Computational tameness of classical non-causal models

We show that the computational power of the non-causal circuit model, i.e., the circuit model where the assumption of a global causal order is replaced by the assumption of logical consistency, is completely characterized by the complexity class~$\operatorname{\mathsf{UP}}\cap\operatorname{\mathsf{coUP}}$. An example of a problem in that class is factorization. Our result implies that classical deterministic closed timelike curves (CTCs) cannot efficiently solve problems that lie outside of that class. Thus, in stark contrast to other CTC models, these CTCs cannot efficiently solve~$\operatorname{\mathsf{NP-complete}}$ problems, unless~$\operatorname{\mathsf{NP}}=\operatorname{\mathsf{UP}}\cap\operatorname{\mathsf{coUP}}=\operatorname{\mathsf{coNP}}$, which lets their existence in nature appear less implausible. This result gives a new characterization of~$\operatorname{\mathsf{UP}}\cap\operatorname{\mathsf{coUP}}$ in terms of fixed points.