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On DNF Approximators for Monotone Boolean Functions
Abstract
We study the complexity of approximating monotone Boolean functions with disjunctive normal form (DNF) formulas, exploring two main directions. First, we construct DNF approximators for arbitrary monotone functions achieving one-sided error: we show that every monotone f can be ε-approximated by a DNF g of size satisfying g(x) ≤ f(x) for all x ∈ {0,1}
n
. This is the first non-trivial universal upper bound even for DNF approximators incurring two-sided error.
Next, we study the power of negations in DNF approximators for monotone functions. We exhibit monotone functions for which non-monotone DNFs perform better than monotone ones, giving separations with respect to both DNF size and width. Our results, when taken together with a classical theorem of Quine [1], highlight an interesting contrast between approximation and exact computation in the DNF complexity of monotone functions, and they add to a line of work on the surprising role of negations in monotone complexity [2,3,4].
... Our analysis applies and builds upon the main structural lemma in [7]. To state it, recall that a DNF is a Boolean function that is formed as an OR of ANDs, and it is monotone if there are no negations. ...
... Their structural lemma shows that each monotone function can be approximated by a DNF with only a constant number of distinct clause widths. Specifically: [7], abridged and restated). For every positive ǫ, for all sufficiently large n, let f be a monotone Boolean function over the domain {0, 1} n . ...
... To overcome these differences, we generalize to the setting of non-Boolean functions the main concept used in the proof of Lemma 1.1: the concept of a minterm of a monotone Boolean function. In [7] the minterm of a monotone Boolean function f is defined as follows: ...
A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions. One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone.
... We say that a CNF formula ϕ ǫ-approximates a function f if Pr[f (x) = ϕ(x)] < ǫ. Initiated by O'Donnell and Wimmer [OW07], a line of research [OW07,BT15,BHST14] has highlighted surprising differences between exact computation of depth-2 formulas and approximation by depth-2 formulas. Two of them are reviewed below. ...
... Quine's Theorem for Approximation. Blais, Håstad, Servedio, and Tan [BHST14] studied whether an analog of Quine's theorem holds. Quine's theorem states that the minimum CNF formula computing a monotone function is achieved by monotone CNF formulas (i.e., CNF formulas without any negated literal). ...
... Given this fact, it is tempting to guess that a smallest CNF formula approximating a monotone function is monotone. However, in [BHST14], it was shown that there is at least a quadratic gap between the size of the smallest CNF formula and the smallest monotone CNF formula approximating a monotone function f . ...
We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n) equal to the inverse of the maximum, over all depth-2 formulas, of one-sided-error correlation bound divided by the size of the depth-2 formula, on a certain hard distribution. We apply this duality to obtain several consequences: 1. Any function f can be approximated by a CNF formula of size with one-sided error and advantage for some , which is tight up to a constant factor. 2. There exists a monotone function f such that f can be approximated by some polynomial-size CNF formula, whereas any monotone CNF formula approximating f requires exponential size. 3. Any depth-3 formula computing the parity function requires gates, which is tight up to a factor of . This establishes a quadratic separation between depth-3 circuit size and depth-3 formula size. 4. We give a characterization of the depth-3 monotone circuit complexity of the majority function, in terms of a natural extremal problem on hypergraphs. In particular, we show that a known extension of Turan's theorem gives a tight (up to a polynomial factor) circuit size for computing the majority function by a monotone depth-3 circuit with bottom fan-in 2. 5. AC0[p] has exponentially small one-sided correlation with the parity function for odd prime p.
... By the lemma, for any χ Z ∈ C, either χ Z or χ Z = χ {0,1} ℓ \Z can be expressed as the logical XOR of at most ℓ monotone boolean functions. On the other hand, for each of these monotone functions, recently Blais, Håstad, Servedio and Tan [1] showed the following result: ...
... Proposition 15 (See [1]). For any 0 < η < 1, every ℓ-bit input monotone boolean function f can be approximated by a DNF formula g of size 2 ℓ−Ω( √ ℓ) satisfying that d(f, g) ≤ η · 2 ℓ (where the dependence on η is omitted in the Ω notation for simplicity). ...
