24 research outputs found
Interpolation in Valiant's theory
We investigate the following question: if a polynomial can be evaluated at
rational points by a polynomial-time boolean algorithm, does it have a
polynomial-size arithmetic circuit? We argue that this question is certainly
difficult. Answering it negatively would indeed imply that the constant-free
versions of the algebraic complexity classes VP and VNP defined by Valiant are
different. Answering this question positively would imply a transfer theorem
from boolean to algebraic complexity. Our proof method relies on Lagrange
interpolation and on recent results connecting the (boolean) counting hierarchy
to algebraic complexity classes. As a byproduct we obtain two additional
results: (i) The constant-free, degree-unbounded version of Valiant's
hypothesis that VP and VNP differ implies the degree-bounded version. This
result was previously known to hold for fields of positive characteristic only.
(ii) If exponential sums of easy to compute polynomials can be computed
efficiently, then the same is true of exponential products. We point out an
application of this result to the P=NP problem in the Blum-Shub-Smale model of
computation over the field of complex numbers.Comment: 13 page
VPSPACE and a Transfer Theorem over the Reals
We introduce a new class VPSPACE of families of polynomials. Roughly
speaking, a family of polynomials is in VPSPACE if its coefficients can be
computed in polynomial space. Our main theorem is that if (uniform,
constant-free) VPSPACE families can be evaluated efficiently then the class PAR
of decision problems that can be solved in parallel polynomial time over the
real numbers collapses to P. As a result, one must first be able to show that
there are VPSPACE families which are hard to evaluate in order to separate over
the reals P from NP, or even from PAR.Comment: Full version of the paper (appendices of the first version are now
included in the text
VPSPACE and a transfer theorem over the complex field
We extend the transfer theorem of [KP2007] to the complex field. That is, we
investigate the links between the class VPSPACE of families of polynomials and
the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of
polynomials is in VPSPACE if its coefficients can be computed in polynomial
space. Our main result is that if (uniform, constant-free) VPSPACE families can
be evaluated efficiently then the class PAR of decision problems that can be
solved in parallel polynomial time over the complex field collapses to P. As a
result, one must first be able to show that there are VPSPACE families which
are hard to evaluate in order to separate P from NP over C, or even from PAR.Comment: 14 page
Symmetry of information and nonuniform lower bounds
12 pagesIn the first part we provide another proof of the result of [Homer, Mocas 1995] that for all constant c, the class EXP is not included in P/(n^c) . The proof is based on a simple diagonalization, whereas it uses resource-bounded Kolmogorov complexity in [Homer, Mocas 1995]. In the second part, we investigate links between resource-bounded Kolmogorov complexity and nonuniform classes in computational complexity. Assuming a version of polynomial-time symmetry of information, we show that exponential-time problems do not have polynomial-size circuits (in symbols, EXP is not included in P/poly)
A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent
We show that the permanent cannot be computed by DLOGTIME-uniform threshold
or arithmetic circuits of depth o(log log n) and polynomial size.Comment: 11 page
Pushdown Compression
The pressing need for eficient compression schemes for XML documents has
recently been focused on stack computation [6, 9], and in particular calls for
a formulation of information-lossless stack or pushdown compressors that allows
a formal analysis of their performance and a more ambitious use of the stack in
XML compression, where so far it is mainly connected to parsing mechanisms. In
this paper we introduce the model of pushdown compressor, based on pushdown
transducers that compute a single injective function while keeping the widest
generality regarding stack computation. The celebrated Lempel-Ziv algorithm
LZ78 [10] was introduced as a general purpose compression algorithm that
outperforms finite-state compressors on all sequences. We compare the
performance of the Lempel-Ziv algorithm with that of the pushdown compressors,
or compression algorithms that can be implemented with a pushdown transducer.
This comparison is made without any a priori assumption on the data's source
and considering the asymptotic compression ratio for infinite sequences. We
prove that Lempel-Ziv is incomparable with pushdown compressors
A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent
11 pagesWe show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size
Cyclotomic Identity Testing and Applications
We consider the cyclotomic identity testing problem: given a polynomial
, decide whether is
zero, for a primitive complex -th root of unity and
integers . We assume that and are
represented in binary and consider several versions of the problem, according
to the representation of . For the case that is given by an algebraic
circuit we give a randomized polynomial-time algorithm with two-sided errors,
showing that the problem lies in BPP. In case is given by a circuit of
polynomially bounded syntactic degree, we give a randomized algorithm with
two-sided errors that runs in poly-logarithmic parallel time, showing that the
problem lies in BPNC. In case is given by a depth-2 circuit
(or, equivalently, as a list of monomials), we show that the cyclotomic
identity testing problem lies in NC. Under the generalised Riemann hypothesis,
we are able to extend this approach to obtain a polynomial-time algorithm also
for a very simple subclass of depth-3 circuits. We complement
this last result by showing that for a more general class of depth-3
circuits, a polynomial-time algorithm for the cyclotomic
identity testing problem would yield a sub-exponential-time algorithm for
polynomial identity testing. Finally, we use cyclotomic identity testing to
give a new proof that equality of compressed strings, i.e., strings presented
using context-free grammars, can be decided in coRNC: randomized NC with
one-sided errors