Automatic generation of data visualizations allows to quickly deploy data visualizations. In visual analytics, the combination of automatic and human analysis increases the effort necessary to achieve similar effects substantially. Where automatic visualization only needs to map the data, in visual analytics the whole data preparation and processing pipeline has to be considered. The user is interested in representations reflecting certain interpretations of the data, for example the idea that different groups represent different clusters in the data. In this paper, we prove that an information-driven automatic design of visual analytics pipelines is feasible. To this end, we prove that the ability of an analysis system to derive and visualize data supporting inquired information is decidable – at least for real-world applications. Having overcome this major obstacle, we outline a general algorithm scheme that can be implemented on a wide range of data and information models.