abstract = "Recent theoretical work has characterised the search
bias of GP sub-tree swapping crossover in terms of
program length distributions, providing an exact fixed
point for trees with internal nodes of identical arity.
However, only an approximate model (based on the notion
of average arity) for the mixed-arity case has been
proposed. This leaves a particularly important gap in
our knowledge because multi-arity function sets are
commonplace in GP and deep lessons could be learnt from
the fixed point. In this paper, we present an accurate
theoretical model of program length distributions when
mixed-arity function sets are employed. The new model
is based on the notion of an arity histogram, a count
of the number of primitives of each arity in a program.
Empirical support is provided and a discussion of the
model is used to place earlier findings into a more
notes = "p42 'This equation (4) has now become a multivariate
Lagrange distribution of the second kind.' p43
'Lagrange distribution of the third kind'