  While feedback loops are known to play important roles in many complex
systems, their existence is ignored in a large part of the causal discovery
literature, as systems are typically assumed to be acyclic from the outset.
When applying causal discovery algorithms designed for the acyclic setting on
data generated by a system that involves feedback, one would not expect to
obtain correct results. In this work, we show that -- surprisingly -- the
output of the Fast Causal Inference (FCI) algorithm is correct if it is applied
to observational data generated by a system that involves feedback. More
specifically, we prove that for observational data generated by a simple and
$\sigma$-faithful Structural Causal Model (SCM), FCI is sound and complete, and
can be used to consistently estimate (i) the presence and absence of causal
relations, (ii) the presence and absence of direct causal relations, (iii) the
absence of confounders, and (iv) the absence of specific cycles in the causal
graph of the SCM. We extend these results to constraint-based causal discovery
algorithms that exploit certain forms of background knowledge, including the
causally sufficient setting (e.g., the PC algorithm) and the Joint Causal
Inference setting (e.g., the FCI-JCI algorithm).
