In this talk, a theory for the diagrammatic representation and computation of finite discrete functions and abstractions is presented. The theory is defined in terms of two basic operations that are computed directly on tables: the functional abstraction and the functional application or reduction. However, unlike the analogous operations of the lambda-calculus, these operations are not fully reversible and the system has an inherent information loss. For this, abstractions have an associated entropy value that measures their degree of indeterminacy or information content. The theory is applied to the definition and construction of an associative memory, where the information is accessed by content, with its associated memory register, recognition and retrieval operations. A case study in visual memory with very promising preliminary results is presented.

The overall theory suggests a comprehensive view or space of possible computations that is defined in relation to (1) the trade-off between extensional and intensional forms of expressing information and (2) the formats employed in computations. This trade-off underlies the knowledge representation trade-off of artificial intelligence and cognitive science.The computing formats, in turn, range from the linguistic format, whose paradigmatic case is the Turing Machine, to fully distributed formats including neural networks and the diagrammatic format.The view suggests that the trade-off between extensions and intensions is the manner in which the entropy of abstractions surface in the linguistic format. It also supports the case of direct representation in AI and the case of images in the imagery debate, and helps to clarify the opposition between symbolic and sub-symbolic computations.