Peter Hertling

Recursively enumerable reals and Chaitin &Ω numbers

A real α is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay (unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, May 1975, 215 pp.) and Chaitin (IBM J. Res. Develop. 21 (1977) 350–359, 496.) we say that an r.e. real α dominates an r.e. real β if from a good approximation of α from below one can compute a good approximation of β from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovays (unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, May 1975, 215 pp.) Ω-like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability of a universal self-delimiting Turing machine (Chaitins Ω number (J. Assoc. Comput. Mach. 22 (1975) 329–340)) is also a random r.e. real. Solovay showed that any Chaitin Ω number is Ω-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

An effective Riemann mapping theorem

The main results of the paper are two effective versions of the Riemann mapping theorem. The first, uniform version is based on the constructive proof of the Riemann mapping theorem by Cheng and by Bishop and Bridges and formulated in the computability framework developed by Kreitz and Weihrauch. It states which topological information precisely one needs about a nonempty, proper, open, connected, and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from the unit disk onto this set, and vice versa, which topological information about the set can be obtained from a description of a holomorphic bijection. The second version, which is derived from the first by considering the sets and the functions with computable descriptions, characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection from the unit disk. This gives a partial answer to a problem posed by Pour-El and Richards. We also show that this class of sets is strictly larger than a class of sets considered by Zhou, which solves an open problem posed by him. In preparation, recursively enumerable open subsets and closed subsets of Euclidean spaces are considered and several effective results in complex analysis are proved.

Topological properties of real number representations

We prove three results about representations of real numbers (or elements of other topological spaces) by infinite strings. Such representations are useful for the description of real number computations performed by digital computers or by Turing machines. First, we show that the so-called admissible representations, a topologically natural class of representations introduced by Kreitz and Weihrauch, are essentially the continuous extensions (with a well-behaved domain) of continuous and open representations. Second, we show that there is no admissible representation of the real numbers such that every real number has only finitely many names. Third, we show that a rather interesting property of admissible real number representations holds true also for a certain non-admissible representation, namely for the naive Cauchy representation: the property that continuity is equivalent to relative continuity with respect to the representation.

Topological Complexity with Continuous Operations

The topological complexity of algorithms is studied in a general context in the first part and for zero-finding in the second part. In the first part thelevel of discontinuityof a functionfis introduced and it is proved that it is a lower bound for the total number of comparisons plus 1 in any algorithm computingfthat uses only continuous operations and comparisons. This lower bound is proved to be sharp if arbitrary continuous operations are allowed. Then there exists even a balanced optimal computation tree forf. In the second part we use these results in order to determine the topological complexity of zero-finding for continuous functionsfon the unit interval withf(0) ·f(1) < 0. It is proved that roughly log2log2??1comparisons are optimal during a computation in order to approximate a zero up to ?. This is true regardless of whether one allows arbitrary continuous operations or just function evaluations, the arithmetic operations {+, ?, *, /}, and the absolute value. It is true also for the subclass of nondecreasing functions. But for the subclass of increasing functions the topological complexity drops to zero even for the smaller class of operations.

Is the Mandelbrot set computable

This work is concerned with the question whether the Mandelbrot set is computable. The computability notions that we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two-sided distance function of the Mandelbrot set is computable if the famous hyperbolicity conjecture is true. We also formulate the question whether the distance function of the Mandelbrot set is computable in terms of the escape time. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Random elements in effective topological spaces with measure

Following a suggestion of Zvonkin and Levin, we generalize Martin-Lofs definition of infinite random sequences over a finite alphabet via randomness tests to effective topological spaces with a measure. We show that under weak computability conditions there is a universal randomness test. We prove a theorem on randomness preserving functions which corrects and extends a result by Schnorr and apply it to a number of examples. In particular, we show that a real number is random if, and only if, it has a random b- ary representation, for any b ≥ 2. We show that many computable, continuously differentiable real functions preserve randomness. Especially, all computable analytic functions which are not constant on any open subset of their domain preserve randomness. Finally, we introduce a new randomness concept for subsets of natural numbers, which we characterize in terms of random sequences. Surprisingly, it turns out that there are infinite co-r.e. random sets.

A Real Number Structure that is Effectively Categorical

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well-known non-appropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers.

Embedding Quantum Universes in Classical Ones

Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Rephrased, How far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result of Kochen and Specker answers the above question in the negative. However, this answer is just one among various possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical ones, more specifically, in terms of embeddings of quantum logics into classical logics; depending upon the type of restrictions imposed on embeddings, the question may get negative or positive answers.

Randomness on full shift spaces

Abstract We give various characterizations for algorithmically random configurations on full shift spaces, based on randomness tests. We show that all nonsurjective cellular automata destroy randomness and surjective cellular automata preserve randomness. Furthermore all one-dimensional cellular automata preserve nonrandomness. The last three assertions are also true if one replaces randomness by richness – a form of pseudorandomness, which is compatible with computability. The last assertion is true even for an arbitrary dimension.

A Banach–Mazur computable but not Markov computable function on the computable real numbers☆

Abstract We consider two classical computability notions for functions mapping all computable real numbers to computable real numbers. It is clear that any function that is computable in the sense of Markov, i.e., computable with respect to a standard Godel numbering of the computable real numbers, is computable in the sense of Banach and Mazur, i.e., it maps any computable sequence of real numbers to a computable sequence of real numbers. We show that the converse is not true. This solves a long-standing open problem posed by Kushner.