99,890 research outputs found
On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
Given a zero-dimensional polynomial system consisting of n integer
polynomials in n variables, we propose a certified and complete method to
compute all complex solutions of the system as well as a corresponding
separating linear form l with coefficients of small bit size. For computing l,
we need to project the solutions into one dimension along O(n) distinct
directions but no further algebraic manipulations. The solutions are then
directly reconstructed from the considered projections. The first step is
deterministic, whereas the second step uses randomization, thus being
Las-Vegas.
The theoretical analysis of our approach shows that the overall cost for the
two problems considered above is dominated by the cost of carrying out the
projections. We also give bounds on the bit complexity of our algorithms that
are exclusively stated in terms of the number of variables, the total degree
and the bitsize of the input polynomials
On Quasi-Interpretations, Blind Abstractions and Implicit Complexity
Quasi-interpretations are a technique to guarantee complexity bounds on
first-order functional programs: with termination orderings they give in
particular a sufficient condition for a program to be executable in polynomial
time, called here the P-criterion. We study properties of the programs
satisfying the P-criterion, in order to better understand its intensional
expressive power. Given a program on binary lists, its blind abstraction is the
nondeterministic program obtained by replacing lists by their lengths (natural
numbers). A program is blindly polynomial if its blind abstraction terminates
in polynomial time. We show that all programs satisfying a variant of the
P-criterion are in fact blindly polynomial. Then we give two extensions of the
P-criterion: one by relaxing the termination ordering condition, and the other
one (the bounded value property) giving a necessary and sufficient condition
for a program to be polynomial time executable, with memoisation.Comment: 18 page
On monotone circuits with local oracles and clique lower bounds
We investigate monotone circuits with local oracles [K., 2016], i.e.,
circuits containing additional inputs that can perform
unstructured computations on the input string . Let be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions , and
be the set of -cliques and the set of complete -partite graphs,
respectively (similarly to [Razborov, 1985]). Our results can be informally
stated as follows.
1. For an appropriate extension of depth- monotone circuits with local
oracles, we show that the size of the smallest circuits separating
(triangles) and (complete bipartite graphs) undergoes two phase
transitions according to .
2. For , arbitrary depth, and , we
prove that the monotone circuit size complexity of separating the sets
and is , under a certain restrictive
assumption on the local oracle gates.
The second result, which concerns monotone circuits with restricted oracles,
extends and provides a matching upper bound for the exponential lower bounds on
the monotone circuit size complexity of -clique obtained by Alon and Boppana
(1987).Comment: Updated acknowledgements and funding informatio
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