Hardness Amplification Proofs Require Majority

Hardness amplification results show that for every Boolean function f, there exists a Boolean function Amp(f) such that if every size s circuit computes f correctly on at most a 1 � � fraction of inputs, then every size s� circuit computes Amp(f) correctly on at most a $${1/2+\epsilon}$$ fraction of inputs. All hardness amplification results in the literature suffer from "size loss" meaning that $${s' \leq \epsilon \cdot s}$$ . We show that proofs using "non-uniform reductions" must suffer from such size loss. A reduction is an oracle circuit $${R^{(\cdot)}}$$ which given oracle access to any function D that computes Amp(f) correctly on a $${1/2+\epsilon}$$ fraction of inputs, computes f correctly on a 1 � � fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string that may depend on both f and D. The well-known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for $${\epsilon < 1/4}$$ . We show that every non-uniform reduction must make at least $${\Omega(1/\epsilon)}$$ queries to its oracle, which implies size loss. Our result is the first lower bound that applies to non-uniform reductions that are adaptive, whereas previous bounds by Shaltiel & Viola (SICOMP 2010) applied only to non-adaptive reductions. We also prove similar bounds for a stronger notion of "function-specific" reductions in which the reduction is only required to work for a specific function f.

We study the task of transforming a hard function f, with which any small circuit disagrees on (1 - /spl delta/)/2 fraction of the input, into a harder function f', with which any small circuit disagrees on (1 - /spl delta//sup k/)/2 fraction of the input, for /spl delta/ /spl isin/ (0,1) and k /spl isin/ /spl Nopf/. We show that this process cannot be carried out in a black-box way by a circuit of depth d and size 2/sup o(k2/d)/ or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth). In particular, for k = 2/sup /spl Omega/(n)/, such hardness amplification cannot be done in ATIME(O(1), 2/sup o(n)/. Therefore, hardness amplification in general requires a high complexity. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently non-uniform in the following sense. Given as an oracle any algorithm which agrees with f' on (1 - /spl delta//sup k/)/2 fraction of the input, we still need an additional advice of length /spl Omega/(k log(1//spl delta/)) in order to compute f correctly on (1 - /spl delta/)/2 fraction of the input. Therefore, to guarantee the hardness, even against uniform machines, of the function f', one has to start with a function f which is hard against non-uniform circuits. Finally, we derive similar lower bounds for any black-box construction of pseudorandom generators from hard functions.

The "direct-product code" of a function $f$ gives its values on all $k$-tuples $(f(x_1),\dots, f(x_k))$. This basic construct underlies "hardness amplification" in cryptography, circuit complexity, and probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29 (2000), pp. 1132--1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622] enabled for the first time testing proximity to this important code in the "list-decoding" regime. In particular, they give a $2$-query test which works for polynomially small success probability $1/k^{\alpha}$ and show that no such test works below success probability $1/k$. Our main result is a $3$-query test which works for exponentially small success probability $\exp({-k^{\alpha}})$. Our techniques (based on recent simplified decoding algorithms for the same code [R. Impagliazzo et al., Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 579--588]) also allow us to considerably simplify the analysis of the 2-query test of [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of $k$, and success probability $1/k^{\alpha}$. Finally, we show the applicability of the new tests to PCPs. Starting with a 2-query PCP with a projection property over an alphabet $\Sigma$ and with soundness error $1-\delta$, Rao [Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 1--10] (building on Raz's ($k$-fold) parallel repetition theorem [R. Raz, SIAM J. Comput., 27 (1998), pp. 763--803] and Holenstein's proof [T. Holenstein, Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, 2007, pp. 411--419] obtains a new 2-query PCP over the alphabet $\Sigma^k$ with soundness error $\exp(-\delta^2 k)$. Our techniques yield a 2-query PCP with soundness error $\exp(-\delta \sqrt{k})$. Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [SIAM J. Comput., 30 (2000), pp. 324--346] but with simpler analysis and exponentially better soundness error.

