Computability of 1-manifolds
نویسندگان
چکیده
A semi-computable set S in a computable metric space need not be computable. However, in some cases, if S has certain topological properties, we can conclude that S is computable. It is known that if a semi-computable set S is a compact manifold with boundary, then the computability of ∂S implies the computability of S. In this paper we examine the case when S is a 1-manifold with boundary, not necessarily compact. We show that a similar result holds in this case under assumption that S has finitely many components.
منابع مشابه
Computability, noncomputability, and hyperbolic systems
In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyperbolic system. We show that (i) locally these manifolds can be computed, but (ii) globally they cannot (though we prove they are semi-computable). We also show that Smale’s horseshoe, the first example of a hyperboli...
متن کاملComputability of semicomputable manifolds in computable topological spaces
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold M is computable if its boundary ∂M is computable. We also show how this result combined with certain construction which compactifies a semicomputable set leads to th...
متن کاملReal Computable Manifolds and Homotopy Groups
Using the model of real computability developed by Blum, Cucker, Shub, and Smale, we investigate the difficulty of determining the answers to several basic topological questions about manifolds. We state definitions of real-computable manifold and of real-computable paths in such manifolds, and show that, while BSS machines cannot in general decide such questions as nullhomotopy and simple conn...
متن کاملThe World Problem: on the Computability of the Topology of 4-manifolds
Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and onl...
متن کاملComputable structures on topological manifolds
We propose a definition of computable manifold by introducing computability as a structure that we impose to a given topological manifold, just in the same way as differentiability or piecewise linearity are defined for smooth and PL manifolds respectively. Using the framework of computable topology and Type-2 theory of effectivity, we develop computable versions of all the basic concepts neede...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Logical Methods in Computer Science
دوره 10 شماره
صفحات -
تاریخ انتشار 2014