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The Log-Rank Conjecture and low degree polynomials
Information Processing Letters 89 (2004) 99–103 www.elsevier.com/locate/ipl
The Log-Rank Conjecture and low degree polynomials Paul Valiant 1 Stanford University, P.O. Box 17308, Stanford, CA 94309, USA Received 17 January 2003; received in revised form 20 July 2003 Communicated by L.A. Hemaspaandra
Abstract We formulate several questions concerning the intersections of sets of Boolean roots of low degree polynomials. Two of these questions we show to be equivalent to the Log-Rank Conjecture from communication complexity. We further exhibit a slightly stronger formulation which we prove to be false, and a weaker formulation which we prove to be true. These results suggest a possible new approach to the Log-Rank Conjecture. 2003 Elsevier B.V. All rights reserved. Keywords: Log-Rank Conjecture; Communication complexity; Boolean polynomials; Computational complexity
1. Introduction The two-party deterministic communication complexity model has been the subject of much research since its introduction by Yao [9]. Here we concern ourselves with the so-called “Log-Rank Conjecture” relating the communication complexity of a function to the rank of the corresponding matrix. Given a function f (x, y) → {0, 1}, with x ∈ X and y ∈ Y , the communication complexity is the minimum amount of communication necessary between two parties to compute f when one party is given the first input x, and the second party is given the second input y. Specifically, we define the communication complexity c(f ) as the length of the shortest deterministic protoE-mail address: [email protected] (P. Valiant). 1 Research supported in part by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Army Research Office under grant DAAD19-00-1-0177 at MIT. 0020-0190/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2003.09.020
col for computing f , where at each stage in the protocol one party is designated to send a single bit of information to the other party. We may also consider f as a matrix M, where Mx,y = f (x, y). From now on, we consider f as being implicit in M. It was shown by Mehlhorn and Schmidt [3] that c(M) log2 rank(M). Subsequently Lovász and Saks [2] asked the question whether a reverse inequality holds to within logarithmic factors. This is now known as the “Log-Rank Conjecture”. We specify the base of the logarithm to be 2 only when it makes a difference. Conjecture 1. There is an α such that, for every M, c(M) [log rank(M)]α . Several authors have found examples where c(m) > log rank(M), but the largest separation known has c(m) equal to about (log rank(M))1.6 [7,6,5]. The conjecture itself was reformulated by Nisan and Wigder-
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son [5] in more combinatorial terms. They considered monochromatic rectangles in M—namely submatrices that are uniformly either 1 or 0. Following their notation, let mono(M) be the maximum value of the fraction |A|/|M| where A is a monochromatic rectangle, and the size of a matrix is the number of elements in it. Their conjecture, which they proved equivalent to Conjecture 1, is as follows. Conjecture 2. There is an α such that, for every M, − log mono(M) (log rank(M))α . Here we present a third formulation which we prove to be equivalent to these two. Consider a set S of linear functions P : Rk → R. We are concerned with the values these functions take on {0, 1}k . Specifically, let S= be the set of points in {0, 1}k where all the functions have equal values, and let S⊥ be the set of points in {0, 1}k where all the functions have values in {0, 1}. For convenience, let S0 and S1 be the sets on which every function is 0 and 1, respectively. Thus S0 ∪ S1 = S⊥ ∩ S= . Conjecture 3. There is an α such that for every nonempty set S of linear functions Rk → R and every set Z ⊂ S⊥ there exists X ⊂ S with log(|S|/|X|) (log k)α such that log |X=|Z|∩Z| (log k)α . As an aside we note that an alternative way of viewing this conjecture is the following. Each function in S can be thought of as describing a k-dimensional affine subspace as a subset of {0, 1}k+1 . The conjecture then states that if the intersection of the projections of these sets into {0, 1}k is large, then there is a large subset of those sets whose intersection in {0, 1}k+1 is large. We know that if we project by log2 k dimensions then the corresponding statement is false. In particular there is a set S of k-dimensional affine subspaces of {0, 1}k+log2 k such that the intersection of the projections of these sets into {0, 1}k is the whole set but no large subset of these sets has a large intersection with {0, 1}k+log2 k . Such a set S over variables x1 . . . xk and z0 . . . zlog2 k−1 is defined by the following set of equations: the left-hand sides are sums of all subsets of k/2 of the x variables; the right-hand sides are all equal to log2 k−1 i 2 zi . i=0 We prove the following theorem.
