18

Recently, I proved that s(f) = bs(f) for unate functions and read-once functions over the Boolean operators AND, OR and EXOR, and my paper including the results was accepted to TCS 2014. (http://dx.doi.org/10.1007/978-3-662-44602-7_9) You may be attacking this problem. If so, I feel sorry, but I started to attack the problem independently before the ...


17

From the paper: what is actually proven is, in Theorem 1.4, $$ \forall f\colon\{0,1\}^n \to \{0,1\}, \qquad s(f) \geq \sqrt{\deg f} \tag{1} $$ which cannot be improved (it is tight for some functions). Then it is combined with the previously known result of Nisan and Szegedy [1], $$ \forall f\colon\{0,1\}^n \to \{0,1\}, \qquad \deg f \geq \sqrt{\frac{1}{2}\...


15

To me, your question seems analogous to saying "I've heard that non-Euclidean geometry requires me to give up Euclid's fifth axiom, which is very useful in many mathematical contexts." You don't have to give up the axiom in the sense of "personally agree, at a basic philosophical level, that it doesn't hold", you just have to think of it in terms like: "As a ...


14

for Question 1, the answer is Yes, and can be shown as follows. (I will also be implicitly sketching an affirmative answer to Q4, since the argument is uniform and will treat all input lengths at once.) Fix any NP-complete language $L$, and a family of good binary error-correcting codes (with rate 1/4 and correcting from a .1 fraction of errors, say). ...


14

You seem to be confusing several things here. First of all, like Alexis said in her answer, I don't see why you would need to accept/reject the principles of a given logical theory in order to study it and learn about it. The fact that your theory is intuitionistic doesn't mean that your meta-theory has to be! You may freely use proof by contradiction or ...


14

In his CCC'17 paper [1], Avishay Tal improved the bound to $$ \left(\log\frac{m}{\varepsilon}\right)^{O(d)}\,. \tag{1} $$ You may want to check p.15:4 for a discussion. It also refers to (see Footnote 30 to a paper of Harsha and Srinivasan, which improves on (1)) and answers Tal's conjecture: $k$-wise independent, for $$ k = \left(\log m \right)^{O(d)}\cdot\...


13

Here is what Scott Aaronson has to say on the subject: What makes this interesting is that block-sensitivity is known to be polynomially related to a huge number of other interesting complexity measures: the decision-tree complexity of $f$, the certificate complexity of $f$, the randomized query complexity of $f$, the quantum query complexity of $f$, the ...


12

Note that formulas using $\land$ and $\lor$ gates (and possibly the constants $0$ and $1$) are known as monotone. The complexity of monotone formula equivalence depends on how complex formulas are allowed. Let me start with the related problem of implication (entailment) between two formulas $\phi\vdash\psi$, which is easier to classify. Note that $\phi$ ...


11

It is important to remark that a specific boolean circuit only treats inputs of a given size, unlike a Turing Machine that can take inputs of any size. So a language with inputs of arbitrary size is usually treated by a family of circuits, one for each size of input. The problem with this definition is that it allows circuit to solve uncomputable problems, ...


11

LMN theorem shows that if f is a boolean function$(f:\{-1,1\}^n \rightarrow \{-1,1\})$ computable by an $\text{AC}^0$ circuit of size M, $$\sum_{S:|S|> k} \hat f(S)^{2} \leq 2^{-\Omega(k/(\log M)^{d-1})}$$ $\Rightarrow \hat f([n])^{2} \leq 2^{-\Omega(n/(\log M)^{d-1})}$ $\Rightarrow |\hat f([n])| \leq 2^{-\Omega(n/(\log M)^{d-1})}$ $|\hat f([n])|$ is ...


11

Here's an algorithm that beats the trivial attempts. The following is a known fact (Exercise 1.12 in O'Donnell's book) : If $f:\{-1,1\}^n\to\{-1,1\}$ is a Boolean function which has degree $\le d$ as a polynomial, then every Fourier coefficient of $f$, $\hat{f}(S)$ is an integer multiple of $2^{-d}$. Using Cauchy-Schwarz and Parseval one gets that there are ...


11

The circuit complexity of any boolean function of $n$ variables is at most $(1+o(1))2^n/n$, so a separation of between circuit and formula complexity of $\Omega(2^n)$ is not possible. This upper bound was established by Lupanov, and his method is known as the Lupanov representation of boolean functions. It also gives an upper bound of $(1+o(1))2^n/\log n$ ...


11

Type theory is a mathematical theory in which we can do very many different things (set theory is like that as well). We can use type theory for computability, or homotopy theory, or use it to express features of a programming language, etc. One of the things that we can use type theory for is as a kind of logic (this is known as Curry-Howard correspondence)....


11

In [1], the paper where Propp & Wilson introduce the "coupling from the past" technique for MCMC sampling, they also outline a "heat bath algorithm" applicable to a certain kind of spin system model (Section 3.1). Then, they further illustrate how to map their spin system model to the problem of uniformly sampling from the order ideals of a poset which, ...


10

Isn't the probability of $\Sigma_{i:h (i) =1} x_{i} \mbox{mod } 2 = 1$ at least 1/2 ? It seems that $1/(10k)$ is a weak lower bound. In fact the answer is no. (It would be that $\Sigma_{i:h (i) =1} x_{i} \mbox{mod } 2 = 1$ holds with probability at least $1/2-\varepsilon$, if we were working with an $\varepsilon$-biased hash family, and indeed using $\...


