P=NP and Bitcoin

The present value of working on conjectures…

Anil Nerode of Cornell University has served over sixty years—he’s believed to be the longest such faculty member in university history. He helped found Cornell’s computer science department, advised more than 55 doctoral students—a Department of Mathematics record—and made important contributions to logic, mathematics, computer science, and more. He and Juris Hartmanis jointly oversaw Ken’s Mathematical Sciences Institute postdoc at Cornell in 1986–89.

I found it interesting that Anil has worked on teaching methods:

“I was educated at Chicago under the business of doing the cleanest, shortest expositions of absolutely everything,”

He has developed his own method: He now teaches in historical order. Translating ancient theories into modern notation allows him to show his students the origins of mathematics. This is a cool method.

Cornell Chronicle source—our felicitations

Anil has done extensive work in complex control systems, which is certainly a challenging area for an approach based on clarity and simplicity. He is often credited for co-creating, together with Wolf Kohn, an approach called hybrid systems theory. His partnership with Kohn has even extended to integrating quantum and macroscopic control mechanisms for magnetic batteries on many scales.

Conjectures Which Way?

Anil once said:

Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required.

I blogged about this point in posts titled “We All Guessed Wrong” and “Even Great Mathematicians Guess Wrong”—and about other cases of surprises and about guessing the right side of conjectures.

But events this week have put a transverse thought into my head. The two sides of a conjecture may not be equal in their ramifications. This leads to a further weird thought:

Maybe the greatest overall value is in working hard on the side that you don’t want to be true—and hoping not to solve it.

The fact of Bitcoin hitting a new all-time high this week leads me to muse further about this.

Trillions

If you think you have a proof that P=NP then it is worth a lot of money. Well for sure, at least, you will get a prize like the Turing Award. But because it would break bitcoin and other digital monies you would possibly get into the thicket of billions of dollars. Maybe more like trillions.

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Exactly what it would be worth to you, I don’t know. That’s a hypothetical. But there is a non-hypothetical that, from the understanding Ken and I have of how hedge transactions are valued, implies that one side of “P=NP” has clear and present monetary value:

What is the present value of not knowing that P=NP, and of exerting work that furthers the general sureness of this status?

A 2022 article titled “P = NP: The Dangling Blackhole Under Crypto” puts the stakes in stark terms:

P=NP is [about] whether a problem (in the case of Cryptocurrencies, a transaction) can be solved procedurally and systemically just as quickly as it can be verified with a hash. The security of the blockchain, such as [for] Bitcoin, is wholly dependent on the assumption that P does not equal NP. However, if P = NP is proven, the Bitcoin can be breached just as quickly as it can be authenticated.

A breach would have the kind of concrete effect we have in mind when we say, “from dollars to donuts…“—although the donuts involved could technically come from work like this:

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Open Problems

Is it sensible to assign a present value to work on one side of the P=NP question? More than work that tries to establish the other side?