  We discuss algorithmic approach to growth of the codimension sequences of
varieties of multilinear algebras, or, equivalently, the sequences of the
component dimensions of algebraic operads. The (exponentional) generating
functions of such sequences are called codimension series of varieties, or
generating series of operads.
  We show that in general there does not exist an algorithm to decide whether
the growth exponent of a codimension sequence of a variety defined by given
finite sets of operations and identities is equal to a given rational number.
In particular, we solve negatively a recent conjecture by Bremner and Dotsenko
by showing that the set generating series of binary quadratic operads with
bounded number of generators is infinite. Then we recall algorithms which in
many cases calculate the codimension series in the form of a defining algebraic
or differential equation. For a more general class of varieties, these
algorithms give upper and lower bounds for the codimensions in terms of
generating functions and asymptotical bounds for the growth of codimensions.
The upper bound (based on an operadic version of the Golod--Shafarevich
theorem) is just a formal power series satisfying an algebraic equation defined
effectively by the generators and the identities of the variety. The first
stage of an algorithm for the lower bound is the construction of a Groebner
basis of the operad. If the Groebner basis happens to be finite and satisfies
mild restrictions, a recent theorem by the author and Anton Khoroshkin
guarantees that the desired generating function is either algebraic or
differential algebraic. We describe algorithms producing such equations. In the
case of infinite Groebner basis, these algorithms applied to its finite subsets
give lower bounds for the generating function of the codimension sequence.
