In 1994, Hava Siegelmann proved that her new (1991) computational model, the Artificial Recurrent Neural Network (ARNN), could perform hypercomputation (using infinite precision real weights for the synapses).
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The term "super-Turing computation" appeared in a 1995 Science paper by Hava Siegelmann.
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Hava Siegelmann and Eduardo Sontag, “Analog Computation via Neural Networks,” Theoretical Computer Science 131, 1994: 331-360.