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A nonlinear time lower bound for boolean branching programs
 In Proc. of 40th FOCS
, 1999
"... Abstract: We give an exponential lower bound for the size of any lineartime Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2way) ..."
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Cited by 55 (0 self)
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Abstract: We give an exponential lower bound for the size of any lineartime Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2way) branching program of size less than 2 εn which, for all inputs X ⊆ {0,1,...,n − 1}, computes in time kn the parity of the number of elements of the set of all pairs 〈x,y 〉 with the property x ∈ X, y ∈ X, x < y, x + y ∈ X. For the proof of this fact we show that if A = (ai, j) n i=0, j=0 is a random n by n matrix over the field with 2 elements with the condition that “A is constant on each minor diagonal,” then with high probability the rank of each δn by δn submatrix of A is at least cδlogδ  −2n, where c> 0 is an absolute constant and n is sufficiently large with respect to δ.
TimeSpace Tradeoffs for Branching Programs
, 1999
"... We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0 ..."
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Cited by 44 (2 self)
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We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0. We also give the first separation result between the syntactic and semantic readk models [BRS93] for k > 1 by showing that polynomialsize semantic readtwice branching programs can compute functions that require exponential size on any syntactic readk branching program. We also show...
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...
Parity graphdriven readonce branching programs and an exponential lower bound for integer multiplication
 In Proc. of 2nd TCS
, 2002
"... Abstract Branching programs are a wellestablished computation model for boolean functions, especially readonce branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic readonce branching programs are known for a long time. On the other hand ..."
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Cited by 5 (3 self)
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Abstract Branching programs are a wellestablished computation model for boolean functions, especially readonce branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic readonce branching programs are known for a long time. On the other hand, the problem of proving superpolynomial lower bounds for parity readonce branching programs is still open. In this paper restricted parity readonce branching programs are considered and an exponential lower bound on the size of wellstructured parity graphdriven readonce branching programs for integer multiplication is proven. This is the first strongly exponential lower bound on the size of a nonoblivious parity readonce branching program model for an explicitly defined boolean function. In addition, more insight into the structure of integer multiplication is yielded.
An Improved Hierarchy Result for Partitioned BDDs
 THEORY OF COMPUTING SYSTEMS
, 2000
"... One of the great challenges of complexity theory is the problem of analyzing the dependence of the complexity of Boolean functions on the resources nondeterminism and randomness. So far, this problem could be solved only for very few models of computation. For socalled partitioned binary decision ..."
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Cited by 3 (2 self)
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One of the great challenges of complexity theory is the problem of analyzing the dependence of the complexity of Boolean functions on the resources nondeterminism and randomness. So far, this problem could be solved only for very few models of computation. For socalled partitioned binary decision diagrams, which are a restricted variant of nondeterministic readonce branching programs, Bollig and Wegener have proven an astonishing hierarchy result which shows that the smallest possible decrease of the available amount of nondeterminism may incur an exponential blowup of the branching program size. They have shown that kpartitioned BDDs which may nondeterministically choose between k alternative subprograms may be exponentially larger than (k + 1)partitioned BDDs for the same function if k = o # (log n/ loglog n) 1/2 # , where n is the input size. In this paper, an improved hierarchy result is established which still works if the number of nondeterministic decisions is O ...
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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Cited by 2 (2 self)
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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...
A Separation of Syntactic and Nonsyntactic (1,+k)Branching Programs
"... For (1; +k)branching programs and readktimes branching programs syntactic and nonsyntactic variants can be distinguished. The nonsyntactic variants correspond in a natural way to sequential computations with restrictions on reading the input while lower bound proofs are easier or only known for t ..."
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Cited by 1 (0 self)
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For (1; +k)branching programs and readktimes branching programs syntactic and nonsyntactic variants can be distinguished. The nonsyntactic variants correspond in a natural way to sequential computations with restrictions on reading the input while lower bound proofs are easier or only known for the syntactic variants. In this paper it is shown that nonsyntactic (1; +k)branching programs are really more powerful than syntactic (1; +k)branching programs by presenting an explicitly defined function with polynomial size nonsyntactic (1; +1)branching programs but only exponential size syntactic (1; +k)branching programs. Another separation of these variants of branching programs is obtained by comparing the complexity of the satisfiability test for both variants. 1 Introduction In complexity theory branching programs and lower bound methods for branching programs are investigated due to the relationship between branching programs and sequential computations. Superpolynomial lower b...
On the Power of Automata Based Proof Systems
, 1999
"... One way to address the NP = co  NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof syste ..."
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One way to address the NP = co  NP question is to consider the length of proofs of tautologies in various proof systems. In this work we consider proof systems defined by appropriate classes of automata. In general, starting from a given class of automata we can define a corresponding proof system in a natural way. An interesting new proof system that we consider is based on the class of push down automata. We present an exponential lower bound for oblivious readonce branching programs which implies that the new proof system based on push down automata is, in a certain sense, more powerful than oblivious regular resolution. 1 Introduction One of the famous open questions of complexity theory is: does NP equal coNP ? Put another way do tautologies always have "short" proofs? If proof is taken in its most general form, i.e. does some nondeterministic polynomial time Turing Machine correctly accept exactly the class of tautologies, then the question seems to be completely beyond ou...