DeepAI AI Chat
Log In Sign Up

On the computability of ordered fields

by   M. V. Korovina, et al.

In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems: whether a generated class is a real closed field and whether there exists a computable presentation of a generated class. We prove a series of theorems that lead to the result that there are no computable presentations neither for polynomial time computable no even for E_n-computable real numbers, where E_n is a level in Grzegorczyk hierarchy, n ≥ 2. We also propose a criterion of computable presentability of an archimedean ordered field.


page 1

page 2

page 3

page 4


Nearly Computable Real Numbers

In this article we call a sequence (a_n)_n of elements of a metric space...

Primitive Recursive Ordered Fields and Some Applications

We establish primitive recursive versions of some known facts about comp...

A recursion theoretic foundation of computation over real numbers

We define a class of computable functions over real numbers using functi...

Relatively acceptable notation

Shapiro's notations for natural numbers, and the associated desideratum ...

On the existence of hidden machines in computational time hierarchies

Challenging the standard notion of totality in computable functions, one...

On the computability properties of topological entropy: a general approach

The dynamics of symbolic systems, such as multidimensional subshifts of ...

On First-order Cons-free Term Rewriting and PTIME

In this paper, we prove that (first-order) cons-free term rewriting with...