  Causal identification in causal Bayes nets (CBNs) is an important tool in
causal inference allowing the derivation of interventional distributions from
observational distributions where this is possible in principle. However, most
existing formulations of causal identification using techniques such as
d-separation and do-calculus are expressed within the mathematical language of
classical probability theory on CBNs. However, there are many causal settings
where probability theory and hence current causal identification techniques are
inapplicable such as relational databases, dataflow programs such as hardware
description languages, distributed systems and most modern machine learning
algorithms. We show that this restriction can be lifted by replacing the use of
classical probability theory with the alternative axiomatic foundation of
symmetric monoidal categories. In this alternative axiomatization, we show how
an unambiguous and clean distinction can be drawn between the general syntax of
causal models and any specific semantic implementation of that causal model.
This allows a purely syntactic algorithmic description of general causal
identification by a translation of recent formulations of the general ID
algorithm through fixing. Our description is given entirely in terms of the
non-parametric ADMG structure specifying a causal model and the algebraic
signature of the corresponding monoidal category, to which a sequence of
manipulations is then applied so as to arrive at a modified monoidal category
in which the desired, purely syntactic interventional causal model, is
obtained. We use this idea to derive purely syntactic analogues of classical
back-door and front-door causal adjustment, and illustrate an application to a
more complex causal model.
