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Boundedwidth polynomialsize branching programs recognize exactly those languages
 in NC’, in “Proceedings, 18th ACM STOC
, 1986
"... We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such prog ..."
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Cited by 215 (14 self)
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We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 196 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
On the Complexity of Branching Programs and Decision Trees for Clique Functions
, 1988
"... Exponential lower bounds on the complexity of computing the clique functions in the Boolean decisiontree model are proved. For onetimeonly branching programs, large polynomial lower bounds are proved for kclique functions if k is fixed, and exponential lower bounds if k increases with n. Finall ..."
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Cited by 41 (4 self)
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Exponential lower bounds on the complexity of computing the clique functions in the Boolean decisiontree model are proved. For onetimeonly branching programs, large polynomial lower bounds are proved for kclique functions if k is fixed, and exponential lower bounds if k increases with n. Finally, the hierarchy of the classes BP&‘) of all sequences of Boolean functions that may be computed by dtimes only branching programs of polynomial size is introduced. It is shown constructively that BP,(P) is a proper subset of BP#).
Superlinear lower bounds for boundedwidth branching programs
 6th Structure in Complexity Theory
, 1991
"... ..."
On Learning Branching Programs and Small Depth Circuits
 Computational Learning Theory: Proc. Third European Conference. Lecture Notes in Articial Intelligence
, 1997
"... This paper studies the learnability of branching programs and small depth circuits with modular and threshold gates in both the exact and PAC learning models with and without membership queries. Some of the results extend earlier works in [GG95, ERR95, BTW95]. The main results are as follows. For ..."
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Cited by 10 (2 self)
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This paper studies the learnability of branching programs and small depth circuits with modular and threshold gates in both the exact and PAC learning models with and without membership queries. Some of the results extend earlier works in [GG95, ERR95, BTW95]. The main results are as follows. For branching programs we show the following.
A TimeSpace Tradeoff for Boolean Matrix Multiplication
"... A timespace tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability nll2 and to be 0 with prob ..."
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Cited by 8 (0 self)
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A timespace tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability nll2 and to be 0 with probability 1 n1/2. Letting S and T denote expected space and time of a deterministic algorithm, the tradeoff is ST = R(n3.5) for T < cln2.5 and ST = R(n3) for T> where c1, c2> 0. The lower bounds are matched to within a logarithmic factor by upper bounds in the branching program model. Thus, the tradeoff possesses a sharp break a.t T = O(n2.5). These expected case lower bounds are also the best known lower bounds for the worst case.
Obfuscating Branching Programs Using BlackBox PseudoFree Groups ∗
, 2013
"... We show that the class of polynomialsize branching programs can be obfuscated according to a virtual blackbox notion akin to that of Barak et.al., in an idealized blackbox group model over pseudofree groups. This class is known to lie between NC 1 and P and includes most interesting cryptographi ..."
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We show that the class of polynomialsize branching programs can be obfuscated according to a virtual blackbox notion akin to that of Barak et.al., in an idealized blackbox group model over pseudofree groups. This class is known to lie between NC 1 and P and includes most interesting cryptographic algorithms. The construction is rather simple and is based on Kilian’s randomization technique for Barrington’s branching programs. The blackbox group model over pseudofree groups is a strong idealization. In particular, in a pseudofree group, the group operation can be efficiently performed, while finding surprising relations between group elements is intractable. A blackbox representation of the group provides an ideal interface which permits prescribed group operations, and nothing else. Still, the algebraic structure and security requirements appear natural and potentially realizable. They are also unrelated to the specific function to be obfuscated. Our modeling should be compared with the recent breakthrough obfuscation scheme of Garg et al. [FOCS 2013]: While the high level structure is similar, some important details differ. It should be stressed however that, unlike Garg et al., we do not provide a candidate concrete instantiation of our abstract structure. 1
LatticeBased FHE as Secure as PKE
"... We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of Õ(n1.5+ɛ)approximation for lattice problems (such as GapSVP) under quantum reductions for any ɛ> 0 (or Õ(n2+ɛ)approximation under classical reductions). This matches the best known hardness for “regular ” ..."
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Cited by 2 (0 self)
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We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of Õ(n1.5+ɛ)approximation for lattice problems (such as GapSVP) under quantum reductions for any ɛ> 0 (or Õ(n2+ɛ)approximation under classical reductions). This matches the best known hardness for “regular ” (nonhomomorphic) lattice based publickey encryption up to the ɛ factor. A number of previous methods had hit a roadblock at quasipolynomial approximation. (As usual, a circular security assumption can be used to achieve a nonleveled FHE scheme.) Our approach consists of three main ideas: Noisebounded sequential evaluation of high fanin operations; Circuit sequentialization using Barrington’s Theorem; and finally, successive dimensionmodulus reduction.
Some Topics in Parallel Computation and Branching Programs
, 1995
"... There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition ..."
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There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition, several restrictions of branching programs, for example ordered binary decision diagrams, have proven to be successful data structures in several VLSI design and verification applications. We construct a branching program of o(n log 3 n) nodes for computing any threshold function on n variables and a branching program of o(n log 4 n) nodes for determining the sum of n variables modulo a fixed divisor. These are improvements over constructions of size 2(n 3=2 ) due to Lupanov [Lup65]. The second p...
On Learning BoundedWidth Branching Programs*
"... In this paper, we study PACleaming algorithms for specialized classes of deterministic finite automata (DFA). Inpartictdar, we study branchingprogrsms, and we investigate the intluence of the width of the branching program on the difficulty of the learning problem. We first present a distributionf ..."
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In this paper, we study PACleaming algorithms for specialized classes of deterministic finite automata (DFA). Inpartictdar, we study branchingprogrsms, and we investigate the intluence of the width of the branching program on the difficulty of the learning problem. We first present a distributionfree algorithm for learning width2 branching programs. We also give an algorithm for the proper learning of width2 branching programs under uniform distribution on labeled samples. We then show that the existence of an efficient algorithm for learning width3 branching programs would imply the existence of an efficient algorithm for learning DNF, which is not known to be the case. Fimlly, we show that the existence of an algorithm for learning width3 branching programs would also yield an algorithm for learning a very restricted version of parity with noise. 1