Dagstuhl Seminar Proceedings, Volume 6111
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
6111
2006
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-6111
06111 Abstracts Collection – Complexity of Boolean Functions
From 12.03.06 to 17.03.06, the Dagstuhl Seminar 06111 ``Complexity of Boolean Functions'' was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Complexity of Boolean functions
Boolean circuits
binary decision diagrams
lower bound proof techniques
combinatorics of Boolean functions
communi algorithmic learning
cryptography
derandomization
1-24
Regular Paper
Matthias
Krause
Matthias Krause
Pavel
Pudlák
Pavel Pudlák
Rüdiger
Reischuk
Rüdiger Reischuk
Dieter
van Melkebeek
Dieter van Melkebeek
10.4230/DagSemProc.06111.1
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06111 Executive Summary – Complexity of Boolean Functions
We briefly describe the state of the art concerning the complexity of discrete functions.
Computational models and analytical techniques are summarized.
After describing the formal organization of the Dagstuhl seminar "Complexity of Boolean Functions"
held in March 2006, we introduce the different topics that have been discussed there
and mention some of the major achievements.
The summary closes with an outlook on the development of discrete computational complexity in the future.
Boolean and quantum circuits
discrete problems
computational complexity
lower bounds
communication complexity
proof and query complexity
randomization
pseudo-randomness
derandomization
approximation
cryptography
computational learning
1-5
Regular Paper
Matthias
Krause
Matthias Krause
Dieter
van Melkebeek
Dieter van Melkebeek
Pavel
Pudlák
Pavel Pudlák
Rüdiger
Reischuk
Rüdiger Reischuk
10.4230/DagSemProc.06111.2
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A Generic Time Hierarchy for Semantic Models With One Bit of Advice
We show that for any reasonable semantic model of computation and for any positive integer $a$ and rationals $1 leq c < d$, there exists a language computable in time $n^d$ with $a$ bits of advice but not in time $n^c$ with $a$ bits of advice. A semantic model is one for which there exists a computable enumeration that contains all machines in the model but may also contain others. We call such a model reasonable if it has an efficient universal machine that can be complemented within the model in exponential time and if it is efficiently closed under deterministic transducers. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one- or zero-sided error, unambiguous machines, symmetric alternation, Arthur-Merlin games of any signature, interactive proof protocols with one or multiple provers, etc.
Time hierarchy
non-uniformity
one bit of advice
probabilistic algorithms
1-39
Regular Paper
Dieter
van Melkebeek
Dieter van Melkebeek
Konstantin
Pervyshev
Konstantin Pervyshev
10.4230/DagSemProc.06111.3
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Approximability of Minimum AND-Circuits
Given a set of monomials, the {sc Minimum AND-Circuit} problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size.
We prove that the problem is not polynomial time approximable within a factor of less than $1.0051$ unless $mathsf{P} = mathsf{NP}$, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of $1.278$. For the general problem, we achieve an approximation ratio of $d-3/2$, where $d$ is the degree of the largest monomial.
In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the {sc Minimum AND-Circuit} problem and several problems from different areas.
Optimization problems
approximability
automated circuit design
1-21
Regular Paper
Jan
Arpe
Jan Arpe
Bodo
Manthey
Bodo Manthey
10.4230/DagSemProc.06111.4
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Bounds on the Fourier Coefficients of the Weighted Sum Function
We estimate Fourier coefficients of a Boolean function
which has recently been introduced in the study of read-once
branching programs. Our bound implies that this function
has an asymptotically ``flat'' Fourier spectrum and thus
implies several lower bounds of its various complexity
measures.
Fourier coefficients
congruences
average sensitivity
decision tree
1-9
Regular Paper
Igor E.
Shparlinski
Igor E. Shparlinski
10.4230/DagSemProc.06111.5
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Computing Shortest Paths in Series-Parallel Graphs in Logarithmic Space
Series-parallel graphs, which are built by repeatedly applying
series or parallel composition operations to paths, play an
important role in computer science as they model the flow of
information in many types of programs. For directed series-parallel
graphs, we study the problem of finding a shortest path between two
given vertices. Our main result is that we can find such a path in
logarithmic space, which shows that the distance problem for
series-parallel graphs is L-complete. Previously, it was known
that one can compute some path in logarithmic space; but for
other graph types, like undirected graphs or tournament graphs,
constructing some path between given vertices is possible in
logarithmic space while constructing a shortest path is
NL-complete.
