Searching for Easy Answers to Hard Computational Questions (preview)

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Whether or not machines can quickly answer yes-or-no questions could affect everything from national security to the limits of human knowledge

By John Pavlus

Image: SET DESIGN: KYLE BEAN; PHOTOGRAPHY: RYAN HOPKINSON

In Brief

• The ?P versus NP? question asks whether tough problems whose solutions can be quickly checked (like a jigsaw puzzle) are, at heart, easily solvable as well.
• Despite decades of investigation, no one has been able to prove that the two categories are different. If they were not, machines would acquire enormous power.
• The problem does not just affect code breakers and Web searches. It suggests a fundamental limitation for biological evolution, physical laws and the nature of knowledge.

On a snowy day in Princeton, N.J., in March 1956, a short, owlish-looking man named Kurt G?del wrote his last letter to a dying friend. G?del addressed John von Neumann formally even though the two had known each other for decades as colleagues at the Institute for Advanced Study in Princeton. Both men were mathematical geniuses, instrumental in establishing the U.S.'s scientific and military supremacy in the years after World War II. Now, however, von Neumann had cancer, and there was little that even a genius like G?del could do except express a few overoptimistic pleasantries and then change the subject:

Dear Mr. von Neumann:

This article was originally published with the title Machines of the Infinite.

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