It is well known that among the three classes of the PRAM models, namely, CRCW, CREW, and EREW, the CRCW models are the weakest, in the sense that, they permit concurrent read/write by processors. Accordingly, algorithms on the CRCW model mainly concentrate on the core computations without much ado about data access. Consequently, this model, at least in principle, allows for the design of the fastest algorithm for a problem. It is intriguing to ask how fast prefixes can be computed on the CRCW models. Since CRCW models are equivalent to the unbounded fan-in circuits (refer to Chapter 2), the task of developing the fastest algorithms for the prefix problems is pursued in the context of the unbounded fan-in circuits. Recall from Chapter 2, that while the standard measures, such as, size and depth are still used to quantify the goodness of unbounded fanin circuits, the size of the circuit is measured by the total number of edges incident on all of its operation nodes, instead of by the number of operations nodes. It turns out that the size and depth of unbounded fanin circuits for computing prefixes, depends critically on the structure of the underlying semigroup from which the input elements are drawn. The principal result of this concluding Chapter may be stated as follows: There exists unbounded fan-in parallel prefix circuits of constant depth and polynomial size if, and only if, the underlying semigroup is group free. The proof of this result involves a very clever synthesis of a number of ideas drawn from different directions — structure of group free semigroups, their relations to a special class of regular sets, called non-counting regular sets, the relation of this latter class of regular sets to yet another class of regular sets defined by star-free regular expressions, and the design of a special class of finite state deterministic automata called RS machines that accept star-free regular expressions. In this context, it is convenient to define the notion of small circuits as the class of circuits with constant depth and polynomial size.

We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any small circuit must have depth superlinear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the first size-depth trade-offs for cutting planes and resolution that are truly supercritical, i.e., in terms of formula size rather than number of variables, and also show supercritical trade-offs between width and size for treelike resolution. Our results build on a new supercritical width-depth trade-off for resolution, obtained by refining and strengthening the compression scheme for the cop-robber game in [Grohe, Lichter, Neuen & Schweitzer 2023]. This yields robust supercritical trade-offs for dimension versus iteration number in the Weisfeiler-Leman algorithm, which also translate into trade-offs between number of variables and quantifier depth in first-order logic. Our other results follow from improved lifting theorems that might be of independent interest.

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen as a midpoint between $ACC$ (where we know nontrivial lower bounds) and depth-two linear threshold circuits (where nontrivial lower bounds remain open). We give an algorithm for evaluating an arbitrary symmetric function of $2^{n^{o(1)}}$ $ACC \circ THR$ circuits of size $2^{n^{o(1)}}$, on all possible inputs, in $2^n \cdot poly(n)$ time. Several consequences are derived: $\bullet$ The number of satisfying assignments to an $ACC \circ THR$ circuit of subexponential size can be computed in $2^{n-n^{\varepsilon}}$ time (where $\varepsilon > 0$ depends on the depth and modulus of the circuit). $\bullet$ $NEXP$ does not have quasi-polynomial size $ACC \circ THR$ circuits, nor does $NEXP$ have quasi-polynomial size $ACC \circ SYM$ circuits. Nontrivial size lower bounds were not known even for $AND \circ OR \circ THR$ circuits. $\bullet$ Every 0-1 integer linear program with $n$ Boolean variables and $s$ linear constraints is solvable in $2^{n-\Omega(n/((\log M)(\log s)^{5}))}\cdot poly(s,n,M)$ time with high probability, where $M$ upper bounds the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in $[-2^{poly(n)},2^{poly(n)}]$ and $poly(n)$ constraints can be solved in $2^{n-\Omega(n/\log^6 n)}$ time.) We also present an algorithm for evaluating depth-two linear threshold circuits (a.k.a., $THR \circ THR$) with exponential weights and $2^{n/24}$ size on all $2^n$ input assignments, running in $2^n \cdot poly(n)$ time. This is evidence that non-uniform lower bounds for $THR \circ THR$ are within reach.

