  We study the identification of causal effects, motivated by two improvements
to identifiability which can be attained if one knows that some variables in a
causal graph are functionally determined by their parents (without needing to
know the specific functions). First, an unidentifiable causal effect may become
identifiable when certain variables are functional. Second, certain functional
variables can be excluded from being observed without affecting the
identifiability of a causal effect, which may significantly reduce the number
of needed variables in observational data. Our results are largely based on an
elimination procedure which removes functional variables from a causal graph
while preserving key properties in the resulting causal graph, including the
identifiability of causal effects.
