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# On the Power of Randomized Branching Programs

Electronic Colloquium on Computational Complexity (ECCC) 01/1995; 2.
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ABSTRACT We define the notion of a randomized branching program in the natural way similar to the definition of a randomized circuit. We exhibit an explicit function fn for which we prove that: 1) f n can be computed by polynomial size randomized read-once ordered branching program with a small one-sided error; 2) fn cannot be computed in polynomial size by deterministic readonce branching programs; 3) fn cannot be computed in polynomial size by deterministic read- k-times ordered branching program for k = o(n= log n) (the required deterministic size is exp GammaOmega Gamma n k DeltaDelta ). 1 Preliminaries Different models of branching programs introduced in [13, 15], have been studied extensively in the last decade (see for example [19]). A survey of known lower bounds for different models of branching programs can be found in [17]. Developments in the field of digital design and verification have led to the introduction of restricted forms of branching programs. In parti...

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ABSTRACT: We investigate the relationship between probabilistic and nondeterministic complexity classes PP, BPP, NP and coNP with respect to ordered read-once branching programs (OBDDs). We exhibit two explicit Boolean functions qn; Rn such that: (1) qn : {0,1}n → { 0,1} belongs to BPP (NP (semi-circle up) coNP) in the context of OBDDs; (2) Rn : {0,1}n → {0,1} belongs to PP \ (BPP(semi-circle up) NP (semi-circle up) coNP) in the context of OBDDs. Both of these functions are not in AC0.
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ABSTRACT: We show that the satisfiability problem for bounded-error probabilistic ordered branching programs is \NP -complete. If the error is very small, however (more precisely, if the error is bounded by the reciprocal of the width of the branching program), then we have a polynomial-time algorithm for the satisfiability problem.
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##### Article: Very narrow quantum OBDDs and width hierarchies for classical OBDDs
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ABSTRACT: We present several results on comparative complexity for di?erent variants of OBDD models. - We present some results on comparative complexity of classical and quantum OBDDs. We consider a partial function depending on pa- rameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2 but any classical OBDD (deter- ministic or stable bounded error probabilistic) needs width 2k+1. - We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more e?cient than classical one. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width but any clas- sical nondeterministic OBDD for this function needs non-constant width. - We also present new hierarchies on widths of deterministic and non- deterministic OBDDs. We focus both on small and large widths.