In this paper, we specify a class of mathematical problems, which we refer to as “Function Density Problems” (FDPs, in short), and point out novel connections of FDPs to the following two cryptographic topics; theoretical security evaluations of keyless hash functions (such as SHA-1), and constructions of provably secure pseudorandom generators (PRGs) with some enhanced security property introduced by Dubrov and Ishai (STOC 2006). Our argument aims at proposing new theoretical frameworks for these topics (especially for the former) based on FDPs, rather than providing some concrete and practical results on the topics. We also give some examples of mathematical discussions on FDPs, which would be of independent interest from mathematical viewpoints. Finally, we discuss possible directions of future research on other crypto-graphic applications of FDPs and on mathematical studies on FDPs themselves.
In 1993, Benaloh and De Mare introduced cryptographic accumulator, a primitive that allows the representation of a set of values by a short object (the accumulator) and offers the possibility to prove that some input values are in the accumulator. For this purpose, so-called asymmetric accumulators require the creation of an additional cryptographic object, called a witness. Through the years, several instantiations of accumulators were proposed either based on number theoretic assumptions, hash functions, bilinear pairings or more recently lattices. In this work, we present the first instantiation of an asymmetric cryptographic accumulator that allows private computation of the accumulator but public witness creation. This is obtained thanks to our unique combination of the pairing based accumulator of Nguyen with dual pairing vector spaces. We moreover introduce the new concept of dually computable cryptographic accumulators, in which we offer two ways to compute the representation of a set: either privately (using a dedicated secret key) or publicly (using only the scheme’s public key), while there is a unique witness creation for both cases. All our constructions of accumulators have constant size accumulated value and witness, and satisfy the accumulator security property of collision resistance, meaning that it is not possible to forge a witness for an element that is not in the accumulated set. As a second contribution, we show how our new concept of dually computable cryptographic accumulator can be used to build a Ciphertext Policy Attribute Based Encryption (CP-ABE). Our resulting scheme permits policies expressed as disjunctions of conjunctions (without “NO” gates), and is adaptively secure in the standard model. This is the first CP-ABE scheme having both constant-size user secret keys and ciphertexts (i.e. independent of the number of attributes in the scheme, or the policy size). For the first time, we provide a way to use cryptographic accumulators for both key management and encryption process.
We establish new separations between the power of monotone and general (non-monotone) Boolean circuits: - For every , there is a monotone function in that requires monotone circuits of depth . This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai and Gurevich (1987). In addition, our separation holds for a monotone graph property, which was unknown even in the context of versus . - For every , there is a monotone function in that requires monotone circuits of size . This makes progress towards a question posed by Grigni and Sipser (1992). These results show that constant-depth circuits can be more efficient than monotone circuits when computing monotone functions. In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an circuit of size s and depth d can be computed by a monotone circuit of size . We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in admits a polynomial size monotone circuit, then is not contained in . Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems established by G\"o\"os et al. (2019) via lifting techniques. Adapting results of Schaefer (1978) and Allender et al. (2009), we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.
We prove tight size bounds on monotone switching networks for the NP-complete problem of k-clique, and for an explicit monotone problem by analyzing a pyramid structure of height h for the P-complete problem of generation. This gives alternative proofs of the separations of m-NC from m-P and of m-NCⁱ from m-NCⁱ⁺¹, different from Raz-McKenzie (Combinatorica 1999). The enumerative-combinatorial and Fourier analytic techniques in this paper are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.
An approximate computation of a Boolean function by a circuit or switching network is a computation in which the function is computed correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many sub areas of complexity theory, such as cryptography and derandomization. Lower bounds for approximate computation are also known as correlation bounds or average case hardness. In this paper, we obtain the first average case monotone depth lower bounds for a function in monotone P. We tolerate errors that are asymptotically the best possible for monotone circuits. Specifically, we prove average case exponential lower bounds on the size of monotone switching networks for the GEN function. As a corollary, we separate the monotone NC hierarchy in the case of errors -- a result which was previously only known for exact computations. Our proof extends and simplifies the Fourier analytic technique due to Potechin, and further developed by Chan and Potechin. As a corollary of our main lower bound, we prove that the communication complexity approach for monotone depth lower bounds does not naturally generalize to the average case setting.
We prove that every monotone circuit which tests st-connectivity of an undirected graph on n nodes has depth &OHgr;(log2n). This implies a superpolynomial (n&OHgr;(log n)) lower bound on the size of any monotone formula for st-connectivity.The proof draws intuition from a new characterization of circuit depth in terms of communication complexity. It uses counting arguments and Extremal Set Theory.Within the same framework, we also give a very simple and intuitive proof of a depth analogue of a theorem of Krapchenko concerning formula size lower bounds.
