12 research outputs found
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
Small space analogues of Valiant\u27s classes and the limitations of skew formula
In the uniform circuit model of computation, the width of a boolean
circuit exactly characterises the ``space\u27\u27 complexity of the
computed function. Looking for a similar relationship in Valiant\u27s
algebraic model of computation, we propose width of an arithmetic
circuit as a possible measure of space. We introduce the class
VL as an algebraic variant of deterministic log-space L. In
the uniform setting, we show that our definition coincides with that
of VPSPACE at polynomial width.
Further, to define algebraic variants of non-deterministic
space-bounded classes, we introduce the notion of ``read-once\u27\u27
certificates for arithmetic circuits. We show that polynomial-size
algebraic branching programs can be expressed as a read-once
exponential sum over polynomials in VL, ie
.
We also show that , ie
VBPs are stable under read-once exponential sums. Further, we
show that read-once exponential sums over a restricted class of
constant-width arithmetic circuits are within VQP, and this is the
largest known such subclass of poly-log-width circuits with this
property.
We also study the power of skew formulas and show that exponential
sums of a skew formula cannot represent the determinant polynomial
Real Interactive Proofs for VPSPACE
We study interactive proofs in the framework of real number complexity as introduced by Blum, Shub, and Smale. The ultimate goal is to give a Shamir like characterization of the real counterpart IP_R of classical IP. Whereas classically Shamir\u27s result implies IP = PSPACE = PAT = PAR, in our framework a major difficulty arises from the fact that in contrast to Turing complexity theory the real number classes PAR_R and PAT_R differ and space resources considered alone are not meaningful. It is not obvious to see whether IP_R is characterized by one of them - and if so by which.
In recent work the present authors established an upper bound IP_R is a subset of MA(Exists)R, where MA(Exists)R is a complexity class satisfying PAR_R is a strict subset of MA(Exists)R, which is a subset of PAT_R and conjectured to be different from PAT_R. The goal of the present paper is to complement this result and to prove interesting lower bounds for IP_R. More precisely, we design interactive real protocols for a large class of functions introduced by Koiran and Perifel and denoted by UniformVSPACE^0. As consequence, we show PAR_R is a subset of IP_R, which in particular implies co-NP_R is a subset of IP_R, and P_R^{Res} is a subset of IP_R, where Res denotes certain multivariate Resultant polynomials.
Our proof techniques are guided by the question in how far Shamir\u27s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVSPACE^0 are extremely helpful
On Annihilators of Explicit Polynomial Maps
We study the algebraic complexity of annihilators of polynomials maps. In
particular, when a polynomial map is `encoded by' a small algebraic circuit, we
show that the coefficients of an annihilator of the map can be computed in
PSPACE. Even when the underlying field is that of reals or complex numbers, an
analogous statement is true. We achieve this by using the class VPSPACE that
coincides with computability of coefficients in PSPACE, over integers.
As a consequence, we derive the following two conditional results. First, we
show that a VP-explicit hitting set generator for all of VP would separate
either VP from VNP, or non-uniform P from PSPACE. Second, in relation to
algebraic natural proofs, we show that proving an algebraic natural proofs
barrier would imply either VP VNP or DSPACE()
P
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
A hitting set construction, with application to arithmetic circuit lower bounds
14 pagesA polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form . From our algorithm we derive an exponential lower bound for representations of polynomials such as under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the ``hardness from derandomization'' approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers
Kolmogorov Complexity Theory over the Reals
Kolmogorov Complexity constitutes an integral part of computability theory,
information theory, and computational complexity theory -- in the discrete
setting of bits and Turing machines. Over real numbers, on the other hand, the
BSS-machine (aka real-RAM) has been established as a major model of
computation. This real realm has turned out to exhibit natural counterparts to
many notions and results in classical complexity and recursion theory; although
usually with considerably different proofs. The present work investigates
similarities and differences between discrete and real Kolmogorov Complexity as
introduced by Montana and Pardo (1998)