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On a Tuesday evening this spring, Sergey Brin, the co-founder of Google, became part man and part machine. About 40 people, all gathered here at a NASA campus for a nine-day, $15,000 course at Singularity University, saw it happen.

While the flesh-and-blood version of Mr. Brin sat miles away at a computer capable of remotely steering a robot, the gizmo rolling around here consisted of a printer-size base with wheels attached to a boxy, head-height screen glowing with an image of Mr. Brin’s face. The BrinBot obeyed its human commander and sputtered around from group to group, talking to attendees about Google and other topics via a videoconferencing system.

The BrinBot was hardly something out of “Star Trek.” It had a rudimentary, no-frills design and was a hodgepodge of loosely integrated technologies. Yet it also smacked of a future that the Singularity University founders hold dear and often discuss with a techno-utopian bravado: the arrival of the Singularity — a time, possibly just a couple decades from now, when a superior intelligence will dominate and life will take on an...