Analysis Of Boolean Functions Research Articles

Hypercontractivity for global functions and sharp thresholds

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the Kahn-Kalai-Linial theorem, Friedgut’s junta theorem and the invariance principle of Mossel, O’Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform ( 1 / 2 1/2 -biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general p p -biased measures. However, simple examples show that when p p is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity inequality for general p p that applies to ‘global functions’, i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain’s sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain’s theorem, making progress on two conjectures of Kahn and Kalai (both these conjectures were open when we arXived this paper in 2019; one of them was solved in 2022; the other is still open), and proving a p p -biased analogue of the seminal invariance principle of Mossel, O’Donnell, and Oleszkiewicz. In this 2023 version of our paper we will also survey many further applications of our results that have been obtained by various authors since we arXived the first version in 2019.

Efficient deterministic preparation of quantum states using decision diagrams

Loading classical data into quantum registers is one of the most important primitives of quantum computing. While the complexity of preparing a generic quantum state is exponential in the number of qubits, in many practical tasks the state to prepare has a certain structure that allows for faster preparation. In this paper, we consider quantum states that can be efficiently represented by (reduced) decision diagrams, a versatile data structure for the representation and analysis of Boolean functions. We design an algorithm that utilizes the structure of decision diagrams to prepare their associated quantum states. Our algorithm has a circuit complexity that is linear in the number of paths in the decision diagram. Numerical experiments show that our algorithm reduces the circuit complexity by up to 31.85% compared to the state-of-the-art algorithm, when preparing generic $n$-qubit states with ${n}^{3}$ nonzero amplitudes. Additionally, for states with sparse decision diagrams, including the initial state of the quantum Byzantine agreement protocol, our algorithm reduces the number of controlled-nots by 86.61--99.9%.

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩=∑j=1rcj|φj⟩ for cj∈C and stabilizer states φj. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states.We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(n/log⁡n). This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\mathrm{AG}(n,q)$.
 We define Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ and $\mathrm{PG}(n,q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets.
 This paper focuses on $\mathrm{AG}(n,q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\mathrm{AG}(3,q)$ was already treated separately.

The junta method for hypergraphs and the Erdős-Chvátal simplex conjecture

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an ‘enlarged’ copy H+ of a fixed hypergraph H. These include well-known problems such as the Erdős-Sós ‘forbidding one intersection’ problem and the Frankl-Füredi ‘special simplex’ problem.We present a general approach to such problems, using a ‘junta approximation method’ that originates from analysis of Boolean functions. We prove that any H+-free hypergraph is essentially contained in a ‘junta’ – a hypergraph determined by a small number of vertices – that is also H+-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a solution of the extremal problem for a large class of H's, which includes the aforementioned problems, and solves them for a large new set of parameters.We apply our method also to the 1974 Erdős-Chvátal simplex conjecture, which asserts that for any d<k≤dd+1n, the maximal size of a k-uniform family that does not contain a d-simplex (i.e., d+1 sets with empty intersection such that any d of them intersect) is (n−1k−1). We prove the conjecture for all d and k, provided n>n0(d).

Boolean Spectral Analysis in Categorical Reservoir Modeling

This work introduces a new method for simulating facies distribution with two categories based on Fourier analysis of Boolean functions. According to this method, two categories of facies distributed along vertical wells are encoded as Boolean functions taking two values. The subsequent simulation process is divided into three consecutive steps. First, Boolean functions of the well data are decomposed into a binary version of a Fourier series. Decomposition coefficients are then simulated over the two-dimensional area as stationary random fields. Finally, synthetic data in the interwell space are reconstructed from simulated coefficients. The described method was implemented experimentally in software and tested on a case of a real oil field and on a case of a synthetic oil field model. Simulations on the synthetic model were used to test the performance of the method for two different bases in the Fourier expansion (Walsh functions and Haar wavelets). The simulation results were compared to those obtained on the same synthetic model via the classical sequential indicator simulation. It was shown that, for both bases, the new method reproduces statistical parameters of the well data better than sequential indicator simulation.

