4,567 research outputs found
Approximation systems for functions in topological and in metric spaces
A notable feature of the TTE approach to computability is the representation
of the argument values and the corresponding function values by means of
infinitistic names. Two ways to eliminate the using of such names in certain
cases are indicated in the paper. The first one is intended for the case of
topological spaces with selected indexed denumerable bases. Suppose a partial
function is given from one such space into another one whose selected base has
a recursively enumerable index set, and suppose that the intersection of base
open sets in the first space is computable in the sense of Weihrauch-Grubba.
Then the ordinary TTE computability of the function is characterized by the
existence of an appropriate recursively enumerable relation between indices of
base sets containing the argument value and indices of base sets containing the
corresponding function value.This result can be regarded as an improvement of a
result of Korovina and Kudinov. The second way is applicable to metric spaces
with selected indexed denumerable dense subsets. If a partial function is given
from one such space into another one, then, under a semi-computability
assumption concerning these spaces, the ordinary TTE computability of the
function is characterized by the existence of an appropriate recursively
enumerable set of quadruples. Any of them consists of an index of element from
the selected dense subset in the first space, a natural number encoding a
rational bound for the distance between this element and the argument value, an
index of element from the selected dense subset in the second space and a
natural number encoding a rational bound for the distance between this element
and the function value. One of the examples in the paper indicates that the
computability of real functions can be characterized in a simple way by using
the first way of elimination of the infinitistic names.Comment: 21 pages, published in Logical Methods in Computer Scienc
Diophantine sets of polynomials over algebraic extensions of the rationals
Let L be a recursive algebraic extension of Q. Assume that, given alpha is an element of L, we can compute the roots in L of its minimal polynomial over Q and we can determine which roots are Aut(L)-conjugate to alpha. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of alpha, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in R, or in a finite extension of Q(p) (with p an odd prime). Then we show that subsets of L[X](k) that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X]
Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra
In many instances in first order logic or computable algebra, classical
theorems show that many problems are undecidable for general structures, but
become decidable if some rigidity is imposed on the structure. For example, the
set of theorems in many finitely axiomatisable theories is nonrecursive, but
the set of theorems for any finitely axiomatisable complete theory is
recursive. Finitely presented groups might have an nonrecursive word problem,
but finitely presented simple groups have a recursive word problem. In this
article we introduce a topological framework based on closure spaces to show
that many of these proofs can be obtained in a similar setting. We will show in
particular that these statements can be generalized to cover arbitrary
structures, with no finite or recursive presentation/axiomatization. This
generalizes in particular work by Kuznetsov and others. Examples from first
order logic and symbolic dynamics will be discussed at length
On Measuring Non-Recursive Trade-Offs
We investigate the phenomenon of non-recursive trade-offs between
descriptional systems in an abstract fashion. We aim at categorizing
non-recursive trade-offs by bounds on their growth rate, and show how to deduce
such bounds in general. We also identify criteria which, in the spirit of
abstract language theory, allow us to deduce non-recursive tradeoffs from
effective closure properties of language families on the one hand, and
differences in the decidability status of basic decision problems on the other.
We develop a qualitative classification of non-recursive trade-offs in order to
obtain a better understanding of this very fundamental behaviour of
descriptional systems
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