can arise in quantum information science. We augment this by introducing three new undecidable problems stated in terms of tensor networks. These relate to ideas of Penrose about the physicality of a spin-network representing a physical process, closed timelike curves, and Boolean relation theory. Seemingly slight modifications of the constraints on the topology or the tensor families generating the networks leads to problems that transition from decidable, to undecidable to even always satisfiable.