Series-parallel graphs
shortest path
logspace
1-9
Regular Paper
Andreas
Jakoby
Andreas Jakoby
Till
Tantau
Till Tantau
10.4230/DagSemProc.06111.6
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Fault Jumping Attacks against Shrinking Generator
In this paper we outline two new cryptoanalytic attacks against hardware implementation of the shrinking generator by Coppersmith et al., a classic design in low-cost, simple-design pseudorandom bitstream generator.
This is a report on work on progress, since implementation and careful adjusting
the attack strategy in order to optimize the atatck is still not completed.
Pseudorandom generator
shrinking generator
fault cryptanalysis
1-6
Regular Paper
Marcin
Gomulkiewicz
Marcin Gomulkiewicz
Miroslaw
Kutylowski
Miroslaw Kutylowski
Pawel
Wlaz
Pawel Wlaz
10.4230/DagSemProc.06111.7
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Graphs and Circuits: Some Further Remarks
We consider the power of single level circuits in the context of
graph complexity. We first prove that the single level conjecture
fails for fanin-$2$ circuits over the basis ${oplus,land,1}$.
This shows that the (surpisingly tight) phenomenon, established by
Mirwald and Schnorr (1992) for quadratic functions, has no analogon
for graphs. We then show that the single level conjecture fails for
unbounded fanin circuits over ${lor,land,1}$. This partially
answers the question of Pudl'ak, R"odl and Savick'y (1986). We
also prove that $Sigma_2
eq Pi_2$ in a restricted version of the
hierarhy of communication complexity classes introduced by Babai,
Frankl and Simon (1986). Further, we show that even depth-$2$
circuits are surprisingly powerful: every bipartite $n imes n$
graph of maximum degree $Delta$ can be represented by a monotone
CNF with $O(Deltalog n)$ clauses. We also discuss a relation
between graphs and $ACC$-circuits.
Graph complexity
single level conjecture
Sylvester graphs
communication complexity
ACC-circuits
1-16
Regular Paper
Stasys
Jukna
Stasys Jukna
10.4230/DagSemProc.06111.8
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Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity
We develop a new method for estimating the discrepancy
of tensors associated with multiparty communication problems
in the ``Number on the Forehead'' model of Chandra, Furst and Lipton.
We define an analogue of the Hadamard property of matrices
for tensors in multiple dimensions and show that any $k$-party communication
problem represented by a Hadamard tensor must have $Omega(n/2^k)$
multiparty communication complexity.
We also exhibit constructions of Hadamard tensors,
giving $Omega(n/2^k)$ lower bounds
on multiparty communication complexity
for a new class of explicitly defined Boolean functions.
Multiparty communication complexity
lower bounds
1-20
Regular Paper
Jeff
Ford
Jeff Ford
Anna
Gál
Anna Gál
10.4230/DagSemProc.06111.9
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Incremental branching programs
We propose a new model of restricted branching programs
which we call {em incremental branching programs}.
We show that {em syntactic}
incremental branching programs capture previously studied
structured models of computation for the problem GEN, namely marking
machines [Cook74].
and Poon's extension [Poon93] of jumping automata
on graphs [CookRackoff80].
We then prove
exponential size lower bounds for our syntactic incremental model,
and for some other restricted branching program models as well.
We further show that nondeterministic syntactic incremental
branching programs are
provably stronger than their deterministic counterpart when solving a
natural NL-complete GEN subproblem.
It remains open if syntactic incremental branching programs are as powerful
as unrestricted branching programs for GEN problems.
Joint work with Anna GÃƒÂ¡l and Michal KouckÃƒÂ½
Complexity theory
branching programs
logarithmic space
marking machines
1-20
Regular Paper
Anna
Gál
Anna Gál
Pierre
McKenzie
Pierre McKenzie
Michal
Koucký
Michal Koucký
10.4230/DagSemProc.06111.10
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On Probabilistic Time versus Alternating Time
Sipser and GÃƒÂ¡cs, and independently Lautemann, proved in '83 that probabilistic polynomial time is contained in the second level of the polynomial-time hierarchy, i.e. BPP is in Sigma_2 P. This is essentially the only non-trivial upper bound that we have on the power of probabilistic computation. More precisely, the Sipser-GÃƒÂ¡cs-Lautemann simulation shows that probabilistic time can be simulated deterministically, using two quantifiers, **with a quadratic blow-up in the running time**. That is, BPTime(t) is contained in Sigma_2 Time(t^2).