Theorem 1. Conjecture 3 is equivalent to Conjectures 1 and 2. We further show that the functions in S need not be linear: the conjecture remains equivalent if the functions are polynomials of degree (log k)β for some constant β. Conjecture 4. For any β there is an α such that for every nonempty set S of polynomials Rk → R of degree (log k)β and every set Z ⊂ S⊥ , there exists X ⊂ S with log(|S|/|X|) (log k)α such that log |X|Z| = ∩Z| (log k)α . Theorem 2. Conjecture 3 is equivalent to Conjecture 4. However, if we allow the polynomials to have degree (log k)ω(1) the corresponding conjecture is false. Example 1. Given a constant α and a function f (k) = ω(1), then for large enough k there is a set S of polynomials of degree (log k)f (k) such that there does not exist X ⊂ S with log(|S|/|X|) (log k)α such that ⊥| (log k)α . log |X|S = ∩S⊥ | The example we construct consists entirely of polynomials with 0–1 values on {0, 1}k , sometimes called Boolean polynomials. In our notation, S⊥ = {0, 1}k . We show however, that such polynomials cannot provide corresponding counterexamples to the above conjectures. Theorem 3. For any β there is an α such that for every set S of Boolean polynomials Rk → R of degree log(k)β and any Z ⊂ S⊥ there exists X ⊂ S with log(|S|/|X|) (log k)α such that log |X=|Z|∩Z| (log k)α . Therefore, among degree (log k)O(1) polynomials, those which may provide counterexamples are those that sometimes take values other than {0, 1} on {0, 1}k . Polynomials of that kind that approximate various natural Boolean functions have been studied in [1] and [8]. The above results hint at neither the truth nor falsity of the “Log-Rank Conjecture”, but rather at its
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elusive nature. Any future proof or counterexample to the “Log-Rank Conjecture” must settle the thorny question of bounding the number of Boolean solutions of sets of non-Boolean polynomials. We withhold judgment. Interesting variants of these problems are obtained if the statements of Conjectures 3 and 4 are restricted to apply to Z = S⊥ rather than every Z ⊂ S⊥ . In principle these may turn out to be equivalent to the conjectures, or possibly provably true independent of them.
2. Proofs Our first theorem establishes the equivalence of the third conjecture with the first two. Proof of Theorem 1. We start by outlining a general relation between sets S of linear functions and 0–1 matrices M, that we use for the rest of this paper. Starting with a set S, let M be the vertical concatenation of a k × m matrix and an n × m matrix, where k is the number of variables as above, n = |S|, and m = |Z|. Let the k × m matrix have columns that are the coordinates of the points in Z, while the corresponding columns in the n × m matrix are the values of each function at that point. Clearly the resulting (k + n) × m matrix M has rank at most k since every row after the first k is a linear combination of the first k. Proceeding in reverse from a matrix M of rank k to a set S, first create a matrix M ∗ by reordering the rows so that the submatrix consisting of the first k rows has full rank. Then each function in S has values defined in rows after the first k at points defined by the first k rows. Proceeding with the theorem, we now assume Conjecture 2 and derive Conjecture 3. Given sets S and Z, find a matrix M corresponding to S as above. From Conjecture 2 we can find a monochromatic submatrix A with log(|M|/|A|) (log rank(M))α . We want such a matrix that does not intersect the first k = rank(M) rows. We note that if the submatrix A has more than k rows, we can remove the intersection of A with the first k rows of M and only reduce |A| by a factor of at most k. Also, if A has k or fewer rows, we may instead consider a new matrix A consisting of any row of M outside the first k. As above, the modified A has area at most 1/k of the
