  Probabilistic models over strings have played a key role in developing
methods allowing indels to be treated as phylogenetically informative events.
There is an extensive literature on using automata and transducers on
phylogenies to do inference on these probabilistic models, in which an
important theoretical question in the field is the complexity of computing the
normalization of a class of string-valued graphical models. This question has
been investigated using tools from combinatorics, dynamic programming, and
graph theory, and has practical applications in Bayesian phylogenetics. In this
work, we revisit this theoretical question from a different point of view,
based on linear algebra. The main contribution is a new proof of a known result
on the complexity of inference on TKF91, a well-known probabilistic model over
strings. Our proof uses a different approach based on classical linear algebra
results, and is in some cases easier to extend to other models. The proving
method also has consequences on the implementation and complexity of inference
algorithms.

Populations and Evolution;Formal Languages and Automata Theory;Computation