10

Any function which has non-zero correlation with parity has degree $n$. That is, if $$\sum_{x \in \{0,1\}^n} (-1)^{\sum_i x_i}f(x) \neq 0$$ then the unique multilinear expansion of $f$ contains the monomial $x_1\cdots x_n$. Indeed, since $(-1)^{x_i} = \frac{1-x_i}{2}$, the Fourier expansion of $f$ (expressed in terms of products of $\frac{1-x_i}{2}$) will ...


10

I'd consider it unlikely that such a trick is easy to find and/or will give you significant gains, as it would give nontrivial satisfiability algorithms. Here's how: First of all, while ostensibly easier, your problem can actually solve the more general problem of, given a circuit $C$ and $N$ inputs $x_0, \ldots, x_{N-1}$, evaluate $C$ at all the inputs ...


10

My thanks to Aryeh for bringing this question to my attention. As others have mentioned, the answer to (1) is Yes, and the simple method of Empirical Risk Minimization in $\mathcal{C}$ achieves the $O((d/\varepsilon)\log(1/\varepsilon))$ sample complexity (see Vapnik and Chervonenkis, 1974; Blumer, Ehrenfeucht, Haussler, and Warmuth, 1989). As for (2), ...


9

Kaveh's answer provides an answer do the question as you have stated it (and this is the usual proof for showing that $\mathsf{TC}^0$ is contained in $\mathsf{NC}^1$). But I was thinking that you might actually have intended to ask a slightly a different question. Namely for an explicit polynomial size monotone formula for majority. Since majority is ...


9

Computing restricted threshold gate ($\sum_i x_i \geq k$) is essentially sorting input bits. If you can sort the bits then it is easy to compare the result to $k$ and compute restricted threshold. On the other hand, assume that we have an circuits to compute restricted threshold. We can do a parallel search to find the number of ones in the input and ...


9

The main thing missing from your list is the beautiful 2006 paper of Klivans and Sherstov. They show there that PAC-learning even depth-2 threshold circuits is as hard as solving the approximate shortest vector problem.


9

[An alternative answer for $BS(f)$] There is a paper related to this problem. Scott Aaronson: Algorithms for Boolean Function Query Properties. SIAM J. Comput. 32(5): 1140-1157 (2003) The paper gives an $O(N^{\log_2 5} \log N)$ algorithm for computing $BS(f)$ if $f$ is given by a truth table of size $N=2^n$. Thus, even if $c$ is not fixed, deciding $BS(f) ...


9

Well done on your independent discovery. This is a Hadamard matrix of Sylvester type, written in a different order. There is a massive literature on this topic. It is used in coding theory, cryptography, is directly related to Reed-Muller codes of degree 1, it can be used to obtain best affine approximations of functions, etc. Ryan O'Donnell's notes on ...


9

Here's just a couple of observations I couldn't fit in a comment: 0) Added because the first answer was deleted: there is an interpretation of $H_n$, namely, indexing the rows and columns by $\{0,1\}^n$, the entry corresponding to $(x,y)$ is $1$ if the Hadamard product $x\odot y=(x_1y_1,\ldots,x_n y_n)$ has even parity, and $-1$ if it has odd parity. 1) In ...


9

An observation too long for a comment (and which also fits well with Jason Gaitonde's observation-too-long-for-comment): As hinted at in the OQ, both of these can in fact be realized by a very simple kind of recursive construction. Namely, we specify $B_0 \in \{(0), (\pm 1)\}$ (a $1 \times 1$ matrix), and then a single recursive formula $$ B_n = \left(\...


8

I agree with Alexis and Damiano, and there is another dimension to $\lambda$-calculus that is not often emphasised, because of the dominance of the Curry-Howard correspondence in thinking about the $\lambda$-calculus. Typed $\lambda$-calculus can be used to represent formulae of classical logic. The most well-known system used for this purpose is the ...


8

It is clear that your language is in DP. In order to show that it is DP-hard, we will give a reduction from SAT-UNSAT to your language, which we can call CRIT-UNSAT. Given a pair of CNFs $(f,g)$, let $x,y$ be fresh variables, and let $$ h = (f \lor \lnot x) \land (g \lor x) \land (g \lor y) \land \lnot x \land (x \lor \lnot y). $$ Here $f \lor \lnot x$ means ...


8

Below I show how to explicity construct an average-case $\varepsilon$-biased space on $n$ bits of size $O(1/\varepsilon)$. In contrast, the best worst-case $\varepsilon$-biased spaces on $n$ bits have size $\tilde\Theta(n/\varepsilon^2)$. So you save a factor of about $n/\varepsilon$ by going from worst-case to average-case. Unfortunately, as you will see,...


8

I don't think parities are the only orthonormal Boolean basis, for example Paley basis provides a Boolean orthonormal basis in some cases. It's natural to ask if such bases, interpreted as Boolean functions, have any interesting applications.


7

The proof (due to Miller and Preparata, 1975) that any symmetric function can be computed by circuits over {AND,OR,NOT} in logarithmic depth can be found, e.g., in Complexity of Boolean Functions by Ingo Wegener (Theorem 4.1, page 76). The corresponding circuit has linear size. And since the depth is logarithmic it can be turned to a formula of polynomial ...


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