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original. Then the rows of the modified A correspond to linear functions in S and the columns correspond to points at which the functions are equal, since A is monochromatic or a single row. The additional factor of k may require the α of Conjecture 3 to be 1 more than that of Conjecture 2. Thus for the set S we have found a suitable subset of functions X defined by the rows of A. Since S was arbitrary, Conjecture 3 follows. Assuming instead Conjecture 3 and deriving Conjecture 2, we start with matrix M. As above, M corresponds to the set S, and from Conjecture 3 we thus have a subset X. Thus the elements of X correspond to rows of M while elements of Z ∩ X= correspond to columns of M. Denote by A the submatrix of M containing these rows and columns. The ratio of |A| to the area of M excluding the k basis rows thus equals the ratio |X|/|S|. Since S is nonempty, the ratio of the area of M to the area of the non-basis portion of M is at most k. Additionally, since each column of A is constant, there is a monochromatic submatrix A that does not intersect the k basis rows and has area at least |A|/2k, where by α Conjecture 3, |M| (2 rank(M))2(log rank(M)) |A | = O(1) 2(log rank(M)) |A | as desired. Since M was arbitrary, Conjecture 2 results. ✷ We now prove the equivalence of Conjectures 3 and 4. Proof of Theorem 2. Clearly Conjecture 4 implies Conjecture 3 since it is more general. We prove the converse. Consider a set S of polynomials as in Conjecture 4. Since the number of monomials on k variables of deβ gree at most (log k)β is at most k (log k) , the matrix M corresponding in the previously described manner to a β given set of polynomials S has rank at most k (log k) . Following the argument in the proof of Theorem 1, we have that Conjecture 4 is implied by Conjecture 2 β with k replaced by k (log k) , which is seen to be trivially equivalent to the original conjecture. ✷ We now present the example of a set of high degree polynomials that disproves a high degree version of Conjecture 4. Example 2. Let S consist of the GF 2 sum function applied to every subset of the k variables of size
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exactly (log k)f (k) . Each of these functions, expressed as a Boolean polynomial over the reals will have degree (log k)f (k) . (See Nisan and Szegedy [4] for further discussion of Boolean functions considered as real polynomials.) |S| Consider a subset X ⊂ S with log |X| (log k)α . Assuming k/2 = (log k)f (k)+ω(1) then k |S| = (log k)f (k) k/2 (log k)f (k) =θ 2 (log k)f (k) |S| k/2 . > f (k) (log k) |X| k/2 Hence |X| > (log k)f (k) . Thus at least half the variables must appear in polynomials in X. To determine the size of X= ∩ X⊥ , we instead consider |X0 | + |X1|. We here bound |X0 |, with similar reasoning identically bounding |X1 |. Since |X0 | is the number of solutions in {0, 1}k of X = 0, we have a system of simultaneous polynomial equations over R. To upperbound the number of solutions, since each polynomial is linear over GF 2 , we may instead consider this as a linear system over GF 2 . Representing this system as a matrix N in the canonical linear algebra manner, we note that N is composed entirely of rows with exactly (log k)f (k) ones in them, and that at least half the columns contain a 1. Consider a maximal linearly independent set of rows R. These rows span the matrix, and the number of these rows is the rank of the matrix (over GF2 ). Clearly each column in N possessing a 1 must also possess a 1 in R, while each row has exactly (log k)f (k) ones. Thus R has at least
k = (log k)ω(1) 2(log k)f (k) rows. Thus there are at most 2 whereby log
Proof of Theorem 3. As in the proof of Theorem 1, map S to a matrix M with rows corresponding to functions in S, columns corresponding to points in Z ⊂ S⊥ , and entries being those functions evaluated at those points. We need only show that M satisfies Conjecture 1—that it has low communication complexity. Since − log(mono(M)) c(M) we would then have the desired result [5]. Consider instead a matrix M ∗ , constructed in a similar manner to M, whose columns instead correspond to points in {0, 1}k . Clearly M is a submatrix of M ∗ , so we need only show that M ∗ has low communication complexity. As shown by Nisan and Szegedy [4], given a Boolean polynomial f , if D(f ) is the decision tree depth necessary to evaluate f then D(f ) 16 deg(f )8 . Thus each polynomial in S corresponds to a decision tree of depth at most 16(log k)8β . Using these decision trees, we explicitly construct a communication protocol for M ∗ . Following the two party communication model, one party is given a row index, and the other a column index. The communicating party who knows the row, hence the function being evaluated, constructs a decision tree, and queries the other party for the bits needed by the tree. Each query takes (log k) + 1 bits, for a total of O(log k)8β+1 bits to evaluate the desired point, thus the conditions of Conjecture 1 apply to M ∗ and hence M, so Theorem 3 holds for S, as desired. ✷
Acknowledgement The author wishes to thank Madhu Sudan for introducing him to the Log-Rank Conjecture.
