[FOM] Gordeev and NP = PSPACE again

Timothy Y. Chow tchow at math.princeton.edu
Mon Jul 29 19:03:51 EDT 2019

Lev Gordeev has recently posted an arXiv preprint purporting to prove that 

   [1]  Lev Gordeev, Proof Compression and NP Versus PSPACE, Part 2,

FOM readers may recall that Gordeev and Haeusler posted an arXiv preprint 
back in 2016, also claiming to give a proof that NP = PSPACE.

   [2]  Lew Gordeev and Edward Hermann Haeusler, NP vs PSPACE,

There was some discussion of [2] on FOM back in October 2016:


To summarize the discussion, some people pointed out that what Gordeev and 
Haeusler were claiming in [2] seemed to contradict known results, but 
Gordeev and Haeusler replied that in fact there was no contradiction. 
Also, Richard Zach said that several crucial points in [2] were unclear.

In addition, there was an amusing interchange on Scott Aaronson's blog, 
where he gave 500:1 odds against [2] being correct, betting with someone 
using the pseudonym "Polymath."  Polymath said, "I'm sure the correctness 
of their proof will be settled one way or another within 2 or 3 months," 
but I have no idea whether Polymath and Aaronson actually settled their 


Skimming [1] briefly, I was surprised to find that [1] makes no mention of 
[2].  Instead, [1] refers to a recently published paper by Gordeev and 

   [3] L. Gordeev and E. H. Haeusler, Proof Compression and NP Versus
       PSPACE, Studia Logica 107 (2019), 55-83.

[3] does not claim to prove that NP = PSPACE.  The abstract of [2] says, 
"We upgrade [3] to a complete proof of the conjecture NP = PSPACE."

Since [2] has not been withdrawn from the arXiv, one might suppose that 
the authors stand by its correctness, but in that case it is curious that

a. the published paper [3], whose abstract and content superficially look 
a lot like [2], conspicuously avoids claiming to prove that NP = PSPACE, 

b. as already mentioned, [1] does not cite [2], but cites only [3].

Note also that [1] is authored only by Gordeev and not by Haeusler.

I believe that Gordeev and Haeusler subscribe to FOM, so perhaps they can 
clarify whether they still maintain that [2] provides a complete and 
correct proof that NP = PSPACE.


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