72,110 research outputs found
Support Sets in Exponential Families and Oriented Matroid Theory
The closure of a discrete exponential family is described by a finite set of
equations corresponding to the circuits of an underlying oriented matroid.
These equations are similar to the equations used in algebraic statistics,
although they need not be polynomial in the general case. This description
allows for a combinatorial study of the possible support sets in the closure of
an exponential family. If two exponential families induce the same oriented
matroid, then their closures have the same support sets. Furthermore, the
positive cocircuits give a parameterization of the closure of the exponential
family.Comment: 27 pages, extended version published in IJA
On the Complexity of Quantum ACC
For any , let \MOD_q be a quantum gate that determines if the number
of 1's in the input is divisible by . We show that for any ,
\MOD_q is equivalent to \MOD_t (up to constant depth). Based on the case
, Moore \cite{moore99} has shown that quantum analogs of AC,
ACC, and ACC, denoted QAC, QACC, QACC respectively,
define the same class of operators, leaving as an open question. Our
result resolves this question, proving that QAC QACC
QACC for all . We also develop techniques for proving upper bounds for QACC
in terms of related language classes. We define classes of languages EQACC,
NQACC and BQACC_{\rats}. We define a notion -planar QACC operators and
show the appropriately restricted versions of EQACC and NQACC are contained in
P/poly. We also define a notion of -gate restricted QACC operators and
show the appropriately restricted versions of EQACC and NQACC are contained in
TC. To do this last proof, we show that TC can perform iterated
addition and multiplication in certain field extensions. We also introduce the
notion of a polynomial-size tensor graph and show that families of such graphs
can encode the amplitudes resulting from apply an arbitrary QACC operator to an
initial state.Comment: 22 pages, 4 figures This version will appear in the July 2000
Computational Complexity conference. Section 4 has been significantly revised
and many typos correcte
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
On Symmetric Circuits and Fixed-Point Logics
We study properties of relational structures such as graphs that are decided
by families of Boolean circuits. Circuits that decide such properties are
necessarily invariant to permutations of the elements of the input structures.
We focus on families of circuits that are symmetric, i.e., circuits whose
invariance is witnessed by automorphisms of the circuit induced by the
permutation of the input structure. We show that the expressive power of such
families is closely tied to definability in logic. In particular, we show that
the queries defined on structures by uniform families of symmetric Boolean
circuits with majority gates are exactly those definable in fixed-point logic
with counting. This shows that inexpressibility results in the latter logic
lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201
Non-Malleable Codes for Small-Depth Circuits
We construct efficient, unconditional non-malleable codes that are secure
against tampering functions computed by small-depth circuits. For
constant-depth circuits of polynomial size (i.e. tampering
functions), our codes have codeword length for a -bit
message. This is an exponential improvement of the previous best construction
due to Chattopadhyay and Li (STOC 2017), which had codeword length
. Our construction remains efficient for circuit depths as
large as (indeed, our codeword length remains
, and extending our result beyond this would require
separating from .
We obtain our codes via a new efficient non-malleable reduction from
small-depth tampering to split-state tampering. A novel aspect of our work is
the incorporation of techniques from unconditional derandomization into the
framework of non-malleable reductions. In particular, a key ingredient in our
analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC
2013), a derandomization of the influential switching lemma from circuit
complexity; the randomness-efficiency of this switching lemma translates into
the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure
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