References k−(log k)ω(1)
solutions,
|S⊥ | = (log k)ω(1) (log k)α |X= ∩ S⊥ |
as desired. We now prove Theorem 3, that Conjecture 4 is true for the limited class of low-degree Boolean polynomials where S⊥ = {0, 1}k .
[1] R. Beigel, N. Reingold, D. Spielman, The perceptron strikes back, in: Proc. 6th IEEE Conf. on Structure in Complexity Theory, 1991, pp. 286–291. [2] L. Lovász, M. Saks, Lattices, Mobius functions and communication complexity, in: Proc. 29th FOCS, 1988, pp. 81–90. [3] K. Mehlhorn, E.M. Schmidt, Las Vegas is better than determinism in VLSI and distributive computing, in: Proc. 14th STOC, 1982, pp. 330–337. [4] N. Nisan, M. Szegedy, On the degree of Boolean functions as real polynomials, in: Proc. 24th STOC, 1992, pp. 462–467.
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[5] N. Nisan, A. Wigderson, On rank vs. communication complexity, in: Proc. 35th FOCS, 1994, pp. 831–836. [6] R. Raz, B. Spiker, On the Log-Rank conjecture in communication complexity, in: Proc. 34th FOCS, 1993, pp. 168–176. [7] A. Razborov, The gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear, Discrete Math. 108, 393–396.
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[8] J. Tarui, Randomized polynomials, threshold circuits, and the polynomial hierarchy, in: Proc. 8th TACS, 1991. [9] A.C.-C. Yao, Some complexity questions related to distributive computing, in: Proc. 11th STOC, 1979, pp. 209–213.
The Log-Rank Conjecture and low degree polynomials Paul Valiant 1 Stanford University, P.O. Box 17308, Stanford, CA 94309, USA Received 17 January 2003; received in revised form 20 July 2003 Communicated by L.A. Hemaspaandra
Abstract We formulate several questions concerning the intersections of sets of Boolean roots of low degree polynomials. Two of these questions we show to be equivalent to the Log-Rank Conjecture from communication complexity. We further exhibit a slightly stronger formulation which we prove to be false, and a weaker formulation which we prove to be true. These results suggest a possible new approach to the Log-Rank Conjecture. 2003 Elsevier B.V. All rights reserved. Keywords: Log-Rank Conjecture; Communication complexity; Boolean polynomials; Computational complexity
1. Introduction The two-party deterministic communication complexity model has been the subject of much research since its introduction by Yao [9]. Here we concern ourselves with the so-called “Log-Rank Conjecture” relating the communication complexity of a function to the rank of the corresponding matrix. Given a function f (x, y) → {0, 1}, with x ∈ X and y ∈ Y , the communication complexity is the minimum amount of communication necessary between two parties to compute f when one party is given the first input x, and the second party is given the second input y. Specifically, we define the communication complexity c(f ) as the length of the shortest deterministic protoE-mail address: [email protected] (P. Valiant). 1 Research supported in part by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Army Research Office under grant DAAD19-00-1-0177 at MIT. 0020-0190/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2003.09.020
