In theory of computation, he proved the first results on computational complexity in nonlinear controllability, and introduced together with his student Hava Siegelmann a new approach to analog computation and super-Turing computing.
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.
Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers (For example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive languages.) or vice versa.
Burgin, as well as other researchers (including Selim Akl, Eugene Eberbach, Peter Kugel, Jan van Leeuwen, Hava Siegelmann, Peter Wegner, and Jiří Wiedermann) who studied different kinds of super-recursive algorithms and contributed to the theory of super-recursive algorithms, have argued that super-recursive algorithms can be used to disprove the Church-Turing thesis, but this point of view has been criticized within the mathematical community and is not widely accepted.