943 research outputs found
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
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
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
Generating Matrix Identities and Proof Complexity
Motivated by the fundamental lower bounds questions in proof complexity, we
initiate the study of matrix identities as hard instances for strong proof
systems. A matrix identity of matrices over a field ,
is a non-commutative polynomial over such that
vanishes on every matrix assignment to its variables.
We focus on arithmetic proofs, which are proofs of polynomial identities
operating with arithmetic circuits and whose axioms are the polynomial-ring
axioms (these proofs serve as an algebraic analogue of the Extended Frege
propositional proof system; and over they constitute formally a
sub-system of Extended Frege [HT12]). We introduce a decreasing in strength
hierarchy of proof systems within arithmetic proofs, in which the th level
is a sound and complete proof system for proving matrix identities
(over a given field). For each level in the hierarchy, we establish a
proof-size lower bound in terms of the number of variables in the matrix
identity proved: we show the existence of a family of matrix identities
with variables, such that any proof of requires
number of lines. The lower bound argument uses fundamental results from the
theory of algebras with polynomial identities together with a generalization of
the arguments in [Hru11].
We then set out to study matrix identities as hard instances for (full)
arithmetic proofs. We present two conjectures, one about non-commutative
arithmetic circuit complexity and the other about proof complexity, under which
up to exponential-size lower bounds on arithmetic proofs (in terms of the
arithmetic circuit size of the identities proved) hold. Finally, we discuss the
applicability of our approach to strong propositional proof systems such as
Extended Frege.Comment: 46 pages, 1 figur
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