A. Beimel, A. Gal, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29--45, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and nondeterministic communication complexity [1] imply that one cannot obtain larger bounds working with 0 1 matrices and monochromatic covers. For the model of monotone span programs two methods have been used for proving superpolynomial lower bounds. The rst method, introduced in [5], uses the size of self avoiding families (see also [3] 4] Using this method, an n 5=2 ) lower bound was proved in [5] for monotone span programs over arbitrary elds computing the 6 clique function, and in [3] 4] n 420 n= log log n) lower bounds were obtained for monotone span ....

....and monochromatic covers. For the model of monotone span programs two methods have been used for proving superpolynomial lower bounds. The rst method, introduced in [5] uses the size of self avoiding families (see also [3] 4] Using this method, an n 5=2 ) lower bound was proved in [5] for monotone span programs over arbitrary elds computing the 6 clique function, and in [3] 4] n 420 n= log log n) lower bounds were obtained for monotone span programs over arbitrary elds computing explicit functions. The other method, introduced in [6] is an extension of Razborov s rank ....

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A. Beimel, A. Gal, M. Paterson, Lower bounds for monotone span programs. Computational Complexity 6 (1996/97), 29-45.

....of W . The spandimension of a Boolean function B is the minimum dimension of W over all linear spaces W over F which support a span program for B. We denote this quantity by sdim F (B) For the signi cance of this complexity measure and its relation to other models of computation, we refer to [11] and [6] In particular, lower bounds for span programs imply lower bounds for formula size and lower bounds for undirected contact schemes, also called symmetric branching programs. Span programs can be viewed as a model of parallel computation. No non linear lower bounds are known for the ....

A. Beimel, A. Gal, M. Paterson, Lower bounds for monotone span programs. Computational Complexity 6 (1996/97) 29-45.

....T (a) a such that for every a; a 0 2 A, 1. if T (a) a 0 then a = a 0 , 2. for every y T (a) a 0 6 [ a 00 2A;a 00 y 6= a 00 n y: Let us de ne S(y) a 00 2A;a 00 y 6= a 00 : Thus the second condition says that no a 0 is contained in S(y) n y. In [3] they proved that every monotone span program that computes a monotone boolean function the minterms of which form a self avoiding family has size at least the size of the family. Their proof actually gives such a lower bound for every monotone span program that accepts all sets a 2 A and rejects ....

....0 intersects at most one of the sets. If a 0 b 1 = then a 0 b 2 6= because otherwise a 0 S(b 1 ) Thus also a 0 intersects at least one of the sets, hence we have condition ( Take z 1 = 0; z 2 = 1, then the matrix R A;B; z has full rank by the following lemma (implicit in [3]) Lemma 1 Let A be a family of subsets that are incomparable by inclusion. Let M be the matrix such that rows are indexed by the elements of A and columns are indexed by sets b such that b a for some a A and the entry corresponding 1 Recall that this is the matrix in which rows are indexed ....

A. Beimel, A. Gal, M. Paterson, Lower bounds for monotone span programs. Computational Complexity 6 (1996/97), 29-45.

....of the basic issue in the area of secret sharing schemes is that of estimating the information rate of the scheme, that is, the ratio between the size of the secret and that of the largest share given to any participant. This problem has received considerable attention in the last few years (e.g. [1, 5, 4, 10, 11, 13, 14, 22]) The practical relevance of this issue is based on the following observations: Firstly, the security of any system tends to degrade as the amount of information that must be kept secret, i.e. the shares of the participants, increases. Secondly, if the shares given to participants are too long, ....

....on the secret. An access structure A is the set of all subsets of P that can recover the secret. Definition 2. 1 Let P be a set of participants, a monotone access structure A on P is a subset A 2 P nf;g, such that A 2 A; A A 0 P ) A 0 2 A: 1 A notable exception is the paper [1], however, the results contained there apply only to linear secret sharing schemes. 2 Definition 2.2 Let P be a set of participants and A 2 P : The closure of A, denoted by cl(A) is the set cl(A) fCjB 2 A and B C Pg: For a monotone access structure A we have A = cl(A) All access ....

A. Beimel, A. Gal, and M. Paterson, Lower Bounds for Monotone Span Programs, to appear in: Proceedings of the 35th IEEE Symp. on Foundations of Computer Science, 1995.

