24,319 research outputs found
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
Evaluating Matrix Circuits
The circuit evaluation problem (also known as the compressed word problem)
for finitely generated linear groups is studied. The best upper bound for this
problem is , which is shown by a reduction to polynomial
identity testing. Conversely, the compressed word problem for the linear group
is equivalent to polynomial identity testing. In
the paper, it is shown that the compressed word problem for every finitely
generated nilpotent group is in . Within
the larger class of polycyclic groups we find examples where the compressed
word problem is at least as hard as polynomial identity testing for skew
arithmetic circuits
A Logical Characterization of Constant-Depth Circuits over the Reals
In this paper we give an Immerman's Theorem for real-valued computation. We
define circuits operating over real numbers and show that families of such
circuits of polynomial size and constant depth decide exactly those sets of
vectors of reals that can be defined in first-order logic on R-structures in
the sense of Cucker and Meer. Our characterization holds both non-uniformily as
well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202
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
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
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