18 research outputs found
Noncomputable functions in the Blum-Shub-Smale model
Working in the Blum-Shub-Smale model of computation on the real numbers, we
answer several questions of Meer and Ziegler. First, we show that, for each
natural number d, an oracle for the set of algebraic real numbers of degree at
most d is insufficient to allow an oracle BSS-machine to decide membership in
the set of algebraic numbers of degree d + 1. We add a number of further
results on relative computability of these sets and their unions. Then we show
that the halting problem for BSS-computation is not decidable below any
countable oracle set, and give a more specific condition, related to the
cardinalities of the sets, necessary for relative BSS-computability. Most of
our results involve the technique of using as input a tuple of real numbers
which is algebraically independent over both the parameters and the oracle of
the machine
Noncomputable Functions in the Blub-Shub-Smale Model
Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more specific condition, related to the cardinalities of the sets, necessary for relative BSS-computability. Most of our results involve the technique of using as input a tuple of real numbers which is algebraically independent over both the parameters and the oracle of the machine
The Cardinality of an Oracle in Blum-Shub-Smale Computation
We examine the relation of BSS-reducibility on subsets of the real numbers.
The question was asked recently (and anonymously) whether it is possible for
the halting problem H in BSS-computation to be BSS-reducible to a countable
set. Intuitively, it seems that a countable set ought not to contain enough
information to decide membership in a reasonably complex (uncountable) set such
as H. We confirm this intuition, and prove a more general theorem linking the
cardinality of the oracle set to the cardinality, in a local sense, of the set
which it computes. We also mention other recent results on BSS-computation and
algebraic real numbers
Computability, noncomputability and undecidability of maximal intervals of IVPs
Let (α, β) ⊆ R denote the maximal interval of existence of solution for
the initial-value problem
dx
dt = f(t, x)
x(t0) = x0,
where E is an open subset of Rm+1, f is continuous in E and (t0, x0) ∈
E. We show that, under the natural definition of computability from
the point of view of applications, there exist initial-value problems with
computable f and (t0, x0) whose maximal interval of existence (α, β) is
noncomputable. The fact that f may be taken to be analytic shows that
this is not a lack of regularity phenomenon. Moreover, we get upper
bounds for the “degree of noncomputability” by showing that (α, β) is
r.e. (recursively enumerable) open under very mild hypotheses. We also
show that the problem of determining whether the maximal interval is
bounded or unbounded is in general undecidable
Strong Turing Degrees for Additive BSS RAM's
For the additive real BSS machines using only constants 0 and 1 and order
tests we consider the corresponding Turing reducibility and characterize some
semi-decidable decision problems over the reals. In order to refine,
step-by-step, a linear hierarchy of Turing degrees with respect to this model,
we define several halting problems for classes of additive machines with
different abilities and construct further suitable decision problems. In the
construction we use methods of the classical recursion theory as well as
techniques for proving bounds resulting from algebraic properties. In this way
we extend a known hierarchy of problems below the halting problem for the
additive machines using only equality tests and we present a further
subhierarchy of semi-decidable problems between the halting problems for the
additive machines using only equality tests and using order tests,
respectively
A Stochastic Complexity Perspective of Induction in Economics and Inference in Dynamics
Rissanen's fertile and pioneering minimum description length principle (MDL) has been viewed from the point of view of statistical estimation theory, information theory, as stochastic complexity theory -.i.e., a computable approximation to Kolomogorov Complexity - or Solomonoff's recursion theoretic induction principle or as analogous to Kolmogorov's sufficient statistics. All these - and many more - interpretations are valid, interesting and fertile. In this paper I view it from two points of view: those of an algorithmic economist and a dynamical system theorist. >From these points of view I suggest, first, a recasting of Jevons's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics.
Computability of Julia sets
In this paper we settle most of the open questions on algorithmic
computability of Julia sets. In particular, we present an algorithm for
constructing quadratics whose Julia sets are uncomputable. We also show that a
filled Julia set of a polynomial is always computable.Comment: Revised. To appear in Moscow Math. Journa
A topological view on algebraic computation models
We investigate the topological aspects of some algebraic computation models, in particular the BSS-model. Our results can be seen as bounds on how different BSS-computability and computability in the sense of computable analysis can be. The framework for this is Weihrauch reducibility. As a consequence of our characterizations, we establish that the solvability complexity index is (mostly) independent of the computational model, and that there thus is common ground in the study of non-computability between the BSS and TTE setting