Natural Proofs is Not the Barrier You Think It Is

If there's a position where I differ from most other complexity theorists it's that I don't believe that natural proofs present a significant barrier to proving circuit results. I wrote about this before but I buried the lead and nobody noticed.

Let's review natural proofs. I'll give a very high-level description. Li-Yang Tan gives a good technical description or you could read the original Razborov-Rudich paper. A natural proof to prove lower bounds against a circuit class \(\cal C\) consists of a collection \(C_n\) of Boolean functions on \(n\) inputs such that

1. No polynomial-size circuit family from \(\cal C\) can compute an element of \(C_n\) for large enough \(n\). 
2. \(C_n\) is a large fraction of all the function on \(n\) inputs.
3. A subset of \(C_n\) is constructive--given the truth-table of a function, you can determine whether it sits in the subset in time polynomial in the length of the truth-table. Note: This is a different than the usual notion of "constructive proof". 

The natural proof theorem states that if all three conditions hold than you can break pseudorandom generators and one-way functions.

My problem is with the third property, constructivity. I haven't seen good reasons why a proof should be constructive. When I saw Rudich give an early talk on the paper, he both had to change the definition of constructivity (allowing subsets instead of requiring an algorithm for \(C_n\) itself) and needed to give heavily modified proofs of old theorems to make them constructive. Nothing natural about it. Compare this to the often maligned relativization where most proofs in complexity relativize without any changes.

Even Razborov and Rudich acknowledge they don't have a good argument for constructivity.

We do not have any formal evidence for constructivity, but from experience it is plausible to say that we do not yet understand the mathematics of \(C_n\) outside exponential time (as a function of \(n\)) well enough to use them effectively in a combinatorial style proof.

Let's call a proof semi-natural if conditions (1) and (2) hold. If you have a semi-natural proof you get the following implication. 

Constructivity \(\Rightarrow\) One-way Functions Fail

In other words, you still get the lower bound, just with the caveat that if an algorithm exists for the property than an algorithm exists to break a one-way function. You still get the lower bound, but you are not breaking a one-way function, just showing that recognizing the proofs would be as hard as breaking one-way functions. An algorithm begets another algorithm. You don't have to determine constructivity either way to get the lower bound. 

Even if they aren't a great barrier to circuit lower bounds, natural proofs can be an interesting, if badly named, concept in their own right. For example the Carmosino-Impagliazzo-Kabanets-Kolokolova paper Learning Algorithms from Natural Proofs.

So if I don't believe in the barrier, why are circuit lower bounds hard? In recent years, we've seen the surprising power of neural nets which roughly correspond to the complexity class TC\(^0\), and we simply don't know how to prove lower bounds against powerful computational models. Blame our limited ability to understand computation, not a natural proof barrier that really isn't there.

Very few problems are in NP intersect coNP but not known to be in P. What to make of that?

Someone once told me:

 I was not surprised when Linear Programming was in P since it was already in \(  NP \cap  coNP  \), and problems in that intersection tend to be in P.

The same thing happened for PRIMALITY.

However FACTORING, viewed as the set 

\( \{ (n,m) \colon \hbox{there is a factor of n that is } \le m \} \),

is in \(  {\rm NP} \cap {\rm coNP} \).  Is that indicative that FACTORING is in P? I do not think so, though that's backwards-since I already don't think FACTORING is in P, I don't think being in the intersection is indicative of being in P.

Same for DISCRETE LOG, viewed as the set

\( \{ (g,b,y,p) : \hbox{p  prime, g is a gen mod p and } (\exists x\le y)[ g^x\equiv b \pmod p] \} \) 

which is in \( {\rm NP} \cap {\rm coNP} \).

1) Sets in  \({\rm NP} \cap {\rm coNP}  \) but not known to be in P:

FACTORING- thought to not be in P, though number theory can surprise you.  

DISCRETE LOG-thought to not be in P, but again number theory...

PARITY GAMES-thought to not be in P. (See Lance's post on the theorem that PARITY GAMES are in Quasi-poly time see here.) 

Darn, only three that I know of. If  you know others, then let me know.