....possibly yield even exponential lower bounds. So far, our largest lower bound for an explicit function on m variables is Omega Gamma m 2:5 ) We obtain this bound for the function that is defined to have the value 1 if and only if the input graph contains a 6 clique. These results appear in [18] Another motivation for studying monotone span programs is their connection to secret sharing schemes. A secret sharing scheme is a cryptographic tool in which a dealer shares a secret, taken from a finite set of possible secrets, among a set of parties such that only some pre defined authorized ....

A. Beimel, A. G'al and M. Paterson. "Lower bounds for monotone span programs ", To appear in Proceedings of FOCS'95.

....In this model, it is not known how to prove large lower bounds for explicit functions even in the monotone case. The Omega Gamma m 2 = log m) lower bound implied by [25] 8 for monotone span program size is the strongest previously known lower bound for an explicit function on m variables. [17] introduced a method that yields quadratic lower bounds for explicit functions, improving on the bound by [25] The methods presented in [17] and [25] cannot give lower bounds larger than Omega Gamma m 2 ) We present a new technique for proving lower bounds for monotone span programs, which ....

....= log m) lower bound implied by [25] 8 for monotone span program size is the strongest previously known lower bound for an explicit function on m variables. 17] introduced a method that yields quadratic lower bounds for explicit functions, improving on the bound by [25] The methods presented in [17] and [25] cannot give lower bounds larger than Omega Gamma m 2 ) We present a new technique for proving lower bounds for monotone span programs, which is a generalization of the method in [17] The new method could possibly yield even exponential lower bounds. So far, our largest lower bound ....

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A. Beimel, A. G'al and M. Paterson, "Lower bounds for monotone span programs ", Technical Report BRICS-RS-94-46, BRICS, Department of Computer Science, University of Aarhus, December 1994.

....refers to the method originated by Razborov in [58] and [60] for proving monotone lower bounds. The approach was continued by [59] and [67] and several others including [6] 68] 9] 69] 72] 78] 40] for general lower bounds, and further used in monotone lower bounds such as [2] 79] and [10]. Other complexity lower bounds that can be generally classified as being based on nonapproximability by low degree or sparse polynomials, or other basis functions include many of the lower bounds on threshold circuit complexity and voting polynomial representations such as [11] 15] 43] ....

.... and corresponding methods for nonapproximability and lower bounds can be found in [21] Finally, our analytic framework is suitable primarily for questions that can be decomposed into linear approximation questions. Many of the lower bounds based on [58] and [60] such as [40] 2] 79] and [10], use distinctly non linear approximation methods. While it is an open question whether these, too, can be treated using purely linear approximation methods, we discuss the current points of difference in Section 3. 1.3. General results. Below, we give an informal description of some of the ....

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R. Beimel, A. Gal, M. Paterson, "Lower bounds for monotone span programs," Proc. 36 th Ann. IEEE Symp. on Foundations of CS, pp. 674-681, 1995.

....the average in some cases. Karchmer and Wigderson in [13] showed that there is a strong connection between the so called (monotone) span programs and certain secret sharing schemes. Thus our result also gives immediately a lower bound for the size of span programs. Beimel, G al, and Paterson in [1] gave general lower bounds for the size of span programs, which implies that for some access structure on n participants, if the scheme is of Karchmer Wigderson type the dealer must use at least c n 2 random bits for each secret bit. Given any, say random, access structure A on a set P of n ....

A. Beimel, A. Gal, M. Paterson, Lower bounds for monotone span programs, Preprint, 1994

....function would solve a major open problem. An important subclass of span programs are monotone span programs; they simulate both monotone formulas and monotone contact switching networks. Recently a superpolynomial lower bound was proved for this model [1, 2] based on a combinatorial condition of [4]) One direction of study of propositional proof systems is to prove lower bounds on the length of proofs in certain restricted proof systems. Exponential lower bounds were obtained for such systems as resolution [12] bounded depth Frege systems [15, 17] cutting planes [6, 18] and ....

.... version of this model was studied in [19] as the projective dimension of graphs (a concept also related to the affine dimension of graphs of [20] We prove an exponential lower bound for monotone dependency programs, using a simplification of the methods used for monotone span programs in [1, 4]. In the non monotone case over finite fields the dependency programs turn out to be equivalent to span programs. However, they may be useful to consider in lower bound proofs, as they are in some sense simpler (as demonstrated by the monotone lower bound, which is much simpler than the analogous ....