In this talk we discuss whether this quadratic blow-up in the running time is necessary. We show that the quadratic blow-up is indeed necessary for black-box simulations that use two quantifiers, such as those of Sipser, GÃƒÂ¡cs, and Lautemann. To obtain this result, we prove a new circuit lower bound for computing **approximate majority**, i.e. computing the majority of a given bit-string whose fraction of 1's is bounded away from 1/2 (by a constant): We show that small depth-3 circuits for approximate majority must have bottom fan-in Omega(log n).
On the positive side, we obtain that probabilistic time can be simulated deterministically, using three quantifiers, in quasilinear time. That is, BPTime(t) is contained in Sigma_3 Time(t polylog t). Along the way, we show that approximate majority can be computed by uniform polynomial-size depth-3 circuits. This is a uniform version of a striking result by Ajtai that gives *non-uniform* polynomial-size depth-3 circuits for approximate majority.
If time permits, we will discuss some applications of our results to proving lower bounds on randomized Turing machines.
Probabilistic time
alternating time
polynomial-time hierarchy
approximate majority
constant-depth circuit
1-0
Regular Paper
Emanuele
Viola
Emanuele Viola
10.4230/DagSemProc.06111.11
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On the Complexity of Numerical Analysis
We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSlp: Given a division-free straight-line program producing an integer N, decide whether N>0. We show that OrdSlp lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.
Blum-Shub-Smale Model
Euclidean Traveling Salesman Problem
Counting Hierarchy
1-9
Regular Paper
Eric
Allender
Eric Allender
Peter
Bürgisser
Peter Bürgisser
Johan
Kjeldgaard-Pedersen
Johan Kjeldgaard-Pedersen
Peter Bro
Miltersen
Peter Bro Miltersen
10.4230/DagSemProc.06111.12
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On the Teachability of Randomized Learners
The present paper introduces a new model for teaching {em randomized learners}.
Our new model, though based on the classical teaching dimension model,
allows to study the influence of various parameters such as the
learner's memory size, its ability to provide or to not provide feedback,
and the influence of the order in which examples are presented.
Furthermore, within the new model it is possible to investigate
new aspects of teaching like teaching from positive data only or
teaching with inconsistent teachers.
Furthermore, we provide characterization theorems for teachability from
positive data for both ordinary teachers and inconsistent teachers with and
without feedback.
Algorithmic Teaching
Complexity of teaching
1-20
Regular Paper
Frank J.
Balbach
Frank J. Balbach
Thomas
Zeugmann
Thomas Zeugmann
10.4230/DagSemProc.06111.13
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Quantum Network Coding
Since quantum information is continuous, its handling is sometimes
surprisingly harder than the classical counterpart. A typical
example is cloning; making a copy of digital information is
straightforward but it is not possible exactly for quantum
information. The question in this paper is whether or not {em
quantum} network coding is possible. Its classical counterpart is
another good example to show that digital information flow can be done
much more efficiently than conventional (say, liquid) flow.
Our answer to the question is similar to the case of cloning, namely,
it is shown that quantum network coding is possible if approximation
is allowed, by using a simple network model called Butterfly. In this
network, there are two flow paths, $s_1$ to $t_1$ and $s_2$ to $t_2$,
which shares a single bottleneck channel of capacity one. In the
classical case, we can send two bits simultaneously, one for each
path, in spite of the bottleneck. Our results for quantum network
coding include: (i) We can send any quantum state $|psi_1
angle$
from $s_1$ to $t_1$ and $|psi_2
angle$ from $s_2$ to $t_2$
simultaneously with a fidelity strictly greater than $1/2$. (ii) If
one of $|psi_1
angle$ and $|psi_2
angle$ is classical, then the
fidelity can be improved to $2/3$. (iii) Similar improvement is also
possible if $|psi_1
angle$ and $|psi_2
angle$ are restricted to
only a finite number of (previously known) states. (iv) Several
impossibility results including the general upper bound of the fidelity
are also given.
