8,903 research outputs found
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 Measure of Space for Computing over the Reals
We propose a new complexity measure of space for the BSS model of
computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the
reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is
small enough for being relevant. We prove that the Real Circuit Decision
Problem is P\_R-complete under LOGSPACE\_W reductions, i.e. that LOGSPACE\_W is
large enough for containing natural algorithms. We also prove that PSPACE\_W is
included in PAR\_R
Static Data Structure Lower Bounds Imply Rigidity
We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space , would already imply a
semi-explicit () construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial () data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime , we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest
Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones
We show that a wide variety of non-linear cellular automata (CAs) can be
decomposed into a quasidirect product of linear ones. These CAs can be
predicted by parallel circuits of depth O(log^2 t) using gates with binary
inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of
inputs are allowed. Thus these CAs can be predicted by (idealized) parallel
computers much faster than by explicit simulation, even though they are
non-linear.
This class includes any CA whose rule, when written as an algebra, is a
solvable group. We also show that CAs based on nilpotent groups can be
predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p''
gates respectively.
We use these techniques to give an efficient algorithm for a CA rule which,
like elementary CA rule 18, has diffusing defects that annihilate in pairs.
This can be used to predict the motion of defects in rule 18 in O(log^2 t)
parallel time
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