  The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set
$X\cap\Gamma$ when $X$ is a subvariety of a semiabelian variety $G$ over a
finite field $\mathbb{F}_q$ and $\Gamma$ is a finitely generated subgroup of
$G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That
description is here made effective, and extended to arbitrary commutative
algebraic groups $G$ and arbitrary finitely generated
$\mathbb{Z}[F]$-submodules $\Gamma$. The approach is to use finite automata to
give a concrete description of $X\cap \Gamma$. These methods and results have
new applications even when specialised to the case when $G$ is an abelian
variety over a finite field, $X\subseteq G$ a subvariety defined over a
function field $K$, and $\Gamma=G(K)$. As an application of the
automata-theoretic approach, a dichotomy theorem is established for the growth
of the number of points in $X(K)$ of bounded height. As an application of the
effective description of $X\cap\Gamma$, decision procedures are given for the
following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does
it contain an infinite coset?
