In these lectures, we finish the proof of the approximate degree lower bound for AND-OR function, then we move to the surjectivity function SURJ. Finally we discuss quasirandom groups.
Lemma 1. Suppose that distributions over are -wise indistinguishable distributions; and distributions over are -wise indistinguishable distributions. Define over as follows:
: draw a sample from , and replace each bit by a sample of (independently).
Then and are -wise indistinguishable.
To finish the proof of the lower bound on the approximate degree of the AND-OR function, it remains to see that AND-OR can distinguish well the distributions and . For this, we begin with observing that we can assume without loss of generality that the distributions have disjoint supports.
Claim 2. For any function , and for any -wise indistinguishable distributions and , if can distinguish and with probability then there are distributions and with the same properties (-wise indistinguishability yet distinguishable by ) and also with disjoint supports. (By disjoint support we mean for any either or .)
Proof. Let distribution be the “common part” of and . That is to say, we define such that multiplied by some constant that normalize into a distribution.
Then we can write and as
where , and are two distributions. Clearly and have disjoint supports.
Then we have
Therefore if can distinguish and with probability then it can also distinguish and with such probability.
Similarly, for all such that , we have
Hence, and are -wise indistinguishable.
Theorem 3. AND-OR.
Proof. Let be -wise indistinguishable distributions for AND with advantage , i.e. . Let be -wise indistinguishable distributions for OR with advantage . By the above claim, we can assume that have disjoint supports, and the same for . Compose them by the lemma, getting -wise indistinguishable distributions . We now show that AND-OR can distinguish :
- : First sample . As there exists a unique such that , . Thus by disjointness of support . Therefore when sampling we always get a string with at least one “”. But then “” is replaced with sample from . We have , and when , AND-OR.
- : First sample , and we know that with probability at least . Each bit “” is replaced by a sample from , and we know that by disjointness of support. Then AND-OR.
Therefore we have AND-OR.
Definition 4. The surjectivity function SURJ, which takes input where for all , has value if and only if .
First, some history. Aaronson first proved that the approximate degree of SURJ and other functions on bits including “the collision problem” is . This was motivated by an application in quantum computing. Before this result, even a lower bound of had not been known. Later Shi improved the lower bound to , see [AS04]. The instructor believes that the quantum framework may have blocked some people from studying this problem, though it may have very well attracted others. Recently Bun and Thaler [BT17] reproved the lower bound, but in a quantum-free paper, and introducing some different intuition. Soon after, together with Kothari, they proved [BKT17] that the approximate degree of SURJ is .
We shall now prove the lower bound, though one piece is only sketched. Again we present some things in a different way from the papers.
For the proof, we consider the AND-OR function under the promise that the Hamming weight of the input bits is at most . Call the approximate degree of AND-OR under this promise AND-OR. Then we can prove the following theorems.
Theorem 6. AND-OR for some suitable .
In our settings, we consider . Theorem 5 shows surprisingly that we can somehow “shrink” bits of input into bits while maintaining the approximate degree of the function, under some promise. Without this promise, we just showed in the last subsection that the approximate degree of AND-OR is instead of as in Theorem 6.
Proof of Theorem 5. Define an matrix s.t. the 0/1 variable is the entry in the -th row -th column, and iff . We can prove this theorem in following steps:
- SURJAND-OR under the promise that each row has weight ;
- let be the sum of the -th column, then AND-OR under the promise that each row has weight , is at least AND-OR under the promise that ;
- AND-OR under the promise that , is at least AND-OR;
- we can change “” into “”.
Now we prove this theorem step by step.
- Let be a polynomial for SURJ, where . Then we have
Then the polynomial for AND-OR is the polynomial with replaced as above, thus the degree won’t increase. Correctness follows by the promise.
- This is the most extraordinary step, due to Ambainis [Amb05]. In this notation, AND-OR becomes the indicator function of . Define
Clearly it is a good approximation of AND-OR. It remains to show that it’s a polynomial of degree in ’s if is a polynomial of degree in ’s.
Let’s look at one monomial of degree in : . Observe that all ’s are distinct by the promise, and by over . By chain rule we have
By symmetry we have , which is linear in ’s. To get , we know that every other entry in row is , so we give away row , average over ’s such that under the promise and consistent with ’s. Therefore
In general we have
which has degree in ’s. Therefore the degree of is not larger than that of .
- Note that , . Hence by replacing ’s by ’s, the degree won’t increase.
- We can add a “slack” variable , or equivalently ; then the condition actually means .
Proof idea for Theorem 6. First, by the duality argument we can verify that if and only if there exists -wise indistinguishable distributions such that:
- can distinguish ;
- and are supported on strings of weight .
Claim 7. OR.
The proof needs a little more information about the weight distribution of the indistinguishable distributions corresponding to this claim. Basically, their expected weight is very small.
Now we combine these distributions with the usual ones for And using the lemma mentioned at the beginning.
What remains to show is that the final distribution is supported on Hamming weight . Because by construction the copies of the distributions for Or are sampled independently, we can use concentration of measure to prove a tail bound. This gives that all but an exponentially small measure of the distribution is supported on strings of weight . The final step of the proof consists of slightly tweaking the distributions to make that measure .
Groups have many applications in theoretical computer science. Barrington [Bar89] used the permutation group to prove a very surprising result, which states that the majority function can be computed efficiently using only constant bits of memory (something which was conjectured to be false). More recently, catalytic computation [BCK14] shows that if we have a lot of memory, but it’s full with junk that cannot be erased, we can still compute more than if we had little memory. We will see some interesting properties of groups in the following.
Some famous groups used in computer science are:
- with bit-wise addition;
- with addition mod ;
- , which are permutations of elements;
- Wreath product , whose elements are of the form where is a “flip bit”, with the following multiplication rules:
- in ;
- is the operation;
An example is . Generally we have
- matrices over with determinant in other words, group of matrices such that .
The group was invented by Galois. (If you haven’t, read his biography on wikipedia.)
Quiz. Among these groups, which is the “least abelian”? The latter can be defined in several ways. We focus on this: If we have two high-entropy distributions over , does has more entropy? For example, if and are uniform over some elements, is close to uniform over ? By “close to” we mean that the statistical distance is less that a small constant from the uniform distribution. For , if uniform over , then is the same, so there is not entropy increase even though and are uniform on half the elements.
Definition 8.[Measure of Entropy] For , we think of for “high entropy”.
Note that is exactly the “collision probability”, i.e. . We will consider the entropy of the uniform distribution as very small, i.e. . Then we have
where is the minimum dimension of irreducible representation of .
By this theorem, for high entropy distributions and , we get , thus we have
If is large, then is very close to uniform. The following table shows the ’s for the groups we’ve introduced.
Here is the alternating group of even permutations. We can see that for the first groups, Equation ((1)) doesn’t give non-trivial bounds.
But for we get a non-trivial bound, and for we get a strong bound: we have .
[BCK14] Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff, and Florian Speelman. Computing with a full memory: catalytic space. In ACM Symp. on the Theory of Computing (STOC), pages 857–866, 2014.