We still don’t know computing’s limits (Image: Louie Psihoyos/Science Faction Jewels/Getty)

Initially hailed as a solution to the biggest question in computer science, the latest attempt to prove P ≠ NP – otherwise known as the “P vs NP” problem – seems to be running into trouble.

Two prominent computer scientists have pointed out potentially “fatal flaws” in the draft proof by Vinay Deolalikar of Hewlett-Packard Labs in Palo Alto, California.

Since the 100-page proof exploded onto the internet a week ago, mathematicians and computer scientists have been racing to make sense of it.


The problem concerns the speed at which a computer can accomplish a task such as factorising a number. Roughly speaking, P is the set of problems that can be computed quickly, while NP contains problems for which the answer can be checked quickly.

Serious hole?

It is generally suspected that P ≠ NP. If this is so, it would impose severe limits on what computers can accomplish. Deolalikar claims to have proved this. If he turns out to be correct, he will earn himself a $1 million Millennium prize from the Clay Mathematics Institute in Cambridge, Massachusetts.

Much of the excited online discussion regarding the proof has taken place on the blog of Richard Lipton, a computer scientist at the Georgia Institute of Technology with over 30 years of experience working on P vs NP.

This morning, however, Lipton posted an email from another computer scientist, Neil Immerman of the University of Massachusetts, who claims to have found a “serious hole” in Deolalikar’s paper.

Immerman’s key point relates to what he says is an error in Deolalikar’s assumptions while attempting to describe the set of P in a particular way. Deolalikar attempts to show that some problems are in NP but not in P (and thus that P ≠ NP) by invoking another mathematical set known as FO(LFP). Immerman says that this set can’t be used in this way, given other methods deployed in the proof.

Professional status

As with all famously unsolved mathematical problems, attempts at tackling P vs NP are common but few are taken seriously because they tend to lack strong theoretical foundations.

Excitement at Deolalikar’s proof seems to have been fuelled by his professional status as a researcher at HP Labs, and also by a comment from Stephen Cook of the University of Toronto, who originally formulated the P vs NP problem, stating that Deolalikar’s work appeared to constitute a “relatively serious claim“.

Commenting on Lipton’s blog earlier this week, Terence Tao at the University of California, Los Angeles, broke down the issue of the proof’s correctness into three scenarios.

Wiki power

In the best-case scenario, minor tweaks would fix Deolalikar’s argument and prove once and for all that P ≠ NP. Alternatively, major repairs might ultimately rescue the proof. Failing that, Deolalikar’s general proof strategy could offer hope for future attempts at a proof.

Tao wrote then that he thought the first two scenarios were unlikely, but the third was still unresolved. Responding to Immerman’s critique today, he revised his assessment to “No”, “No” and “Probably not”.

Even if these criticisms stand, the whole incident might still help galvanise mathematical discovery indirectly. Following the Deolalikar claim, a wiki was set up where people could post opinions on Deolalikar’s proof. Comments range from corrections of minor typos to detailed mathematical critiques of his methods.

This flurry of online activity suggests that a new way of doing mathematics might be emerging, with blogs and wikis rivalling blackboards and journals – a potentially positive outcome, even if P vs NP remains unresolved.

“The internet is making a huge difference to the way mathematicians operate” says Timothy Gowers of the University of Cambridge . “A process that might have taken weeks and weeks has taken place extremely quickly.”