Towards a Proof of the Fourier-Entropy Conjecture?

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is \(o(\log n)\). However, both results become useless when the total influence of the function is \(\omega (\log n)\). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of \(S_n\). In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1. A quantitative improvement of the Bourgain–Kalai result regarding the total influence of functions that are transitively symmetric. 2. A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on \(2^{O(I[f]\log I[f])}\) characters, improving an earlier result of Friedgut. Removing the \(\log I[f]\) factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time \(2^{O(K\log K)}\) using membership queries. Thus, the class of functions with total influence \(O(\log n/\log \log n)\) is agnostically learnable in \(\mathsf{poly}(n)\) time.

Quantum algorithms for learning the algebraic normal form of quadratic Boolean functions

Quantum algorithms for the analysis of Boolean functions have received a lot of attention over the last few years. The algebraic normal form (ANF) of a linear Boolean function can be recovered by using the Bernstein–Vazirani (BV) algorithm. No research has been carried out on quantum algorithms for learning the ANF of general Boolean functions. In this paper, quantum algorithms for learning the ANF of quadratic Boolean functions are studied. We draw a conclusion about the influences of variables on quadratic functions, so that the BV algorithm can be run on them. We study the functions obtained by inversion and zero-setting of some variables in the quadratic function and show the construction of their quantum oracle. We introduce the concept of “club” to group variables that appear in quadratic terms and study the properties of clubs. Furthermore, we propose a bunch of algorithms for learning the full ANF of quadratic Boolean functions. The most efficient algorithm, among those we propose, provides an O(n) speedup over the classical one, and the number of queries is independent of the degenerate variables.

Random tree recursions: Which fixed points correspond to tangible sets of trees?

Let be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children u and v such that the subtrees rooted at u and v belong to it. Let p be the probability that a Galton‐Watson tree falls in . The metaproperty makes p satisfy a fixed‐point equation, which can have multiple solutions. One of these solutions is p, but what is the meaning of the others? In particular, are they probabilities of the Galton‐Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed‐point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton‐Watson trees and the analysis of Boolean functions.

Public Projects, Boolean Functions, and the Borders of Border’s Theorem

Border’s theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border’s theorem either restrict attention to relatively simple settings or resort to approximation. This article identifies a complexity-theoretic barrier that indicates, assuming standard complexity class separations, that Border’s theorem cannot be extended significantly beyond the state of the art. We also identify a surprisingly tight connection between Myerson’s optimal auction theory, when applied to public project settings, and some fundamental results in the analysis of Boolean functions.

The Junta Method in Extremal Hypergraph Theory and Chvátal's Conjecture

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an ‘enlarged’ copy H+ of a fixed hypergraph H. These include well-known open problems such as the Erdős matching conjecture, the Erdős-Sós ‘forbidding one intersection’ problem, the Frankl-Füredi ‘special simplex’ problem, etc.We present a general approach to such problems, using a ‘junta approximation method’ that originates from analysis of Boolean functions. We prove that any H+-free hypergraph is essentially contained in a ‘junta’ – a hypergraph determined by a small number of vertices – that is also H+-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a complete solution of the extremal problem for a large class of H's, which includes all the aforementioned problems.We apply our method to the 1974 Chvátal's conjecture, which asserts that for any d<k≤dd+1n, the maximal size of a k-uniform family that does not contain a d-simplex (i.e., d+1 sets with empty intersection such that any d of them intersect) is (n−1k−1). We prove the conjecture for all d and k, provided n>n0(d).

