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The semantics of neural networks can be analyzed mathematically as a distributed system of knowledge and as systems of possible worlds expressed in the knowledge. Learning in a neural network can be analyzed as an attempt to acquire a representation of knowledge. We express the knowledge system, systems of possible worlds, and neural architectures at different stages of learning as categories. Diagrammatic constructs express learning in terms of pre-existing knowledge representations. Functors express structure-preserving associations between the categories. This analysis provides a mathematical vehicle for understanding connectionist systems and yields design principles for advancing the state of the art.
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