43 research outputs found
A polynomial-time computable curve whose interior has a nonrecursive measure
AbstractA polynomial-time computable simple curve is constructed such that its measure in the two-dimensional plane is positive. This construction is applied to prove the following two results: 1.1) there exists a polynomial-time computable simple closed curve in the two-dimensional plane such that the measure of its interior region is a nonrecursive real number;2.(2) there exists a polynomial-time computable simple curve in the two-dimensional plane such that its length is finite but is a nonrecursive real number
Jordan Areas and Grids
AbstractJordan curves can be used to represent special subsets of the Euclidean plane, either the (open) interior of the curve or the (compact) union of the interior and the curve itself. We compare the latter with other representations of compact sets using grids of points and we are able to show that knowing the length of a rectifiable curve is sufficient to translate from the grid representation to the Jordan curve
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations
The outcomes of this paper are twofold.
Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side.
This result gives a purely continuous (time and space) elegant and simple characterization of P. We believe it is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis.
Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations.
Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog model of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both at the computability and complexity level, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both at a computability and at a computational complexity level
The power of backtracking and the confinement of length
We show that there is a point on a computable arc that does not belong to any computable rectifiable curve. We also show that there is a point on a computable rectifiable curve with computable length that does not belong to any computable arc
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
We provide an implicit characterization of polynomial time computation in
terms of ordinary differential equations: we characterize the class
of languages computable in polynomial time in terms of
differential equations with polynomial right-hand side.
This result gives a purely continuous (time and space) elegant and simple
characterization of . This is the first time such classes
are characterized using only ordinary differential equations. Our
characterization extends to functions computable in polynomial time over the
reals in the sense of computable analysis. This extends to deterministic
complexity classes above polynomial time.
This may provide a new perspective on classical complexity, by giving a way
to define complexity classes, like , in a very simple
way, without any reference to a notion of (discrete) machine. This may also
provide ways to state classical questions about computational complexity via
ordinary differential equations, i.e.~by using the framework of analysis
Point-Separable Classes of Simple Computable Planar Curves
In mathematics curves are typically defined as the images of continuous real
functions (parametrizations) defined on a closed interval. They can also be
defined as connected one-dimensional compact subsets of points. For simple
curves of finite lengths, parametrizations can be further required to be
injective or even length-normalized. All of these four approaches to curves are
classically equivalent. In this paper we investigate four different versions of
computable curves based on these four approaches. It turns out that they are
all different, and hence, we get four different classes of computable curves.
More interestingly, these four classes are even point-separable in the sense
that the sets of points covered by computable curves of different versions are
also different. However, if we consider only computable curves of computable
lengths, then all four versions of computable curves become equivalent. This
shows that the definition of computable curves is robust, at least for those of
computable lengths. In addition, we show that the class of computable curves of
computable lengths is point-separable from the other four classes of computable
curves
Study of numeric Saturation Effects in Linear Digital Compensators
Saturation arithmetic is often used in finite precision digital compensators to circumvent instability due to radix overflow. The saturation limits in the digital structure lead to nonlinear behavior during large state transients. It is shown that if all recursive loops in a compensator are interrupted by at least one saturation limit, then there exists a bounded external scaling rule which assures against overflow at all nodes in the structure. Design methods are proposed based on the generalized second method of Lyapunov, which take the internal saturation limits into account to implement a robust dual-mode suboptimal control for bounded input plants. The saturating digital compensator provides linear regulation for small disturbances, and near-time-optimal control for large disturbances or changes in the operating point. Computer aided design tools are developed to facilitate the analysis and design of this class of digital compensators