33,022 research outputs found
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Self-Consistent Effective-Medium Approximations with Path Integrals
We study effective-medium approximations for linear composite media by means
of a path integral formalism with replicas. We show how to recover the
Bruggeman and Hori-Yonezawa effective-medium formulas. Using a replica-coupling
ansatz, these formulas are extended into new ones which have the same
percolation thresholds as that of the Bethe lattice and Potts model of
percolation, and critical exponents s=0 and t=2 in any space dimension d>= 2.
Like the Bruggeman and Hori-Yonezawa formulas, the new formulas are exact to
second order in the weak-contrast and dilute limits. The dimensional range of
validity of the four effective-medium formulas is discussed, and it is argued
that the new ones are of better relevance than the classical ones in dimensions
d=3,4 for systems obeying the Nodes-Links-Blobs picture, such as
random-resistor networks.Comment: 18 pages, 6 eps figure
Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances
This paper first analyzes the resolution complexity of two random CSP models
(i.e. Model RB/RD) for which we can establish the existence of phase
transitions and identify the threshold points exactly. By encoding CSPs into
CNF formulas, it is proved that almost all instances of Model RB/RD have no
tree-like resolution proofs of less than exponential size. Thus, we not only
introduce new families of CNF formulas hard for resolution, which is a central
task of Proof-Complexity theory, but also propose models with both many hard
instances and exact phase transitions. Then, the implications of such models
are addressed. It is shown both theoretically and experimentally that an
application of Model RB/RD might be in the generation of hard satisfiable
instances, which is not only of practical importance but also related to some
open problems in cryptography such as generating one-way functions.
Subsequently, a further theoretical support for the generation method is shown
by establishing exponential lower bounds on the complexity of solving random
satisfiable and forced satisfiable instances of RB/RD near the threshold.
Finally, conclusions are presented, as well as a detailed comparison of Model
RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively,
exhibit three different kinds of phase transition behavior in NP-complete
problems.Comment: 19 pages, corrected mistakes in Theorems 5 and
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