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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 Restricted Branching Programs and Applications (Extended Abstract)
"... We present a new lower bound technique for two types of restricted Branching Programs (BPs), namely for readonce BPs (BP1s) with restricted amount of nondeterminism and for (1, +k)BPs. For this technique, we introduce the notion of (strictly) kwise lmixed Boolean functions, which generalizes t ..."
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We present a new lower bound technique for two types of restricted Branching Programs (BPs), namely for readonce BPs (BP1s) with restricted amount of nondeterminism and for (1, +k)BPs. For this technique, we introduce the notion of (strictly) kwise lmixed Boolean functions, which generalizes the concept of lmixedness defined by Jukna in 1988 [3]. We prove that if a Boolean function f ∈ Bn is (strictly) kwise lmixed, then any nondeterministic BP1 with at most k − 1 nondeterministic nodes and any (1, +k)BP representing f has a size of at least 2 Ω(l). While leading to new exponential lower bounds of wellstudied functions (e.g. linear codes), the lower bound technique also shows that the polynomial size hierarchy for BP1s with respect to the available amount of nondeterminism is strict. More precisely, we present a class of functions g k n ∈ Bn which can be represented by polynomial size BP1s with k nondeterministic nodes, but require superpolynomial size if only k − 1 nondeterministic nodes are available (for k = o(n 1/3 / log 2/3 n)). This is the first hierarchy result of this kind where the BP1 does not obey any further restrictions. We also obtain a hierarchy result with respect to k for (1, +k)BPs as long as k = o ( � n / log n). This extends the hierarchy result of Savick´y and ˇ Zák [9], where k was bounded above by 1 2 n1/6 / log 1/3 n.