  Neurally-parameterized Structural Causal Models in the Pearlian notion to
causality, referred to as NCM, were recently introduced as a step towards
next-generation learning systems. However, said NCM are only concerned with the
learning aspect of causal inference but totally miss out on the architecture
aspect. That is, actual causal inference within NCM is intractable in that the
NCM won't return an answer to a query in polynomial time. This insight follows
as corollary to the more general statement on the intractability of arbitrary
SCM parameterizations, which we prove in this work through classical 3-SAT
reduction. Since future learning algorithms will be required to deal with both
high dimensional data and highly complex mechanisms governing the data, we
ultimately believe work on tractable inference for causality to be decisive. We
also show that not all ``causal'' models are created equal. More specifically,
there are models capable of answering causal queries that are not SCM, which we
refer to as \emph{partially causal models} (PCM). We provide a tabular taxonomy
in terms of tractability properties for all of the different model families,
namely correlation-based, PCM and SCM. To conclude our work, we also provide
some initial ideas on how to overcome parts of the intractability of causal
inference with SCM by showing an example of how parameterizing an SCM with SPN
modules can at least allow for tractable mechanisms.
  We hope that our impossibility result alongside the taxonomy for tractability
in causal models can raise awareness for this novel research direction since
achieving success with causality in real world downstream tasks will not only
depend on learning correct models as we also require having the practical
ability to gain access to model inferences.