Network coding
quantum computation
quantum information
1-17
Regular Paper
Masahito
Hayashi
Masahito Hayashi
Kazuo
Iwama
Kazuo Iwama
Harumichi
Nishimura
Harumichi Nishimura
Rudy
Raymond
Rudy Raymond
Shigeru
Yamashita
Shigeru Yamashita
10.4230/DagSemProc.06111.14
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Quantum vs. Classical Read-Once Branching Programs
A simple, explicit boolean function on 2n input bits is presented that is
computable by errorfree quantum read-once branching programs of size
O(n^3), while each classical randomized read-once branching program
and each quantum OBDD for this function with bounded two-sided error
requires size 2^{omega(n)}.
Quantum branching program
randomized branching program
read-once
1-13
Regular Paper
Martin
Sauerhoff
Martin Sauerhoff
10.4230/DagSemProc.06111.15
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Secure Linear Algebra Using Linearly Recurrent Sequences
In this work we present secure two-party protocols for
various core problems in linear algebra.
Our main building block is a protocol to obliviously decide singularity
of an encrypted matrix:
Bob holds an $n imes n$ matrix $M$, encrypted with Alice's secret
key, and wants to learn whether
the matrix is singular or not (and nothing beyond that).
We give an interactive protocol between Alice and Bob that solves the
above problem
with optimal communication complexity while at the same time achieving
low round complexity.
More precisely, the number of communication rounds in our protocol
is $polylog(n)$ and
the overall communication is roughly $O(n^2)$ (note that the input size is $n^2$).
At the core of our protocol we exploit some nice mathematical
properties of linearly recurrent sequences and their
relation to the characteristic polynomial of the matrix $M$, following [Wiedemann, 1986].
With our new techniques we are able to improve the round complexity of
the communication efficient solution of [Nissim and Weinreb, 2006] from $n^{0.275}$ to $polylog(n)$.
Based on our singularity protocol we further
extend our result to the problems of securely computing the rank of an
encrypted matrix and solving systems of linear equations.
Secure Linear Algebra
Linearly Recurrent Sequences
Wiedemann's Algorithm
1-19
Regular Paper
Eike
Kiltz
Eike Kiltz
Enav
Weinreb
Enav Weinreb
10.4230/DagSemProc.06111.16
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The Cell Probe Complexity of Succinct Data Structures
In the cell probe model with word size 1 (the bit probe model), a
static data structure problem is given by a map
$f: {0,1}^n imes {0,1}^m
ightarrow {0,1}$,
where ${0,1}^n$ is a set of possible data to be stored,
${0,1}^m$ is a set of possible queries (for natural problems, we
have $m ll n$) and $f(x,y)$ is
the answer to question $y$ about data $x$.
A solution is given by a
representation $phi: {0,1}^n
ightarrow {0,1}^s$ and a query algorithm
$q$ so that $q(phi(x), y) = f(x,y)$. The time $t$ of the query algorithm
is the number of bits it reads in $phi(x)$.
In this paper, we consider the case of {em succinct} representations
where $s = n + r$ for some {em redundancy} $r ll n$.
For
a boolean version of the problem of polynomial
evaluation with preprocessing of coefficients, we show a lower bound on
the redundancy-query time tradeoff of the form
[ (r+1) t geq Omega(n/log n).]
In particular, for very small
redundancies $r$, we get an almost optimal lower bound stating that the
query algorithm has to inspect almost the entire data structure
(up to a logarithmic factor).
We show similar lower bounds for problems satisfying a certain
combinatorial property of a coding theoretic flavor.
Previously, no $omega(m)$ lower bounds were known on $t$
in the general model for explicit functions, even for very small
redundancies.
By restricting our attention to {em systematic} or {em index}
structures $phi$ satisfying $phi(x) = x cdot phi^*(x)$ for some
map $phi^*$ (where $cdot$ denotes concatenation) we show
similar lower bounds on the redundancy-query time tradeoff
for the natural data structuring problems of Prefix Sum
and Substring Search.
Cell probe model
data structures
lower bounds
time-space tradeoffs
1-13
Regular Paper
Anna
Gál
Anna Gál
Peter Bro
Miltersen
Peter Bro Miltersen
10.4230/DagSemProc.06111.17
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The complexity of Boolean functions from cryptographic viewpoint
Cryptographic Boolean functions must be complex to satisfy Shannon's principle of confusion. But the cryptographic viewpoint on complexity is not the same as in circuit complexity.