Say that f:{0,1}n
→{0,1} ε-approximates g : {0,1}n
→{0,1} if the functions disagree on at most an ε fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits:
(1) For every constant 0 < ε< 1/2 there is a DNF of size that ε-approximates the Majority function on n bits, and this is optimal up to the constant in the exponent.
(2) There is a monotone function with total influence (AKA average sensitivity) such that any DNF or CNF that .01-approximates requires size 2Ω(n / logn) and such that any unbounded fan-in AND-OR-NOT circuit that .01-approximates requires size Ω(n/ logn). This disproves a conjecture of Benjamini, Kalai, and Schramm (appearing in [BKS99,Kal00,KS05]).
. The sensitivity of a point is dist, i.e. the number of neighbors of the point in the discrete cube on which the value of differs. The average sensitivity of is the average of the sensitivity of all points in . (This can also be interpreted as the sum of the influences of the variables on , or as a measure of the edge boundary of the set which is the characteristic function of.) We show here that if the average sensitivity of is then can be approximated by a function depending on coordinates where is a constant depending only on the accuracy of the approximation but not on . We also present a more general version of this theorem, where the sensitivity is measured with respect to a product measure
which is not the uniform measure on the cube.
Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(m
s
/(logm)2s
) for fixeds, and sizem
Ω(logm) form/4].
In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)2/3 requires monotone circuits exp (Ω((m/logm)1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm)s
) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n
1/4 · (logn)1/2)), improving a recent result of exp (Ω(n
1/8-ε)) due to Andreev.
In connection with the least fixed point operator the following question was raised: Suppose that a first-order formula P(P) is (semantically) monotone in a predicate symbol P on finite structures. Is P(P) necessarily equivalent on finite structures to a first-order formula with only positive occurrences of P? In this paper, this question is answered negatively. Moreover, the counterexample naturally gives a uniform sequence of constant-depth, polynomial-size, monotone Boolean circuits that is not equivalent to any (however nonuniform) sequence of constant-depth, polynomial-size, positive Boolean circuits.
In this paper, monotone Boolean functions are studied using harmonic analysis on the cube. The main result is that any monotone Boolean function has most of its power spectrum on its Fourier coefficients of "degree" at most O( p n) under any product distribution. This is similar to a result of Linial, Mansour, and Nisan [LMN93] which showed that AC 0 functions have almost all of its power spectrum on the coefficients of degree at most (log n) O(1) under the uniform distribution. As a consequence of the main result the following two corollaries are obtained: ffl For any ffl ? 0, monotone Boolean functions are PAC learnable with error ffl under product distributions in time 2 ~ O( 1 ffl p n) . ffl Any monotone Boolean function can be approximated within error ffl under product distributions by a non-monotone Boolean circuit of size 2 ~ O( 1 ffl p n) and depth ~ O( 1 ffl p n). The learning algorithm runs in time subexponential as long as the required error is \Om...
The average sensitivity of a Boolean function is the expectation,
given a uniformly random input, of the number of input bits
which when flipped change the output of the function.
Answering a question by O'Donnell, we show
that every Boolean function represented by a k-CNF (or a k-DNF)
has average sensitivity at most k. This
bound is tight since
the parity function on k variables has average sensitivity k.
We study the complexity of approximating Boolean functions with DNFs and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function. In the first direction, our main positive results are the first non-trivial universal upper bounds on appropriability by DNFs: : Every Boolean function can be ε-approximated by a DNF of size Oε(2n/ log n). : Every Boolean function can be ε-approximated by a DNF of width cε n, where cε
We prove tight size bounds on monotone switching networks for the k-clique problem, and for an explicit monotone problem by analyzing the generation problem with a pyramid structure of height h. This gives alternative proofs of the separations of m-NC from m-P and of m-NCi from m-NCi+1, different from Raz-McKenzie (Combinatorica '99). The enumerative-combinatorial and Fourier analytic techniques in this work are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.
Monotone Boolean functions are an important object in discrete mathematics and mathematical cybernetics. Topics related to these functions have been actively studied for several decades. Many results have been obtained, and many papers published. However, until now there has been no sufficiently complete monograph or survey of results of investigations concerning monotone Boolean functions. The object of this survey is to present the main results on monotone Boolean functions obtained during the last 50 years.
A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastn
c losn
. We observe that this lower bound can be improved to exp(cn
1/6–o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grtschel—Lovsz—Schrijver on the Lovsz — capacity, of a graph.
