* x_i is defined (that looks like the i’th “bit” of x?)

* “circuit complexity at most m” is defined, but then the existence of an NP problem with superpolynomial complexity is claimed to give P != NP. It may be clear to most readers that the latter definition of complexity is worst-case, but maybe it should be made explicit?

]]>The problem I was suggesting is the following:

Let MAJ denote the class of functions of the form , where the ‘s are integers with .

Let THR denote the same class, except there is no upper bound on the magnitude of the ‘s. (Although it is known [Muroga-Toda-Takasu’61] that the ‘s need not have more than bits.) A function in THR is precisely the indicator of a halfspace in intersected with the discrete cube.

Let’s define circuit classes. Let MAJ o MAJ denote the class of functions computable as , where are in MAJ and . Let THR o THR o THR denote the functions computable as a depth-3 poly-size circuit where the layers are functions from THR. Etc.

is normally defined as the functions computable in MAJ o MAJ o … o MAJ (constantly many times).

It is a theorem [Goldmann-Hastad-Razborov’92] that THR o THR o … o THR (d times) is contained in MAJ o MAJ o … o MAJ (d+1 times). So sometimes you see people defining in terms of THR rather than MAJ, because it doesn’t really matter.

It *does* matter though if you’re being very careful about depth.

The original paper defining [Hajnal-Maass-Pudlak-Szegedy-Turan’87] gave an explicit function, IP2, which is not in MAJ o MAJ. This function IP2 = “Inner product mod 2” is nothing more than:

.

If you write bits as elements of the field of size 2, then IP2 is just . This function is the canonical example of a function with a perfectly uniform Fourier spectrum; all coefficients are . Using this property is one of the several relatively easy ways to prove that IP2 is not in MAJ o MAJ. (Of course, IP2 is in P.)

[Nisan’94] gave a nice proof (not very hard) that IP2 is not in MAJ o THR.

[Forster2000] gave a very beautiful (and a bit hard) proof that IP2 is not in THR o MAJ.

It is open whether IP2 is in THR o THR; I think it’s universally conjectured not to be. In fact, as I mentioned, there is no known explicit function not in THR o THR. For the complexity-theory lovers, I would add that as far as we know, not just but indeed could be contained in uniform depth-2 arbitrary-threshold-gate circuit families. People often point to “bounded-depth AND-OR-NOT-Mod6-gate circuits” as the most ridiculously small class we can’t separate from NP, but I prefer the class THR o THR. It’s pretty galling that we can’t find an explicit function not in THR o THR.

But I don’t feel too pessimistic about it (certainly I feel the AND-OR-NOT-Mod6 problem is much harder). I kind of feel like “IP2 not in THR o THR” should be doable; I mean, we managed MAJ o THR and THR o MAJ, so why not?

]]>That’s the correct definition of TC0; note that you don’t need to explicitly allow AND gates and OR gates, because they are just special types of threshold gates. You are allowed to use NOT gates as well.

]]>Maybe that could be a suitable Polymath project instead, though I’d want to think about it for a little while before getting started. Just to check, does depth-2 TC0 mean functions computable by a circuit of polynomial size, unbounded fanin, and threshold gates as well as AND and OR gates, where a threshold gate means “output 1 if at least r inputs are 1”? I had thought that it was at depth 3 that nothing was known.

]]>I certainly support trying to separate P (or NP or NEXP or ) from TC0, and heartily recommend starting with depth-2 TC0. (I mean here general thresholds of thresholds, not just majority of majorities.) I feel the depth-2 case should be doable (even though I’ve worked on it unsuccessfully for a really long time).

]]>Sorry, the above anonymous comment was mine.

]]>It now seems to me that if the strong conjecture (conjecture 3 in Ryan’s terms) then it would give a non-naturalizing circuit lower bound. Or, at least, I cannot see any natural proof obstacle to the strong conjecture.

– The paper I cite above does not construct a family of functions that are indistinguishable from random monotone functions: rather it considers (as a starting point for a more elaborate construction) a family of monotone functions which are indistinguishable from monotone functions where the “middle slice” of inputs of weights n/2 are mapped randomly, and the function is fixed to majority on the remaining inputs. (A family of pseudorandom functions is easily modified to be indistinguishable from this distribution of monotone functions.) Both random and pseudorandom families as above are noise-stable because they agree with majority almost everywhere

– Even if a family of functions in TC0 could be constructed to be indistinguishable from the uniform distribution over monotone functions, I find Gil’s claim convincing that both families would mostly contain functions close to majority and hence be mostly noise-stable

– More generally, I don’t see a way to exploit the ability to compute pseudorandom functions in TC0 in order to compute a monotone noise-sensitive function in TC0. In particular, the following super-strong conjecture seems plausible: “with probability 1, relative to a random oracle TC0 contains no monotone function which is noise-sensitive for 1/polylog n noise.” (And it better be, because if factoring is hard then the strong and the superstrong conjecture are equivalent.)

– Overall it seems entirely plausible that the hypothetical TC0 P proof given by the strong conjecture would be non-naturalizable. Interestingly, this would be so because the largeness condition would be violated. (Razborov and Rudich have a heuristic argument for why it would be strange to have a lower bound proof based on a property that fails the largeness condition, but the argument assumes that the property used to prove the lower bound is well defined for non-monotone functions.)

]]>The last sentence should be: if you want a monotone function in *C0 that cannot be approximated by a function in monotone *C0 I believe it is open for AC0 as well.

