  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.