Improved NP-Inapproximability for $2$-Variable Linear Equations

$ \newcommand{\twolin}{2\textnormal{-Lin}(2)\xspace} \newcommand{\eps}{\epsilon} \newcommand{\NP}{\textsf{NP}} $ An instance of the $2-\text{Lin}(2)$ problem is a system of equations of the form “$x_i + x_j = b \pmod{2}$.” Given such a system in which it is possible to satisfy all but an $\eps$ fraction of the equations, we show it is $\NP$-hard to satisfy all but a $C \eps$ fraction of equations, for any $C < {11}/{8} = 1.375$ (and any $0 < \eps \leq 1/8$). The previous best result, standing for over $15$ years, had ${5}/{4}$ in place of ${11}/{8}$. Our result provides the best known $\NP$-hardness even for the $\text{Unique-Games}$ problem, and it also holds for the special case of $\text{Max-Cut}$. The precise factor ${11}/{8}$ is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of ${3}/{2}$. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor $C$ greater than $2.54$. Previously, no such limitations on gadget reductions was known. A preliminary version of this paper appeared in the Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2015).

Upper bounds on Fourier entropy

Given a function f:{0,1}n→{+1,−1}, its Fourier Entropy is defined to be −∑Sfˆ(S)2log⁡fˆ(S)2, where fˆ denotes the Fourier transform of f. In the analysis of Boolean functions, an outstanding open question is a conjecture of Friedgut and Kalai, 1996 [3], called the Fourier Entropy Influence (FEI) Conjecture, asserting that the Fourier Entropy of any Boolean function f is bounded above, up to a constant factor, by the total influence (=average sensitivity) of f.In this paper we give several upper bounds on the Fourier Entropy. We first give upper bounds on the Fourier Entropy of Boolean functions in terms of several complexity measures that are known to be bigger than the influence. These complexity measures include, among others, the logarithm of the number of leaves and the average depth of a parity decision tree. We then show that for the class of Linear Threshold Functions (LTF), the Fourier Entropy is O(n). It is known that the average sensitivity for the class of LTF is Θ(n). We also establish a bound of Od(n1−14d+6) for general degree-d polynomial threshold functions. Our proof is based on a new upper bound on the derivative of noise sensitivity. Next we proceed to show that the FEI Conjecture holds for read-once formulas that use AND, OR, XOR, and NOT gates. The last result is independent of a result due to O'Donnell and Tan [1] for read-once formulas with arbitrary gates of bounded fan-in, but our proof is completely elementary and very different from theirs. Finally, we give a general bound involving the first and second moments of sensitivities of a function (average sensitivity being the first moment), which holds for real-valued functions as well.

An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube

We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young's orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form $\{(x_1,\ldots,x_n) \in \{0,1\}^n : \sum_i x_i = k\}$), which refines the eigenspaces of the Johnson association scheme; our basis is orthogonal with respect to any exchangeable measure. More concretely, our basis is an orthogonal basis for all multilinear polynomials $\mathbb{R}^n \to \mathbb{R}$ which are annihilated by the differential operator $\sum_i \partial/\partial x_i$. As an application of the last point of view, we show how to lift low-degree functions from a slice to the entire Boolean hypercube while maintaining properties such as expectation, variance and $L^2$-norm.As an application of our basis, we streamline Wimmer's proof of Friedgut's theorem for the slice. Friedgut's theorem, a fundamental result in the analysis of Boolean functions, states that a Boolean function on the Boolean hypercube with low total influence can be approximated by a Boolean junta (a function depending on a small number of coordinates). Wimmer generalized this result to slices of the Boolean hypercube, working mostly over the symmetric group, and utilizing properties of Young's orthogonal representation. Using our basis, we show how the entire argument can be carried out directly on the slice.

Logic circuits from zero forcing.

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of “back forcing” as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we show that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity. Moreover, in the light of applications of zero forcing in quantum mechanics, the link with Boolean functions may suggest a new directions in quantum control theory and in the study of engineered quantum spin systems. It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits.

An Entropic Proof of Chang's Inequality

Chang's lemma is a useful tool in additive combinatorics and the analysis of Boolean functions. Here we give an elementary proof using entropy. We obtain a tight constant and give a slight improvement in the case where the variables are highly biased.