]]>Dear Ryan,

Indeed the two conjectures I stated were your conjectures 2 and 3. The theorem of Linial Mansour and Nisan was an inspiration. It gives a much stronger property than noise stability for 1/polylog noise for AC0 functions.

Conjecture 2 will imply that recussive majority of 3 is not a or close to a function in monotone TC0. I agree that it is not that scary and that depth 2 is a place to start.

I agree that trying to prove conjecture 3 is scary as it will gives that reccursive-majority-of 3 is not in TC0 (which is widely believed) and not even very close to a function in TC0.

Seperating TC0 from P is a formidable task.

What I do not know is if Conjecture 3 being true is already scary.

If Conjecture 3 amounts to a natural proof seperating TC0 from P then Conjecture 3 being true implies that factoring is easy (by Naor Reingold or perhaps the result Luca cited). This means that not only it will be very hard to prove Conjecture 3 but that we may expect that this conjecture is false.

If in order to present a natural proof we need that most monotone functions are not noise stable to polylogarithmically small noise than I suspect this is not true. (So maybe, in general, restricting to monotone functions is a way, in princi-le, to overcome the natural proof barrier.) Again I am not sure about it and I do not know if the statement of conjecture 3 will have unreasonable consequences that natural proofs give.

BTW You are right there is a distinction between monotone functions in TC0 and functions in monotone TC0. But I do not know of a function showing that. For AC0 this is a result by Ajtai and Gurevich; if you want a monotone function in *C0 that cannot be approximated by a function in *C0 I believe it is open for AC0 as well.

]]>Sorry, I still can’t quite catch what the conjectures are 🙂

Let me try to restate them:

1. Monotone functions in AC0 are stable wrt 1/polylog(n) noise.

Comment: Isn’t this true even for nonmonotone functions, by LMN (Linial-Mansour-Nisan)?

2. Functions in monotone-TC0 are stable wrt 1/polylog(n) noise.

This is a very nice conjecture, and if I am thinking straight this morning I might guess that it’s true. Furthermore, it does not look impossibly scary.

This is in distinction with another conjecture (“Conjecture 3”?): Monotone functions in TC0 are stable wrt 1/polylog(n) noise. (There’s a distinction, right, between functions computable by bounded-depth polynomial-size circuits of monotone threshold gates — i.e. monotone-TC0 — and monotone functions computable by bounded-depth polynomial-size circuits of general threshold gates.)

Conjecture 3 looks too scary to try to prove, since as Luca points out the existence of Recursive-Majority-of-3 means this conjecture would imply TC0 differs from P.

But as I said, the conjecture that monotone-TC0 is noise stable for 1/polylog(n) noise looks nice, and not so bad given that (as Gil points out) Yao has a lower bound against monotone-TC0. Maybe the first step would be to look at depth 2?

]]>I would suppose a random monotone function looks a lot like a majority function.

]]>My reaction to David’s calculation was, “ah so the construction **is** pseudorandom up to density after all, except that the density is either 0 or 1.”

Indeed the definition is useful/non-trivial only if the density is noticeably bounded away from 0 and 1 with noticeable probability.

In such a case, there is another way to see that you can convert pseudorandom-up-to-density constructions into standard pseudorandom functions: the former imply one-way functions and from one-way functions you get pseudorandom functions in the standard sense.

If you let F be distribution of functions which is pseudorandom up to density, and A(.,.) be the evaluation algorithm that given randomness r and input x computes A(r,x) = f_r(x), then g(r):= (A(r,0),A(r,1),…,A(r,m)) must be a one-way function for large enough m=poly(n). (Here the number i is identified with the i-th binary string.) Otherwise, suppose you have an inverter I(.) that succeeds with noticeable probability; then, given an oracle function f(.) which might be either random up to density or coming from F, compute I(f(0),…,f(m)). If f is pseudorandom, then with noticeable probability the inverter finds an r such that f(.) = A(r,.); if f is random, then this happens with negligible probability (because the pseudorandom function has too much entropy to be well approximated by a short representation).

Once you have one-way functions, you can certainly construct standard pseudorandom functions.

]]>Actually I don’t know if a random monotone function is noise sensitive, it just sounded like the kind of thing that should be true.

What is your intuition for thinking that a random monotone function is stable?

]]>Dear Luca, The first conjecture (which I still think is correct) talks about monotone TC_0 circuits. (Sorry I omitted the word monotone).

It will give that recursive majority of 3 is not expressible by (or very close to a function expressible by) a bounded depth monotone size threshold circuit.

The second conjecture is precisely as you stated. it would prove TC0 not equal P.

But I do not see why it distinguish easy function from a random monotone function because I do not thing a random monotone function is noise sensitive. Is it??? (Still I would not be surprised if the stronger conjecture may represent a natural proof of hardness somehow.)

]]>Gil, I did not understand the difference between the first conjecture and the second conjecture.

If the second conjecture is “every monotone function in TC0 is stable against polylog noise,” then would it prove TC0 P? (because recursive majority-of-3 is monotone, in P, and noise sensitive?)

If so, your noise-related property would distinguish monotone easy (in TC0) functions from random monotone functions. Something like monotone pseudorandom functions (providing a natural-proof-type obstacle to your conjecture) might be constructed here:

Dana Dachman-Soled, Homin K. Lee, Tal Malkin, Rocco A. Servedio, Andrew Wan, Hoeteck Wee:

Optimal Cryptographic Hardness of Learning Monotone Functions.

ICALP (1) 2008: 36-47