Characterizations of Recognizable Picture Series∗

We investigate power series on pictures which map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We will assign weights to devices, ranging from tiling systems and domino systems to picture automata. We prove that, for commutative semirings, the behaviors of weighted picture automata are precisely alphabetic projections of series defined by rational operations and also coincide with the families of projections of tile-local and projections of hv-local series. Furthermore, we establish properties of unambiguous picture languages by characterizations in terms of unambiguous rational operations with injective projections and devices such as unambiguous (quadrapolic) picture automata or unambiguous domino systems.