Results 1  10
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15
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2350 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
On span programs
 In Proc. of the 8th IEEE Structure in Complexity Theory
, 1993
"... We introduce a linear algebraic model of computation, the Span Program, and prove several upper and lower bounds on it. These results yield the following applications in complexity and cryptography: • SL ⊆ ⊕L (a weak Logspace analogue of N P ⊆ ⊕P). • The first superlinear size lower bounds on branc ..."
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Cited by 120 (6 self)
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We introduce a linear algebraic model of computation, the Span Program, and prove several upper and lower bounds on it. These results yield the following applications in complexity and cryptography: • SL ⊆ ⊕L (a weak Logspace analogue of N P ⊆ ⊕P). • The first superlinear size lower bounds on branching programs that count. • A broader class of functions which posses informationtheoretic secret sharing schemes. The proof of the main connection, between span programs and counting branching programs, uses a variant of Razborov’s general approximation method. 1
Algorithmic SelfAssembly of DNA
, 1998
"... How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a heteropolymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, ..."
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Cited by 104 (6 self)
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How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a heteropolymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, using an entirely different approach, has led to a wealth of ideas for how to build DNAbased computers in the laboratory, whose energy efficiency, information density, and parallelism may have potential to surpass conventional electronic computers for some purposes. In this thesis, I examine one mechanism used in all designs for DNAbased computer  the selfassembly of DNA by hybridization and formation of the double helix  and show that this mechanism alone in theory can perform universal computation. To do so, I borrow an important result in the mathematical theory of tiling: Wang showed how jigsawshaped tiles can be designed to simulate the operation of any Turing Machine. I propose...
Lower Bounds for Deterministic and Nondeterministic Branching Programs
 in Proceedings of the FCT'91, Lecture Notes in Computer Science
, 1991
"... We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switchingandrectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networ ..."
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Cited by 57 (4 self)
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We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switchingandrectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, boundedwidth devices , oblivious devices and readk times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...
Complexity of Restricted and Unrestricted Models of Molecular Computation
 DNA Based Computers 1, volume 27 of DIMACS
, 1995
"... In [Li1] and [Ad2] a formal model for molecular computing was proposed, which makes focused use of affinity purification. The use of PCR was suggested to expand the range of feasible computations, resulting in a second model. In this note, we give a precise characterization of these two models in te ..."
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Cited by 32 (4 self)
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In [Li1] and [Ad2] a formal model for molecular computing was proposed, which makes focused use of affinity purification. The use of PCR was suggested to expand the range of feasible computations, resulting in a second model. In this note, we give a precise characterization of these two models in terms of recognized computational complexity classes, namely branching programs (BP) and nondeterministic branching programs (NBP) respectively. This allows us to give upper and lower bounds on the complexity of desired computations. Examples are given of computable and uncomputable problems, given limited time. 1 Introduction Molecular computation, as introduced by [Ad1], provides a new approach to solving combinatorial inverse problems, where we are interested in computing f \Gamma1 (1) for nbit strings x and boolean function f . Instances of NPcomplete problems can be expressed in this form; for example 3SAT. Adleman's technique involves using individual DNA strands to represent poten...
Superlinear Lower Bounds For BoundedWidth Branching Programs
, 1995
"... We use algebraic techniques to obtain superlinear lower bounds on the size of boundedwidth branching programs to solve a number of problems. In particular, we show that any boundedwidth branching program computing a nonconstant threshold function has length \Omega\Gamma n log log n); improving on ..."
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Cited by 20 (5 self)
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We use algebraic techniques to obtain superlinear lower bounds on the size of boundedwidth branching programs to solve a number of problems. In particular, we show that any boundedwidth branching program computing a nonconstant threshold function has length \Omega\Gamma n log log n); improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over a finite solvable monoid computing products in a nonsolvable group has length\Omega\Gamma n log log n): This result is a step toward proving the conjecture that the circuit complexity class ACC 0 is properly contained in NC 1 : A preliminary version of this paper appeared in the Proceedings of the 1991 Structure in Complexity Theory Symposium. 1. The Main Results In this paper we describe a general algebraic technique for obtaining superlinear lower bounds on the length of boundedwidth branching programs to solve certain problems. Our method is based on the interpretation, ...
Complexity Theoretical Results for Randomized Branching Programs
, 1998
"... This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straigh ..."
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Cited by 9 (8 self)
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This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps readonce branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...
The multiparty communication complexity of ExactT: Improved bounds and new problems
 In Proc. of 31st MFCS
, 2006
"... Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 ..."
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Cited by 7 (3 self)
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Abstract Let x1,..., xk be nbit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 case, and a lower bound of!(1) for k> = 3 when T = \Theta (2n). We obtain(1) for general k> = 3 an upper bound of k + O(n1/(k1)), (2) for k = 3, T = \Theta (2n), a lowerbound of \Omega (log log n), (3) a generalization of the protocol to abelian groups, (4) lower boundson the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case. 1 Introduction Multiparty communication complexity was first defined by Chandra, Furst, and Lipton [8] and used to obtain lower bounds on branching programs. Since then it has been used to get additional lower bounds and tradeoffs for branching programs [1, 5], lower bounds on problems in data structures [5], timespace tradeoffs for restricted Turing machines [1], and unconditional pseudorandom generators for logspace [1]. Def 1.1 Let f: {{0, 1}n}k! {0, 1}. Assume, for 1 < = i < = k, Pi has all of the inputs except xi. Let d(f) be the total number of bits broadcast in the optimal deterministic protocol for f. This is called the multiparty communication complexity of f. The scenario is called the forehead model.
On The Complexity Of The Hidden Weighted Bit Function For Various BDD Models
 Theoretical Informatics and Applications
, 1998
"... Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verication, timing analysis, test pattern generation or combinatorial optimization. The hidden weighted bit function (HWB) is ..."
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Cited by 4 (1 self)
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Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verication, timing analysis, test pattern generation or combinatorial optimization. The hidden weighted bit function (HWB) is of particular interest, since it seems to be the simplest function with exponential OBDD size. The complexity of this function with respect to dierent circuit models, formulas, and various BDD models is discussed. Supported by DFG grant We 1066/81. 1. INTRODUCTION If one likes to have short representations of Boolean functions, circuits are the most powerful model. But if one likes to work with these representations, one additionally needs eÆcient algorithms for certain problems, among them satisability test, equivalence test, and synthesis, i. e., the combination of two or more representations by a Boolean operation. For this purpose ordered binary decision diagrams (OBDDs) introd...
Relating Branching Program Size and Formula Size over the Full Binary Basis
 In Proc. of 16th STACS, LNCS 1563
, 1998
"... Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L (f) over the restricted basis f; ; :g is considered. It is wellknown that C(f) 3 BP(f), ..."
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Cited by 3 (3 self)
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Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L (f) over the restricted basis f; ; :g is considered. It is wellknown that C(f) 3 BP(f), BP(f) L (f), L (f) L(f) 2 , and C(f) L(f) \Gamma 1. These estimates are optimal. But the inequality BP(f) L(f) 2 can be improved to BP(f) 1:360 L(f) fi , where fi = log 4 (3 + p 5) ! 1:195. 1