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A Hierarchy Result for ReadOnce Branching Programs with Restricted Parity Nondeterminism
 IN MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE: 25TH INTERNATIONAL SYMPOSIUM, VOLUME 1893 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2000
"... Restricted branching programs are considered in complexity theory in order to study the space complexity of sequential computations and in applications as a data structure for Boolean functions. In this paper (⊕, k)branching programs and (#, k)branching programs are considered, i.e., branching pro ..."
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Restricted branching programs are considered in complexity theory in order to study the space complexity of sequential computations and in applications as a data structure for Boolean functions. In this paper (⊕, k)branching programs and (#, k)branching programs are considered, i.e., branching programs starting with a ⊕ (or #)node with a fanout of k whose successors are k readonce branching programs. This model is motivated by the investigation of the power of nondeterminism in branching programs and of similar variants that have been considered as a data structure. Lower bound methods and hierarchy results for polynomial size (⊕, k) and (#, k)branching programs with respect to k are presented.
A Lower Bound Technique for Nondeterministic GraphDriven ReadOnceBranching Programs and its Applications (Extended Abstract)
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Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds
"... Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argum ..."
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Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argument can prove that a problem requires circuits of superpolynomial size, even for some very restricted classes of circuits (under reasonable cryptographic assumptions). This barrier is so daunting, that some researchers have decided to focus their attentions elsewhere. Yet the goal of proving circuit lower bounds is of such importance, that some in the community have proposed concrete strategies for surmounting the obstacle. This lecture will discuss some of these strategies, and will dwell at length on a recent approach proposed by Michal Koucky and the author.
TriangleFreeness is Hard to Detect
, 2002
"... We show that recognizing the K3freeness and K4freeness of graphs is hard, respectively, for twoplayer nondeterministic communication protocols using exponentially many partitions and for nondeterministic syntactic readr times branching programs. The key ingredient is a generalization of a colo ..."
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We show that recognizing the K3freeness and K4freeness of graphs is hard, respectively, for twoplayer nondeterministic communication protocols using exponentially many partitions and for nondeterministic syntactic readr times branching programs. The key ingredient is a generalization of a coloring lemma, due to Papadimitriou and Sipser, which says that for every balanced redblue coloring of the edges of the complete nvertex graph there is a set of ffln2 triangles, none of which is monochromatic and no triangle can be formed by picking edges from different triangles. We extend this lemma to exponentially many colorings and to partial colorings.
On Uncertainty versus Size in Branching Programs
 SUBMITTED TO TCS
"... We propose an informationtheoretic approach to proving lower bounds on the size of branching programs. The argument is based on Kraft type inequalities for the average amount of uncertainty about (or entropy of) a given input during the various stages of computation. The uncertainty is measured ..."
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We propose an informationtheoretic approach to proving lower bounds on the size of branching programs. The argument is based on Kraft type inequalities for the average amount of uncertainty about (or entropy of) a given input during the various stages of computation. The uncertainty is measured by the average depth of socalled `splitting trees' for sets of inputs reaching particular nodes of the program. We first
On the NonApproximability of Boolean Functions by OBDDs and ReadKTimes Branching Programs
"... Branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e. g., verification), exact representations are required. For learning boolean functions f on the basis of clas ..."
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Branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e. g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f . This motivates the investigation of the size of the smallest branching program approximating f . Although several nonapproximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following nonapproximability results are presented. (1) It is proven that a simple functions from the branching program literature requires exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic readktimes branching programs with respect to the uniform distribution and error probability 1/22 # n) , n the input size, is shown. In order to improve upon the so far best results for error probabilities smaller than 1/3, a strong combinatorial lemma from a recent paper of Ajtai on linearlength branching programs is exploited. Keywords: Computational complexity, branching programs, binary decision diagrams, approximations, lower bounds. # Supported in part by DFG We 1066/9. 1
Approximation of Boolean Functions by Combinatorial Rectangles
 Electr. Coll. on Comp. Compl
, 2000
"... This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of ..."
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This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of rectangles required for the approximation of Boolean functions in this model is very sensitive to the allowed error: There is an explicitly defined sequence of functions f n : {0, 1} n # {0, 1} such that f n has rectangle approximations with a constant number of rectangles and onesided error 1/3+o(1) or twosided error 1/4+2 #(n) , but, on the other hand, f n requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant. Rectangle partitions and rectangle approximations with the same partition of the input variables for all rectangles have been thoroughly investigated in communication complexity theory. The complexity measures where each r...
Binary Decision Diagrams
"... Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and timespace tradeoffs. Due to the lack of powerfu ..."
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Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and timespace tradeoffs. Due to the lack of powerful lower bound techniques, various types of restricted decision diagrams are investigated. They lead to new lower bound techniques and some of them allow efficient algorithms for a list of operations on boolean functions. Indeed, restricted decision diagrams like ordered binary decision diagrams (OBDDs) are the most common data structure for boolean functions with many applications in verification, model checking, CAD tools, and graph problems. From a complexity theoretical point of view also randomized and nondeterministic decision diagrams are of interest.
On MultiPartition Communication Complexity
 In Proc. of 18th Int. Symp. on Theoretical Aspects of Computer Science vol. 2010 of Springer Lecture Notes in Computer Science
, 2001
"... We study kpartition communication protocols, an extension of the standard twoparty bestpartition model to k input partitions. The main results are as follows. ..."
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We study kpartition communication protocols, an extension of the standard twoparty bestpartition model to k input partitions. The main results are as follows.