Deolalikar P vs NP paper

Note: This is currently an UNOFFICIAL page on Deolalikar's P!=NP paper, and is not yet affiliated with a Polymath project.

The paper

• Vinay Deolalikar's web page
• First draft (date?)
• Second draft (date?)

Proof strategy

(Excerpted from this comment of Ken Regan.)

Deolalikar has constructed a vocabulary V such that:

1. Satisfiability of a k-CNF formula can be expressed by NP-queries over V—in particular, by an NP-query Q over V that ties in to algorithmic properties.
2. All P-queries over V can be expressed by FO+LFP formulas over V.
3. NP = P implies Q is expressible by an LFP+FO formula over V.
4. If Q is expressible by an LFP formula over V, then by the algorithmic tie-in, we get a certain kind of polynomial-time LFP-based algorithm.
5. Such an algorithm, however, contradicts known statistical properties of randomized k-SAT when k >= 9.

Possible issues

Issues with LFP

There appear to be three issues related to the use of the characterization of P in terms of first order logic, an ordering and a least fixed point operator. All of these are discussed in the Lipton/Regan post

1. Is the lack of ordering in the logical structures used to define the LFP structure a problem ? On the surface, it appears to be, since it is not known whether FO(LFP) can be used to characterize P without ordering.
2. The paper requires that a certain predicate in the FO(LFP) formula be unary, and forces this by expanding neighborhoods and constructing k-tuples of parameters to act as single parameters. It is not clear how this affects the arguments about the propagation of local neighborhoods.
3. Does the logical vocabulary created to express the LFP operation suffice to capture all P-time operations ?

Barriers

Any P vs NP proof must deal with the three known barriers described below. The concerns around this paper have not yet reached this stage yet.

Terminology

• Least fixed point (LFP)
• The complexity class NP
• The complexity class P

Online reactions

Theory blogs

• P ≠ NP Greg Baker, Greg and Kat’s blog, August 7 2010
• A proof that P is not equal to NP? Richard Lipton, Godel's lost letter and P=NP, August 8 2010
• On the Deolalikar proof: Crowdsourcing the discussion ? Suresh Venkatasubramanian, The Geomblog, August 9 2010
• Putting my money where my mouth isn’t Scott Aaronson, Shtetl-Optimized, August 9 2010
• That P ne NP proof- whats up with that? Bill Gasarch, Computational Complexity, August 9 2010
• Issues In The Proof That P≠NP Richard Lipton and Ken Regan, Godel's lost letter and P=NP, August 9 2010

Other

• Twitter Lance Fortnow, August 8 2010
• P ≠ NP Hacker News, August 8 2010
• Claimed Proof That P != NP Slashdot, August 8 2010
• Google Buzz Terence Tao, August 9 2010

Additions to the above list of links are of course very welcome.

Timeline

• August 6: Vinay Deolalikar sends out his manuscript to several experts in the field.

Other links

• P versus NP problem - Wikipedia
• Vinay Deolalikar - Wikipedia