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FROM SYLVESTERGALLAI CONFIGURATIONS TO RANK BOUNDS: IMPROVED BLACKBOX IDENTITY TEST FOR DEPTH3 CIRCUITS
"... Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over ratio ..."
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Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d O(k2)time. Our structure theorem essentially says that the number of independent variables in a real depth3 identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for blackbox identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional SylvesterGallai theorems and the rank of depth3 identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of SylvesterGallai rank bounds for any field, and show the intimate connection between this and depth3 identity rank bounds. We also prove the first ever theorem about high dimensional SylvesterGallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional SylvesterGallai configuration. 1.
Blackbox identity testing for bounded top fanin depth3 circuits: the field doesn’t matter
 In Proceedings of the 43rd annual ACM Symposium on Theory of Computing (STOC
, 2011
"... Abstract. Let C be a depth3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed ..."
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Cited by 18 (5 self)
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Abstract. Let C be a depth3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(n)dk, regardless of the base field. The only field for which polynomial time algorithms were previously known is F = Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ΣΠΣ(k, d, n) circuit to k variables, but preserves the identity structure. Key words. depth3 circuits; polynomial identity testing; derandomization; blackbox; Chinese remaindering; algebra homomorphism
Quasipolynomial hittingset for setdepth formulas
 In STOC
, 2013
"... Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1 ..."
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Cited by 13 (4 self)
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Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1
Shallow circuits with highpowered inputs
 PROCEEDINGS OF THE SECOND SYMPOSIUM ON INNOVATIONS IN COMPUTER SCIENCE
, 2011
"... A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic blackbox identity testing algorithm for (highdegree) univariate polynomials would imply a lower b ..."
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A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic blackbox identity testing algorithm for (highdegree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (lowdegree) multivariate identity testing are weaker. To obtain a lower bound for the permanent it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the ShubSmale τconjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a polynomial upper bound on the number of real roots of sums of products of sparse polynomials (Descartes ’ rule of signs gives such a bound for sparse polynomials and products thereof). In fact the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent. These results suggest the intriguing possibility that tools from real analysis might be brought to bear on a longstanding open problem: what is the arithmetic complexity of the permanent polynomial?
Algorithmics on SLPcompressed strings: a survey,
 Groups Complex. Cryptol.
, 2012
"... Abstract Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straightline program is a contextfree grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pat ..."
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Abstract Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straightline program is a contextfree grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.
The Limited Power of Powering: Polynomial Identity Testing and a Depthfour Lower Bound for the Permanent
, 2011
"... Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a “real τconjecture” which is inspired by this ..."
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Cited by 7 (5 self)
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Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a “real τconjecture” which is inspired by this connection. The real τconjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real τconjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.
A Wronskian approach to the real τconjecture. arXiv:1205.1015, 2012. Accepted for oral presentation at MEGA
, 2013
"... According to the real τconjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the per ..."
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Cited by 5 (2 self)
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According to the real τconjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent. In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse polynomials of a special form. We focus on the case where the number of distinct sparse polynomials is small, but each polynomial may be repeated several times. We also give a deterministic polynomial identity testing algorithm for the same class of polynomials. Our proof techniques are quite versatile; they can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems).
Constructing Small Tree Grammars and Small Circuits for Formulas
, 2014
"... Abstract It is shown that every tree of size n over a fixed set of σ different ranked symbols can be decomposed into O( n log σ n ) = O( n log σ log n ) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straightline linear contextfree tree grammar o ..."
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Abstract It is shown that every tree of size n over a fixed set of σ different ranked symbols can be decomposed into O( n log σ n ) = O( n log σ log n ) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straightline linear contextfree tree grammar of size O( n log σ n ), which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammarbased tree compressors were not analyzed for the worstcase size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of O( n log 0.19 n ) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size n, in which only m ≤ n different variables occur, can be transformed (in time O(n log n)) into an arithmetical circuit of size O( n·log m log n ) and depth O(log n). This refines a classical result of Brent, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n). Missing proofs can be found in the long version ACM Subject Classification E.4 Data compaction and compression Keywords and phrases grammarbased compression, tree compression, arithmetical circuits Introduction Grammarbased compression has emerged to an active field in string compression during the past 20 years. The idea is to represent a given string s by a small contextfree grammar that generates only s; such a grammar is also called a straightline program, briefly SLP. For instance, the word (ab) 1024 can be represented by the SLP with the productions A 0 → ab and A i → A i−1 A i−1 for 1 ≤ i ≤ 10 (A 10 is the start symbol). The size of this grammar is much smaller than the size (length) of the string (ab) 1024 . In general, an SLP of size n (the size of an SLP is usually defined as the total length of all righthand sides of the productions) can produce a string of length 2 Ω(n) . Hence, an SLP can be seen indeed as a succinct representation of the generated string. The goal of grammarbased string compression is to construct from a given input string s a small SLP that produces s. Several algorithms for this have been proposed and analyzed. Prominent grammarbased string compressors are for instance LZ78, RePair, and BISECTION, see To evaluate the compression performance of a grammarbased compressor C, two different approaches can be found in the literature: A first approach is to analyze the size of the SLP produced by C for an input string x compared to the size of a smallest SLP for x. This leads to the approximation ratio for C, see