We present a new lower bound technique for two types of restricted Branching Programs (BPs), namely for read-once BPs (BP1s) with restricted amount of nondeterminism and for (1, +k)-BPs. For this technique, we introduce the notion of (strictly) k-wise l-mixed Boolean functions, which generalizes the concept of l-mixedness defined by Jukna in 1988 . We prove that if a Boolean function f ∈ Bn is (strictly) k-wise l-mixed, 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 well-studied 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 , where k was bounded above by 1 2 n1/6 / log 1/3 n
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