  Clift and Murfet (2019) introduced a naive Bayesian smooth relaxation of
Turing machines motivated by work in differential linear logic; this was
subsequently used to endow spaces of program codes of bounded length with a
smooth manifold structure via the staged-pseudo universal Turing machine
introduced by Clift, Murfet and Wallbridge (2021).
  In this paper, we give a general construction for simulating n-tape Turing
machines on a single tape Turing machine such that the (naive Bayesian)
uncertainty is propagated in an equivalent manner. This result suggests a
stronger kind of equivalence between single tape and n-tape Turing machines
than that established by classical results, however, the clarification of these
implications is open to future work. We then construct a pseudo universal
Turing machine which similarly preserves the propagation of uncertainty in its
simulations, and observe that this gives rise to a particularly natural smooth
relaxation of the space of programs.