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent to {\em depth-$d$ formulas of size $n^{o(d)}$} (moreover, this is optimal in that $n^{o(d)}$ cannot be improved to $n^{O(d)}$). This result is obtained by a combination of new lower and upper bounds for {\em Approximate Majorities}, the class of Boolean functions $\{0,1\}^n \to \{0,1\}$ that agree with the Majority function on $3/4$ fraction of inputs. $\mathrm{AC}^0[\oplus]$ formula lower bound: We show that every depth-$d$ $\mathrm{AC}^0[\oplus]$ formula of size $s$ has a {\em $1/8$-error polynomial approximation} over $\mathbb{F}_2$ of degree $O(\frac{1}{d}\log s)^{d-1}$. This strengthens a classic $O(\log s)^{d-1}$ degree approximation for \underline{circuits} due to Razborov. Since the Majority function has approximate degree $\Theta(\sqrt n)$, this result implies an $\exp(\Omega(dn^{1/2(d-1)}))$ lower bound on the depth-$d$ $\mathrm{AC}^0[\oplus]$ formula size of all Approximate Majority functions for all $d(n) \le O(\log n)$. Monotone $\mathrm{AC}^0$ circuit upper bound: For all $d(n) \le O(\frac{\log n}{\log\log n})$, we give a randomized construction of depth-$d$ monotone $\mathrm{AC}^0$ circuits (without NOT or MOD$_2$ gates) of size $\exp(O(n^{1/2(d-1)}))$ that compute an Approximate Majority function. This strengthens a construction of \underline{formulas} of size $\exp(O(dn^{1/2(d-1)}))$ due to Amano.

The purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing AND’s, OR’s and NOT’s, with no bound placed on the fan-in of AND-gates and OR-gates. Two such reducibilities are defined, and reductions and equivalences among several common problems such as parity, sorting, integer multiplication, graph connectivity, bipartite matching and network flow are given. Certain problems are shown to be complete, with respect to these reducibilities, in the complexity classes deterministic logarithmic space, nondeterministic logarithmic space, and deterministic polynomial time. New upper bounds on the size-depth (unbounded fan-in) circuit complexity of symmetric Boolean functions are established.

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in mathsf {Quasi}text {-}mathsf {NP} = mathsf {NTIME}[n^{(log n)^{O(1)}}] and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes mathcal { C}, by showing that mathcal { C} admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (f circ mathcal { C})-circuits of polynomial size.Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

A finite function f is a mapping of {0,1}n into {0,1}m∪{#}, where “#” is a symbol to be thought of as “undefined.” A family of finite functions is said to be one-way (in a circuit complexity sense) if it can be computed with polynomial size circuits, but no family of inverses of these functions can be computed with polynomial size circuits. In this paper we show that (provided functions that are not one-to-one are allowed) one-way functions exist if and only if the satisfiability problem SAT does not have polynomial size circuits.A family of functions f i (x) can be checked if some family of polynomial size circuits with inputs x and y can determine if f i (x)=y. A family of functions f i (x) can be evaluated if some family of polynomial size circuits with input x can compute f i (x). Can all families of total functions that can be checked also be evaluated? We show that this is true if and only if the nonuniform versions of the complexity classes P and UP∩co-UP are equal.A family of functions f i is one-way for constant depth circuits if f i can be computed with unbounded fan-in circuits of polynomial size and constant depth, but every family of inverses f −1i cannot. In this paper we give two provably one-way functions (in fact permutations) for constant depth circuits. The second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth.

On uniform circuit complexity

Circuit and Decision Tree Complexity of Some Number Theoretic Problems

AC0∘MOD2 lower bounds for the Boolean Inner Product

AC^0 o MOD_2 circuits are AC^0 circuits augmented with a layer of parity gates just above the input layer. We study AC^0 o MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC^0 o MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ~Omega(n^2) lower bound for the special case of depth-4 AC^0 o MOD_2. Our proof of the depth-4 lower bound employs a new moment-matching inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match.

\n Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a small circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erds and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called robust sunflowers, to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossmans bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain the most significant progress on the sunflower conjecture since its inception in 1960. In this work, we will show how the recent progresses in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, thus breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes.\n