col for computing f , where at each stage in the protocol one party is designated to send a single bit of information to the other party. We may also consider f as a matrix M, where Mx,y = f (x, y). From now on, we consider f as being implicit in M. It was shown by Mehlhorn and Schmidt [3] that c(M) log2 rank(M). Subsequently Lovász and Saks [2] asked the question whether a reverse inequality holds to within logarithmic factors. This is now known as the “Log-Rank Conjecture”. We specify the base of the logarithm to be 2 only when it makes a difference. Conjecture 1. There is an α such that, for every M, c(M) [log rank(M)]α . Several authors have found examples where c(m) > log rank(M), but the largest separation known has c(m) equal to about (log rank(M))1.6 [7,6,5]. The conjecture itself was reformulated by Nisan and Wigder-
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son [5] in more combinatorial terms. They considered monochromatic rectangles in M—namely submatrices that are uniformly either 1 or 0. Following their notation, let mono(M) be the maximum value of the fraction |A|/|M| where A is a monochromatic rectangle, and the size of a matrix is the number of elements in it. Their conjecture, which they proved equivalent to Conjecture 1, is as follows. Conjecture 2. There is an α such that, for every M, − log mono(M) (log rank(M))α . Here we present a third formulation which we prove to be equivalent to these two. Consider a set S of linear functions P : Rk → R. We are concerned with the values these functions take on {0, 1}k . Specifically, let S= be the set of points in {0, 1}k where all the functions have equal values, and let S⊥ be the set of points in {0, 1}k where all the functions have values in {0, 1}. For convenience, let S0 and S1 be the sets on which every function is 0 and 1, respectively. Thus S0 ∪ S1 = S⊥ ∩ S= . Conjecture 3. There is an α such that for every nonempty set S of linear functions Rk → R and every set Z ⊂ S⊥ there exists X ⊂ S with log(|S|/|X|) (log k)α such that log |X=|Z|∩Z| (log k)α . As an aside we note that an alternative way of viewing this conjecture is the following. Each function in S can be thought of as describing a k-dimensional affine subspace as a subset of {0, 1}k+1 . The conjecture then states that if the intersection of the projections of these sets into {0, 1}k is large, then there is a large subset of those sets whose intersection in {0, 1}k+1 is large. We know that if we project by log2 k dimensions then the corresponding statement is false. In particular there is a set S of k-dimensional affine subspaces of {0, 1}k+log2 k such that the intersection of the projections of these sets into {0, 1}k is the whole set but no large subset of these sets has a large intersection with {0, 1}k+log2 k . Such a set S over variables x1 . . . xk and z0 . . . zlog2 k−1 is defined by the following set of equations: the left-hand sides are sums of all subsets of k/2 of the x variables; the right-hand sides are all equal to log2 k−1 i 2 zi . i=0 We prove the following theorem.