A Fourier-Analytic Approach to Reed–Muller Decoding

We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We use this to show that quadratic forms over any field are locally list-decodable up to their minimum distance. The analogous statement for linear polynomials was proved in the celebrated works of Goldreich Previously, tight bounds for quadratic polynomials were known only for q = 2 and 3; the best bound known for other fields was the Johnson radius. Departing from previous work on Reed-Muller decoding which relies on some form of self-corrector, our work applies ideas from Fourier analysis of Boolean functions to low-degree polynomials over finite fields, in conjunction with results about the weight-distribution. We believe that the techniques used here could find other applications, we present some applications to testing and learning.

Functions classification approach to generate reconfigurable fine-grain logic based on Ambipolar Independent Double Gate FET (Am-IDGFET)

Am-IDGFET is a new family of particular devices in view of the fact that it associates three benefits: (i) it is usually a 1-D electronic device (CNT or SiNW), meaning high mobility, achievable current density and high ION/IOFF ratio; (ii) Independently controlled gates which offers the device extra logic options; (iii) ambipolar behaviour opens the way for N- and P-type polarities in the same device via its back gate. The creativity of this work consists of looking at this new class of emerging technology as an opportunity for new design paradigms with no equivalent counterparts in CMOS technology. Nevertheless, to build a feasible and complete picture of ambipolar logic, innovative design approaches and tools are required. In this paper, we exploit functional classification, a powerful tool for the construction and analysis of Boolean functions, to build reconfigurable logic blocks by defining a hierarchical correlation between structures of functions classes with ambipolar devices. We demonstrate how this approach enables us to build Am-I DGFET-based n-input reconfigurable cells. Several dynamically reconfigurable 2-inputs logic cells with partial and full functionality are designed in this paper. We evaluate the performances of circuits designed from this approach in a case study focused on Double Gate Carbon Nanotube FET (DG-CNTFET) technology. Simulations results show efficiency to build fine grain reconfigurable cells with partial functionality. In the case of 9-functions reconfigurable cell, an improvement of 1.8X in terms of power delay product (PDP) is proved when compared to a CMOS-16nm technology. Fewer control signals are required and the area is reduced by 35% over CMOS technology.

Noise stability of functions with low influences: Invariance and optimality

In this paper, we study functions with low influences on product probability spaces. The analysis of Boolean functions f {-1, 1}/sup n/ /spl rarr/ {-1, 1} with low influences has become a central problem in discrete Fourier analysis. is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly smoothed; this extension is essential for our applications to noise stability -type problems. In particular; as applications of the invariance principle we prove two conjectures: the Majority Is Stablest conjecture [29] from theoretical computer science, which was the original motivation for this work, and the It Ain't Till It's Over conjecture [27] from social choice theory. The Majority Is Stablest conjecture and its generalizations proven here, in conjunction with the Unique Games Conjecture and its variants, imply a number of (optimal) inapproximability results for graph problems.

Harmonic Analysis of Polynomial Threshold Functions

The analysis of linear threshold Boolean functions has recently attracted the attention of those interested in circuit complexity as well as ofthose interested in neural networks. Here a generalization oflinear threshold functions is defined, namely, polynomial threshold functions, and its relation to the class of linear threshold functions is investigated. A Boolean function is polynomial threshold if it can be represented as a sign function ofa polynomial that consists ofa polynomial (in the number ofvariables) number ofterms. The main result ofthis paper is showing that the class ofpolynomial threshold functions (which is called PT1 is strictly contained in the class ofBoolean functions that can be computed by a depth 2, unbounded fan-in polynomial size circuit of linear threshold gates (which is called LT2). Harmonic analysis ofBoolean functions is used to derive a necessary and sufficient condition for a function to be an S-threshold function for a given set S of monomials. This condition is used to show that the number of different S-threshold functions, for a given S, is at most 2 t'/ 1)lsl. Based on the necessary and sufficient condition, a lower bound is derived on the number of terms in a threshold function. The lower bound is expressed in terms of the spectral representation of a Boolean function. It is found that Boolean functions having an exponentially small spectrum are not polynomial threshold. A family of functions is exhibited that has an exponentially small spectrum; they are called functions. A function is constructed that is both semibent and symmetric to prove thatPT is properly contained in LT2.