271,227 research outputs found
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
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
On the Weak Computability of Continuous Real Functions
In computable analysis, sequences of rational numbers which effectively
converge to a real number x are used as the (rho-) names of x. A real number x
is computable if it has a computable name, and a real function f is computable
if there is a Turing machine M which computes f in the sense that, M accepts
any rho-name of x as input and outputs a rho-name of f(x) for any x in the
domain of f. By weakening the effectiveness requirement of the convergence and
classifying the converging speeds of rational sequences, several interesting
classes of real numbers of weak computability have been introduced in
literature, e.g., in addition to the class of computable real numbers (EC), we
have the classes of semi-computable (SC), weakly computable (WC), divergence
bounded computable (DBC) and computably approximable real numbers (CA). In this
paper, we are interested in the weak computability of continuous real functions
and try to introduce an analogous classification of weakly computable real
functions. We present definitions of these functions by Turing machines as well
as by sequences of rational polygons and prove these two definitions are not
equivalent. Furthermore, we explore the properties of these functions, and
among others, show their closure properties under arithmetic operations and
composition
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