Theorem 1. Conjecture 3 is equivalent to Conjectures 1 and 2. We further show that the functions in S need not be linear: the conjecture remains equivalent if the functions are polynomials of degree (log k)β for some constant β. Conjecture 4. For any β there is an α such that for every nonempty set S of polynomials Rk → R of degree (log k)β and every set Z ⊂ S⊥ , there exists X ⊂ S with log(|S|/|X|) (log k)α such that log |X|Z| = ∩Z| (log k)α . Theorem 2. Conjecture 3 is equivalent to Conjecture 4. However, if we allow the polynomials to have degree (log k)ω(1) the corresponding conjecture is false. Example 1. Given a constant α and a function f (k) = ω(1), then for large enough k there is a set S of polynomials of degree (log k)f (k) such that there does not exist X ⊂ S with log(|S|/|X|) (log k)α such that ⊥| (log k)α . log |X|S = ∩S⊥ | The example we construct consists entirely of polynomials with 0–1 values on {0, 1}k , sometimes called Boolean polynomials. In our notation, S⊥ = {0, 1}k . We show however, that such polynomials cannot provide corresponding counterexamples to the above conjectures. Theorem 3. For any β there is an α such that for every set S of Boolean polynomials Rk → R of degree log(k)β and any Z ⊂ S⊥ there exists X ⊂ S with log(|S|/|X|) (log k)α such that log |X=|Z|∩Z| (log k)α . Therefore, among degree (log k)O(1) polynomials, those which may provide counterexamples are those that sometimes take values other than {0, 1} on {0, 1}k . Polynomials of that kind that approximate various natural Boolean functions have been studied in [1] and [8]. The above results hint at neither the truth nor falsity of the “Log-Rank Conjecture”, but rather at its
P. Valiant / Information Processing Letters 89 (2004) 99–103
elusive nature. Any future proof or counterexample to the “Log-Rank Conjecture” must settle the thorny question of bounding the number of Boolean solutions of sets of non-Boolean polynomials. We withhold judgment. Interesting variants of these problems are obtained if the statements of Conjectures 3 and 4 are restricted to apply to Z = S⊥ rather than every Z ⊂ S⊥ . In principle these may turn out to be equivalent to the conjectures, or possibly provably true independent of them.
2. Proofs Our first theorem establishes the equivalence of the third conjecture with the first two. Proof of Theorem 1. We start by outlining a general relation between sets S of linear functions and 0–1 matrices M, that we use for the rest of this paper. Starting with a set S, let M be the vertical concatenation of a k × m matrix and an n × m matrix, where k is the number of variables as above, n = |S|, and m = |Z|. Let the k × m matrix have columns that are the coordinates of the points in Z, while the corresponding columns in the n × m matrix are the values of each function at that point. Clearly the resulting (k + n) × m matrix M has rank at most k since every row after the first k is a linear combination of the first k. Proceeding in reverse from a matrix M of rank k to a set S, first create a matrix M ∗ by reordering the rows so that the submatrix consisting of the first k rows has full rank. Then each function in S has values defined in rows after the first k at points defined by the first k rows. Proceeding with the theorem, we now assume Conjecture 2 and derive Conjecture 3. Given sets S and Z, find a matrix M corresponding to S as above. From Conjecture 2 we can find a monochromatic submatrix A with log(|M|/|A|) (log rank(M))α . We want such a matrix that does not intersect the first k = rank(M) rows. We note that if the submatrix A has more than k rows, we can remove the intersection of A with the first k rows of M and only reduce |A| by a factor of at most k. Also, if A has k or fewer rows, we may instead consider a new matrix A consisting of any row of M outside the first k. As above, the modified A has area at most 1/k of the
101
original. Then the rows of the modified A correspond to linear functions in S and the columns correspond to points at which the functions are equal, since A is monochromatic or a single row. The additional factor of k may require the α of Conjecture 3 to be 1 more than that of Conjecture 2. Thus for the set S we have found a suitable subset of functions X defined by the rows of A. Since S was arbitrary, Conjecture 3 follows. Assuming instead Conjecture 3 and deriving Conjecture 2, we start with matrix M. As above, M corresponds to the set S, and from Conjecture 3 we thus have a subset X. Thus the elements of X correspond to rows of M while elements of Z ∩ X= correspond to columns of M. Denote by A the submatrix of M containing these rows and columns. The