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 $mbox{VBP}inSigma^R cdotmbox{VL}$. We also show that $Sigma^R cdot mbox{VBP} =mbox{VBP}$, 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 $\neq$ VNP or DSPACE($\log^{\log^{\ast}n} n$) $\not\subset$ 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 $\sum_{j=0}^t c_j X^{\alpha_j} (a + b X)^{\beta_j}$. From our algorithm we derive an exponential lower bound for representations of polynomials such as $\prod_{i=1}^{2^n} (X^i-1)$ 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)