  We consider weighted tiling systems to represent functions from graphs to a
commutative semiring such as the Natural semiring or the Tropical semiring. The
system labels the nodes of a graph by its states, and checks if the
neighbourhood of every node belongs to a set of permissible tiles, and assigns
a weight accordingly. The weight of a labeling is the semiring-product of the
weights assigned to the nodes, and the weight of the graph is the semiring-sum
of the weights of labelings. We show that we can model interesting algorithmic
questions using this formalism - like computing the clique number of a graph or
computing the permanent of a matrix. The evaluation problem is, given a
weighted tiling system and a graph, to compute the weight of the graph. We
study the complexity of the evaluation problem and give tight upper and lower
bounds for several commutative semirings. Further we provide an efficient
evaluation algorithm if the input graph is of bounded tree-width.