ratio of |A| to the area of M excluding the k basis rows thus equals the ratio |X|/|S|. Since S is nonempty, the ratio of the area of M to the area of the non-basis portion of M is at most k. Additionally, since each column of A is constant, there is a monochromatic submatrix A that does not intersect the k basis rows and has area at least |A|/2k, where by α Conjecture 3, |M| (2 rank(M))2(log rank(M)) |A | = O(1) 2(log rank(M)) |A | as desired. Since M was arbitrary, Conjecture 2 results. ✷ We now prove the equivalence of Conjectures 3 and 4. Proof of Theorem 2. Clearly Conjecture 4 implies Conjecture 3 since it is more general. We prove the converse. Consider a set S of polynomials as in Conjecture 4. Since the number of monomials on k variables of deβ gree at most (log k)β is at most k (log k) , the matrix M corresponding in the previously described manner to a β given set of polynomials S has rank at most k (log k) . Following the argument in the proof of Theorem 1, we have that Conjecture 4 is implied by Conjecture 2 β with k replaced by k (log k) , which is seen to be trivially equivalent to the original conjecture. ✷ We now present the example of a set of high degree polynomials that disproves a high degree version of Conjecture 4. Example 2. Let S consist of the GF 2 sum function applied to every subset of the k variables of size
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exactly (log k)f (k) . Each of these functions, expressed as a Boolean polynomial over the reals will have degree (log k)f (k) . (See Nisan and Szegedy [4] for further discussion of Boolean functions considered as real polynomials.) |S| Consider a subset X ⊂ S with log |X| (log k)α . Assuming k/2 = (log k)f (k)+ω(1) then k |S| = (log k)f (k) k/2 (log k)f (k) =θ 2 (log k)f (k) |S| k/2 . > f (k) (log k) |X| k/2 Hence |X| > (log k)f (k) . Thus at least half the variables must appear in polynomials in X. To determine the size of X= ∩ X⊥ , we instead consider |X0 | + |X1|. We here bound |X0 |, with similar reasoning identically bounding |X1 |. Since |X0 | is the number of solutions in {0, 1}k of X = 0, we have a system of simultaneous polynomial equations over R. To upperbound the number of solutions, since each polynomial is linear over GF 2 , we may instead consider this as a linear system over GF 2 . Representing this system as a matrix N in the canonical linear algebra manner, we note that N is composed entirely of rows with exactly (log k)f (k) ones in them, and that at least half the columns contain a 1. Consider a maximal linearly independent set of rows R. These rows span the matrix, and the number of these rows is the rank of the matrix (over GF2 ). Clearly each column in N possessing a 1 must also possess a 1 in R, while each row has exactly (log k)f (k) ones. Thus R has at least
k = (log k)ω(1) 2(log k)f (k) rows. Thus there are at most 2 whereby log
Proof of Theorem 3. As in the proof of Theorem 1, map S to a matrix M with rows corresponding to functions in S, columns corresponding to points in Z ⊂ S⊥ , and entries being those functions evaluated at those points. We need only show that M satisfies Conjecture 1—that it has low communication complexity. Since − log(mono(M)) c(M) we would then have the desired result [5]. Consider instead a matrix M ∗ , constructed in a similar manner to M, whose columns instead correspond to points in {0, 1}k . Clearly M is a submatrix of M ∗ , so we need only show that M ∗ has low communication complexity. As shown by Nisan and Szegedy [4], given a Boolean polynomial f , if D(f ) is the decision tree depth necessary to evaluate f then D(f ) 16 deg(f )8 . Thus each polynomial in S corresponds to a decision tree of depth at most 16(log k)8β . Using these decision trees, we explicitly construct a communication protocol for M ∗ . Following the two party communication model, one party is given a row index, and the other a column index. The communicating party who knows the row, hence the function being evaluated, constructs a decision tree, and queries the other party for the bits needed by the tree. Each query takes (log k) + 1 bits, for a total of O(log k)8β+1 bits to evaluate the desired point, thus the conditions of Conjecture 1 apply to M ∗ and hence M, so Theorem 3 holds for S, as desired. ✷
Acknowledgement The author wishes to thank Madhu Sudan for introducing him to the Log-Rank Conjecture.
References k−(log k)ω(1)
solutions,
|S⊥ | = (log k)ω(1) (log k)α |X= ∩ S⊥ |
as desired. We now prove Theorem 3, that Conjecture 4 is true for the limited class of low-degree Boolean polynomials where S⊥ = {0, 1}k .
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