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TR01-101 | 14th December 2001 00:00
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#### A Lower Bound Technique for Restricted Branching Programs and Applications

**Abstract:**
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 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^Omega(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 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 n)^(2/3)) ). 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)^(1/2)). This extends the hierarchy result of

Savicky and Zak (2000), where k was bounded above by

(1/2)n^(1/6)/(log n)^(1/3).