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A. Beimel, A. G'al, and M. Paterson. Lower bounds for monotone span programs. In Proc. of the 36th Ann. IEEE Symp. on Foundations of Computer Sci., pages 674--681. IEEE, 1995.

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A. Beimel, A. Gal, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29--45, 1997.

.... NC and even monotone symmetric logspace [11, 12, 43] 2) it is contained in algebraic NC (as follows from [13, 18, 46, 22] implying that it is contained in NC when log F is polynomially bounded; and (3) there are explicit monotone functions that are provably not in this class [9, 2, 36]. As opposed to linear secret sharing schemes, nearly nothing is known for general (i.e. possiblynonlinear) schemes. Several constructions of nonlinear secret sharing schemes have been suggested, both for the threshold case [56, 30, 48] and for general access structures [34] The question of ....

A. Beimel, A. Gal, and M. Paterson. Lower bounds for monotone span programs. ComputationalComplexity, 6(1):29--45, 1997. Conference version: FOCS '95.

.... NC 1 and even monotone symmetric logspace [11, 12, 42] 2) it is contained in algebraic NC 2 (as follows from [13, 17, 45, 21] implying that it is contained in NC 3 when log jF j is polynomially bounded; and (3) there are explicit monotone functions that are provably not in this class [9, 2, 35]. As opposed to linear secret sharing schemes, nearly nothing is known for general (i.e. possibly nonlinear) schemes. Several constructions of nonlinear secret sharing schemes have been suggested, both for the threshold case [55, 29, 47] and for general access structures [19, 33] The question of ....

A. Beimel, A. Gal, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29-- 45, 1997. Conference version: FOCS '95.

No context found.

A. Beimel, A. G'al and M. Paterson. Lower bounds for monotone span programs. In Proceedings of the 36th IEEE Symposium on the Foundations of Computer Science, (1995) 674--681.

....25, 19, 7, 34, 36, 12] and all the schemes described in the survey [37] The Omega Gamma m 2 = log m) lower bound implied by [15, 16] for monotone span program size is the strongest previously known lower bound for an explicit function on m variables. In a preliminary version of this paper ([5]) we presented a method that yields quadratic lower bounds for explicit functions, improving on the bound by [16] The methods presented in [5] and [15, 16] cannot give lower bounds larger than Omega Gamma m 2 ) In this paper we present a new technique for proving lower bounds for monotone ....

....monotone span program size is the strongest previously known lower bound for an explicit function on m variables. In a preliminary version of this paper ( 5] we presented a method that yields quadratic lower bounds for explicit functions, improving on the bound by [16] The methods presented in [5] and [15, 16] cannot give lower bounds larger than Omega Gamma m 2 ) In this paper we present a new technique for proving lower bounds for monotone span programs, which is a generalization of the method in [5] We present an Omega Gamma m 2:5 ) lower bound for an explicit function on m ....

[Article contains additional citation context not shown here]

A. Beimel, A. G'al and M. Paterson. Lower bounds for monotone span programs. Technical Report BRICS-RS-94-46, BRICS, Department of Computer Science, University of Aarhus, December 1994.

....p span programs over the finite fields GF(p) have the same power as modular branching programs. That is, a function can be computed by a polynomial size mod p branching program if and only if it can be computed by a polynomial size span program over GF(p) Span programs were considered also in [5, 3, 4, 2, 12, 20]. Pudl ak and Sgall [20] defined a similar model called dependency programs. They proved that span programs are at least as strong as dependency programs over every field, and for fixed finite fields the reverse also holds, that is over fixed finite fields span programs and dependency programs are ....

A. Beimel, A. G'al, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29--45, 1997. Conference version: FOCS '95.

....defined span programs and showed, using results of [7] that over fixed finite fields they are equivalent to modular branching programs. That is, a function has a small mod p branching program if and only if the function has a small span program over GF(p) Span programs were considered also in [5, 3, 4, 2]. Pudl ak and Sgall [14] defined a similar model called dependency programs. They proved that span programs are at least as strong as dependency programs. It is not known whether span programs are actually stronger. We give precise definitions of the above models in the next section. We denote the ....

A. Beimel, A. G'al, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29--45, 1997. Conference version: FOCS '95.

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