Predicting When P=NP is Resolved



Has it outlasted the ability to estimate when?



Composite of src1, src2

Ryohei Hisano and Didier Sornette wrote in 2012 a paper titled, “On the distribution of time-to-proof of mathematical conjectures.”

Today Ken and I discuss predicting the end to mathematical conjectures. This is apart from considering odds on which way they will go, which we also talk about.



Nine years ago the Christmas issue of the New Scientist magazine analyzed a small set of solved mathematical conjectures and used it to forecast when the P vs. NP conjecture would be solved. The article estimated that the “probability for the P vs. NP problem to be solved by the year 2024 is roughly 50%”.

New Scientist 2019 “biggest problem” source

This was done without any special insight into P vs. NP, just its prominence among the world’s best mathematicians and time since inception. How can one make the case that this argument is not silly, that it is not a wild guess?

Time to Solve a Conjecture

The 2010 New Scientist article could be considered a bagatelle since they looked at only 18 conjectures, but Hisano and Sornette took the idea seriously. They widened the data field to Wikipedia’s list of mathematical conjectures. They removed a dozen-plus conjectures whose origin dates and/or solution dates are not firmly known, leaving 144 conjectures: 60 solved and 84 still open as of 2012.

It should be understood that their conclusions about the whole enterprise are negative. They say in their abstract:

We find however evidence that the mathematical process of creation is too much non-stationary, with too little data and constraints, to allow for a meaningful conclusion. … In conclusion we cannot really reject the simplest model of an exponential rate of conjecture proof with a rate of 0.01/year for the dataset that we have studied, translating into an average waiting time to proof of 100 years.

They found exponential growth in notable conjectures being formulated, which they ascribe to growth in the number of mathematicians. They try adjusting for this by invoking the overall growth curve of the world population.

Then they find a dearth of conjectures with midrange solution times, compared to the number needed to give strong fits to simple models. They try cutting quickly-resolved conjectures (ones solved in under 20 years) from the dataset to improve the fits, and obtain only an enhancement of evidence for a slow exponential curve of solution time. Here is a figure from their paper showing the exponential curves:







In the end they are not able to find many hard distinctions between possible models. Most in particular, they conclude by calling the simple -solution-chance-per-year model “our best model.”

But that model in turn, without memory of the age of the conjecture, struck me—Ken writing here—as running counter to a famous controversial argument of recent vintage. So Dick and I put our heads together again…

Doomsday For Conjectures?

The Doomsday Argument can be phrased as a truism:

If your hearing about is a uniformly random event in the lifespan of , which began years ago, then the odds are 50-50 that the future lifetime of will be no less than years and no more than years.

This extends to say the probability is 90% that will be no less than and no more than . The numbers come from considering the middle half or middle 90% of the span.

The key word in our phrasing is if—if your cognizance is a uniformly random event of the lifespan. The argument’s force comes from taking your birthday as a random event in the span of humanity. We can adjust for the population growth curve and our ignorance of prehistoric times by taking to be your ordinal in human lives ordered by birthday and projecting in the same units. If there were 50 billion people before you, this speaks 95% confidence that the sum of human lives will not reach a trillion.

We can try the same reasoning on conjectures. Suppose you just learned about the P vs. NP problem from catching a reference to this blog. By Kurt Gödel’s letter to John von Neumann we date the problem to 1956, which makes years. Presuming your encounter is random, you can conclude a 50-50 chance of the conjecture being solved between the years 2040 (which is 21 years from now) and 2208 (which is still shorter than Fermat’s Last Theorem lasted). The less you knew about P vs. NP, the more random the encounter—and the stronger the inference.

Perhaps Wikipedia’s 1971 origin for P vs. NP from its precise statement by Steve Cook is more realistic. Was my learning about it in 1979 ( ) a random event? If so, I should have been 75% confident of a solution by 2003. Its lasting 16 more years and counting busts the 83.3% confidence interval. Well, Dick learned about it within weeks, at most months, after inception. If that was random then its longevity counts as a 99.9% miss. Of course, Dick’s learning about it was far from random. But.

Doomsday Conflict

Our point, however, is this: Consider any conjecture that has lasted years. That’s the point at which the exponential and Doomsday models come into conflict:

By Doomsday, there is 75% confidence of the conjecture lasting at least more years.

more years. But since , by the H-S best model, one should have less confidence.

The longer a conjecture lasts, the greater this conflict. We are not addressing the issue of uniform sampling over the lifespan of the conjecture, but this conflict applies to all observers past the 87-year mark. The honeycomb conjecture was documented by 36 BCE and finally solved by Thomas Hales in 1999.

Note that since , one cannot evade our model conflict by saying only a small fraction of conjectures last that long. Despite H-S noting that the honeycomb conjecture and Fermat and some others break their curve, their curve at left below shows excellent agreement up to and beyond the 100-year mark.



The curve at right shows that a rate fits even better when short-lived conjectures are deleted. Since , that makes the conflict set in at years old. Then it applies even for P vs. NP dated to Cook. At the conflict is considerable: Doomsday 66.7% of its lasting at least 50 more years, versus chance of its lasting that long by H-S.

This conflict says there must be a factor that invalidates either or both lines of reasoning. But the Hisano-Sornette estimates are supported by data, while our Doomsday representation is quite generic.

Do Odds Matter?

A further wrinkle is whether the collective balance of opinion on whether a conjecture is true or false has any bearing. If the field is evenly divided, does that argue that the conjecture will take longer to resolve?

The idea of “odds” is hard to quantify and has come into play only recently. Despite lots of chat about betting on P vs. NP, and Bill Gasarch’s wonderful polls on which side is true, data for other problems is scant.

This prompts us to mention that the Springer LNCS 10,000 anniversary volume came out in October. It includes a contribution by Ken titled, “Rating Computer Science Via Chess”—the title refers both to how the chess ratings of the best chess computers mirrored Moore’s Law and how they draw on many general CS achievements in both hardware and software. For now we note also the contribution by Ryan Williams titled, “Some Estimated Likelihoods for Computational Complexity.” Ryan places only 80% on and puts versus at 50-50.

There are also non-betting and non-opinion senses in which conjectures are regarded as “almost certainly true.” We recently posted about this. Perhaps something more can be gleaned from highlighting those conjectures in the H-S data set to which such criteria apply.

All of this speculation may be less idle and less isolated if it can be carried over to unsolved problems in other walks of life. Corporations and consortia doing open-ended research may already need to quantify the average time to achieve potential breakthroughs—or dismiss them—because they deal with many “feelers” at a time. The Doomsday assumption of random encounters probably doesn’t apply there. But there could be large-scale tests of it by “punters” seeking out random lifespans they can bet on, such as runs of shows or the time to make a movie of Cats, and profit by applying it.

Open Problems

What do you think about attempts to forecast the time to solve P vs. NP, or to forecast breakthroughs on larger scales?

There are many rebuttals to the Doomsday Argument, some general and some context-specific. Is our argument that it conflicts generally with memoryless exponential models subsumed by any of them?