The two main criteria evaluating the cryptographic complexity of Boolean functions on $F_2^n$ are the nonlinearity (and more generally the $r$-th order nonlinearity, for every positive $r< n$) and the algebraic degree. Two other criteria have also been considered: the algebraic thickness and the non-normality. After recalling the definitions of these criteria and why, asymptotically, almost all Boolean functions are deeply non-normal and have high algebraic degrees, high ($r$-th order) nonlinearities and high algebraic thicknesses, we study the relationship between the $r$-th order nonlinearity and a recent cryptographic criterion called the algebraic immunity. This relationship strengthens the reasons why the algebraic immunity can be considered as a further cryptographic complexity criterion.
Boolean function
nonlinearity
Reed-Muller code
1-15
Regular Paper
Claude
Carlet
Claude Carlet
10.4230/DagSemProc.06111.18
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The optimal sequence compression
This paper presents the optimal compression for sequences with
undefined values.
Let we have $(N-m)$ undefined and $m$ defined positions in the
boolean sequence $vv V$ of length $N$. The sequence code length
can't be less then $m$ in general case, otherwise at least two
sequences will have the same code.
We present the coding algorithm which generates codes of almost $m$
length, i.e. almost equal to the lower bound.
The paper presents the decoding circuit too. The circuit has low
complexity which depends from the inverse density of defined values
$D(vv V) = frac{N}{m}$.
The decoding circuit includes RAM and random logic. It performs
sequential decoding. The total RAM size is proportional to the
$$logleft(D(vv V)
ight) ,$$
the number of random logic cells is proportional to
$$log logleft(D(vv V)
ight) * left(log log logleft(D(vv V)
ight)
ight)^2 .$$
So the decoding circuit will be small enough even for the very low
density sequences. The decoder complexity doesn't depend of the
sequence length at all.
Compression
partial boolean function
1-11
Regular Paper
Alexander E.
Andreev
Alexander E. Andreev
10.4230/DagSemProc.06111.19
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Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines
In this talk, we establish lower bounds for the running time of randomized
machines with two-sided error which use a small amount of workspace to
solve complete problems in the polynomial-time hierarchy. In particular,
we show that for integers $l > 1$, a randomized machine with two-sided error
using subpolynomial space requires time $n^{l - o(1)}$ to solve QSATl, where
QSATl denotes the problem of deciding the validity of a Boolean first-order
formula with at most $l-1$ quantifier alternations. This represents the first
time-space lower bounds for complete problems in the polynomial-time
hierarchy on randomized machines with two-sided error.
Corresponding to $l = 1$, we show that a randomized machine with one-sided
error using subpolynomial space requires time $n^{1.759}$ to decide the set
of Boolean tautologies. As a corollary, this gives the same lower bound for
satisfiability on deterministic machines, improving on the previously best
known such result.
Time-space lower bounds
lower bounds
randomness
polynomial-time hierarchy
1-33
Regular Paper
Scott
Diehl
Scott Diehl
Dieter
van Melkebeek
Dieter van Melkebeek
10.4230/DagSemProc.06111.20
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Using Quantum Oblivious Transfer to Cheat Sensitive Quantum Bit Commitment
We define $(varepsilon,delta)$-secure quantum computations
between two parties that can play dishonestly to maximise
advantage $delta$, however keeping small the probability
$varepsilon$ that the computation fails in evaluating correct value.
We present a simple quantum protocol for computing
one-out-of-two oblivious transfer that is
$(O(sqrt{varepsilon}),varepsilon)$-secure.
Using the protocol as a black box we construct a scheme for
cheat sensitive quantum bit commitment which guarantee that
a mistrustful party has a nonzero probability of detecting a
cheating.
Two-Party Computations
Quantum Protocols
Bit Commitment
Oblivious Transfer.
1-12
Regular Paper
Andreas
Jakoby
Andreas Jakoby
Maciej
Liskiewicz
Maciej Liskiewicz
Aleksander
Madry
Aleksander Madry
10.4230/DagSemProc.06111.21
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Very Large Cliques are Easy to Detect
It is known that, for every constant $kgeq 3$, the presence of a
$k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex
graph cannot be detected by a monotone boolean circuit using fewer
than $Omega((n/log n)^k)$ gates. We show that, for every constant
$k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be
detected by a monotone circuit using only $O(n^2log n)$ gates.
Moreover, if we allow unbounded fanin, then $O(log n)$ gates are
enough.
Clique function
monotone circuits
perfect hashing
1-7
Regular Paper
Alexander E.
Andreev
Alexander E. Andreev
Stasys
Jukna
Stasys Jukna
10.4230/DagSemProc.06111.22
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