2) Candidates for sets in \( {\rm  NP} \cap {\rm coNP} \) but not known to be in P:

Graph Isomorphism. Its in NP and its in co-AM. Under the standard derandomization assumptions GI is in AM=NP so then GI would be in \( {\rm NP} \cap {\rm coNP} \). Not known to be in P. Is it in P? I do not think there is a consensus on this. 

Darn, only one that I know of. If you know of others, then let me know, but make sure they are not in the third category below. 

3) Problems with an \( {\rm NP} \cap  {\rm coNP} \) feel to them but not known to be in P.

A binary relation B(x,y) is in TFNP if,  B(x,y)\in P and, for all x there is a y such that

|y| is bounded by a poly in |x|

B(x,y) holds.

So these problems don't really fit into the set-framework of P and NP.

TFNP has the feel of \( {\rm NP} \cap  {\rm coNP} \).

Nash Eq is in TFNP and is not known to be in P, and indeed thought to not be in P. There are other problems in here as well, and some complexity classes, and some completeness results. But these problems are not sets so not in the intersection. (I will prob have a future post on those other classes.)

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So what to make of this? Why are so few problems in the intersection? Does the intersection = P (I do not think so)?

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I close with another quote: 

as a former recursion theorist I hope that  \(NP \cap coNP\)=P, but as someone whose credit card numbers are on the web, I hope not. 

Favorite Theorems: Parity Games

August Edition

A quasipolynomial-time algorithm for a long standing open problem. Yes, we have two of them this decade.

Deciding Parity Games in Quasi-polynomial Time

Cristian Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li and Frank Stephan

I covered this theorem in 2017. In a parity game, Alice and Bob take turns walking along a directed graph with integer weights on the vertices and no sinks. Alice wins if the largest weight seen infinitely often is even. While it's not hard to show computing the winner sits in NP\(\cap\)co-NP and even UP\(\cap\)co-UP, the authors give the surprising result that you can determine the winner in near polynomial time.

The result has implications for modal logics, for example that model checking for the \(\mu\)-calculus can now be solved in quasipolynomial-time.

In follow-up work, Hugo Gimbert and Rasmus Ibsen-Jensen give a short proof of correctness of the parity games algorithm. Marcin Jurdziński and Ranko Lazić give an alternative algorithm that reduces the space complexity from quasipolynomial to nearly linear.

Six degrees of separation has been proven. Really?

There is a paper (see here for an article about the paper, the link to the paper itself is later) that claims to PROVE that, on average, the distance (for some definition of distance) between any two people is 6.

1) We've blogged about this kind of thing:

My Pope Number is 2

What is your Erdos Number?

The six degrees of VDW

2) The paper's link is here and in case link rot sets in, my copy of it is here.

3) The paper has 14 authors. AH- so that's why we are, on the average, 6 degrees apart- because papers have so many authors. (Actually papers in Biology  have LOTS more than 14.)

4) The paper defines a mathematical model for social networks and analyzes a persons cost and benefit of forming a connection. 

5) Is the mathematical model realistic? I think so. But its always tricky since empirical evidence already gave the answer of six. The true test of a mathematical model is to predict something we didn't already know. 

6) One thing about applying this to the real world: What is a connection? Friendship? (also hard to define), handshakes? I like the will respond to my emails metric, though that may leave out half of my colleagues and even some of my friends.

(How come people do not respond to important emails? Perhaps a topic for a later blog.) 

7) My Jeopardy number is 1 in two different ways:

a)  My co-author Erik Demaine was mentioned in a question on Jeopardy, see here and look at the 800 dollar question in Double Jeopardy, Category Boy Genius.

b) My cousin Adam Winkler's book, Gunfight,  was mentioned in a question Jeopardy, see here. It was a 400 dollar question.

In both cases the question was easy, hence my inside information did not give me an advantage. 

(Note: they were actually mentioned in an answer on Jeop since Jeop has that weird format where they give the answer and you need to find the question. For example:

Game show that has a format that Bill Gasarch thinks is stupid

What is Jeopardy?