Given a DNF formula on n variables, the two natural size measures are the
number of terms or size s(f), and the maximum width of a term w(f). It is
folklore that short DNF formulas can be made narrow. We prove a converse,
showing that narrow formulas can be sparsified. More precisely, any width w DNF
irrespective of its size can be -approximated by a width w DNF with
at most terms.
We combine our sparsification result with the work of Luby and Velikovic to
give a faster deterministic algorithm for approximately counting the number of
satisfying solutions to a DNF. Given a formula on n variables with poly(n)
terms, we give a deterministic time algorithm
that computes an additive approximation to the fraction of
satisfying assignments of f for \epsilon = 1/\poly(\log n). The previous best
result due to Luby and Velickovic from nearly two decades ago had a run-time of
.
We present a membership-query algorithm for efficiently learning DNF with respect to the uniform distribution. In fact, the algorithm properly learns with respect to uniform the class TOP of Boolean functions expressed as a majority vote over parity functions. We also describe extensions of this algorithm for learning DNF over certain nonuniform distributions and for learning a class of geometric concepts that generalizes DNF. Furthermore, we show that DNF is weakly learnable with respect to uniform from noisy examples. Our strong learning algorithm utilizes one of Freund's boosting techniques and relies on the fact that boosting does not require a completely distribution-independent weak learner. The boosted weak learner is a nonuniform extension of a parity-finding algorithm discovered by Goldreich and Levin.
We prove that a monotone circuit of size nd recognizing connectivity must have depth . For formulas this implies depth . For polynomial-size circuits the bound becomes which is optimal up to a constant.
The question of whether it is easier to solve two communication problems together rather than separately is related to the complexity of the composition of Boolean functions. Based on this relationship, an approach to separating NC1 from P is outlined. Furthermore, it is shown that the approach provides a new proof of the separation of monotone NC1 from monotone P
We investigate the computational power of depth two circuits consisting of MOD r --gates at the bottom and a threshold gate with arbitrary weights at the top (for short, threshold--MOD r circuits) and circuits with two levels of MOD gates (MOD p -MOD q circuits). In particular, we will show the following results. (i) For all prime numbers p and integers q; r, it holds that if p divides r but not q then all threshold--MOD q circuits for MOD r have exponentially many nodes. (ii) For all integers r, all problems computable by depth two fAND;OR;NOTg-- circuits of polynomial size have threshold--MOD r circuits with polynomially many edges. (iii) There is a problem computable by depth 3 fAND;OR;NOTg--circuits of linear size and constant bottom fan--in which for all r needs threshold--MOD r circuits with exponentially many nodes. (iv) For p; r different primes, and q 2; k positive integers, where r does not divide q; every MOD p k -MOD q circuit for MOD r has e...
We prove that any monotone switching network solving directed connectivity on
N vertices must have size at least N^(Omega(logN)).
We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC 6= monotone-P. 2. For every i 1, monotone-NC i 6= monotone-NC i+1 . 3. More generally: For any integer function D(n), up to n ffl (for some ffl ? 0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const Delta D(n) (for some constant Const). Only a separation of monotone-NC 1 from monotone-NC 2 was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In...
We prove that monotone circuits computing the perfect matching function on n-vertex graphs requireOmegaGamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits. CR categories: F.1.1, Models of Computation, Unbounded Action Devices. This research was partially supported by the American-Israeli Binational Science Foundation grant number 87-00082 1 General Terms: THEORY Key words: circuit depth, monotone computation, perfect matching 1 Introduction In 1985 Razborov [R1] proved a superpolynomial lower bound on the size of monotone circuits computing the perfect matching function. As matching is in P , this result first showed a superpolynomial gap between the size of monotone and non-monotone circuits. This gap was shown to be exponential by Tardos [T] (with a different monotone function in P ). Another exponential gap for size, when the circuits are restricted to have constant depth, was proved by Ajtai and Gurevich [AG]. Karchm...
this paper we show that monotone logarithmic depth circuits are strictly weaker than monotone polynomial size logarithmic width circuits, i.e. we prove that mNC
Monotone versus positiveView ArticleMATH
- M Ajtai
- Y Grevich
Natural proofsView ArticleMATHMathSciNet
- A Razborov
- S Rudich
Approximation by DNF: examples and counterexamplesView Article
- R T Donnell
- K Wimmer
Noise stability and threshold circuits. Gil Kalai’s, Combinatorics and more, blog
- G Kalai
- A. Razborov