A Proof System Based on Ramsey Theory
Possible quasi-polynomial size proofs for tautologies based on Ramsey theory
Endre Szemerédi is one of the greatest combinatorists in the world. He is famous, of course, for solving the Erdös-Turán arithmetic progression conjecture, proving the powerful regularity lemma, and for a many other terrific results. He has won many prizes, including the Pólya prize, Steele Prize, and the Schock Prize.
Today I plan to talk about a potential relationship between combinatorial theorems and proof systems for tautologies. As you know I am not afraid, to appear foolish, while suggesting approaches to problems that most think are false. In this post, it is an approach to obtain quasi-polynomial sized proofs for tautologies. I thought for a while that this could potentially succeed, but now I am less confident that it is on the right track. However, I would like to share it with you, and let you decide for yourself.
Since Szemerédi plays a role in my proposed proof system, I thought I would say something about him first. Years ago I looked at Szemerédi’s paper that proves his famous arithmetic progression theorem. I do not claim to have understood the proof, but I wanted to see the masterpiece for myself.
An aside: I think we all can benefit from at least looking at some the great masterpieces. While the details may not be easy to follow, especially for a non-expert, looking at these papers is an experience that is valuable. Take a look at Walter Feit and John Thompson’s paper, Andrew Wiles’s paper, Ben Green and Terence Tao’s paper, the proof of the PCP theorem, and so on. Really.
Endre’s paper was so complex that I recall it has a figure that is a flow-chart, which shows the dependencies of the various parts of the proof. However, the coolest part of the paper is–in my opinion– in the acknowledgment, where Szemerédi said:
“I would like to thank Ron Graham for writing the paper.”
Amazing. It was all Endre’s work, but Ron wrote the paper.
When I was at Princeton working with Dan Boneh, we once needed a constructive version of Andy Yao’s XOR lemma. At the time the proofs of this lemma were not strong enough for our application. So for a COLT paper we proved what we needed from scratch–luckily we only needed a special case, so the proof was not too difficult.
However, Dan and I still wanted to be able to prove the full case, and after getting nowhere we asked Szemerédi for help. He was kind enough to come down from Rutgers and visit me at my Princeton office to discuss the problem. After a while, he understood the problem and said to me that either he could solve it in the next day or two or the problem was hopeless.
Two days later I got a phone call from Endre, who said that he had proved what we wanted. I was thrilled and immediately responded that I would be happy to come up to Rutgers to hear the details. Endre said no, that would not be necessary, since the proof was “so easy.” In the next few minutes he spoke fast and outlined a proof using an entropy type argument. I furiously wrote down all I could follow. Then, he said good-bye.
I remember running down to Dan’s office and excitedly telling him we had the solution. The trouble was that my notes were meager and we were not Endre. But, in a mere three hours or so, we re-constructed his argument. As I recall, Dan and I wrote up a draft of the result with all three as authors, but we never tried to publish it. Does that make our Szemerédi number ?
Ramsey Theory and PCP
The famous Ramsey theorem will play an important role in the suggested proof system. For a graph let as usual be the size of the largest clique of . Also define , if in every coloring of the edges of , there is a set of vertices of size at least , so that forms a clique of and all the edges in are the same color. The famous Ramsey theorem states that for fixed, tends to infinity as tends to infinity.
There are other Ramsey style theorems that are also needed. The following is a generalization of the above to “hypergraphs.” These concern coloring of not edges, but coloring of -cliques of the graph. Define , if in every coloring of the -cliques of , there is a set of vertices of size at least , so that forms a clique of , and all -subsets of have the same color. Also for fixed and , tends to infinity as tends to infinity.
Another important ingredient, for our approach is the PCP gap theorem for cliques. One of the best bounds is due to David Zuckerman:
Theorem: It is NP-hard to approximate the max-clique problem to within a factor of
Actually, under stronger assumptions can tend to slowly.
A Proof System
Suppose that is a graph that either has only small cliques all of size at most , or has at least one large clique of size . We do not know how in polynomial time to determine whether or not a graph is in either of these categories: small or large. The goal here is to present a simple proof system based on Ramsey theory that might be able to supply a quasi-polynomial size proof that does not have a large clique.
This is the proof system:
- Let where ;
- Guess a -coloring of the edges of ;
- For each set of vertices of , check if they form a mono-chromatic clique of ;
- If all fail to be mono-chromatic, then state that has no large clique.
There are several issues with this “proof system.” Let’s skip the discussion of step (1), and return to that later on. The size of the proof is dominated by the cost of step (3): this takes at most quasi-polynomial time, since is at most order . Next the proof system is sound: if it states that has no large clique, then this is true. Note, if has a large clique, then just that part of the graph will always have a mono-chromatic clique of size at least . So the system is sound.
The central question is simple: is the proof system complete? The only way that it could fail to be complete is if there were a graph so that , and yet for all -colorings, there is a mono-chromatic clique of size . If there is no such graph, then the proof system is complete. Thus, a key question, that I do not know the answer to, is: Do such graphs exist?
There is one loose end. In step (1) we need the value ? The way we fix this is to allow the proof system bits of advice. This follows since it is well known that is always at most logarithmic in the size of the graph.
Another Proof System
We can also extend the proof system via hypergraph Ramsey theory. Again suppose that is a graph that either has only small cliques all of size at most , or has at least one large clique of size .
This is the second proof system:
- Let where ;
- Guess a -coloring of the -cliques of ;
- For each set of vertices of , check that is a clique of , so that all the subsets of are labeled with the same color.
- If all fail this test, then state that has no large clique.
As before, step (1) will be done through advice bits. The size of the proof is dominated by the cost of step (3): this takes at most quasi-polynomial time, since and are at most order . Next the proof system is sound: if it states that has no large clique, then this is true. Note, if has a large clique, then just that part of the graph will always have a mono-chromatic clique of size at least . So the system is sound.
Again completeness is the open problem, and the role of Szemerédi was as an informal oracle. I ran similar ideas by him, years ago, and he seemed to think that it was not obvious whether or not the proof systems worked. Any mistake is mine, but I thought that his comments were at least comforting.
The reason for the second system is that the first might be incomplete, while the second is complete. The advantage of the second system is our lack of knowledge of the behavior of hypergraph Ramsey numbers. In a funny way, the fact that these numbers are so poorly understood, means that the second proof system might be harder to show incomplete. It might even be complete.
Of course the obvious question is to show that these proof systems are incomplete, especially if you believe that tautologies are hard. Or try to show that one of them is complete. Note, the second proof system is really a family of systems, since we are free to select and .
Another possibility is to create other similar proof systems, perhaps based on other Ramsey type theorems. Other examples, would, in my opinion, be quite interesting.
I think using the PCP theorem as part of a proof system seems to be a powerful idea. Again, while the proof system outlined is likely not to be complete, its incompleteness would shed light on Ramsey theory itself. That would be another example of Reverse Complexity Theory.
Lastly, can oracle results shed any light here? Can the known oracle results show that this type of proof systems cannot work? You know how I feel about oracle results, but can they say something here?%d bloggers like this: