Computable Real Functions Research Articles

Nearly computable real numbers

We call a sequence ( a n ) n of elements of a metric space nearly computably Cauchy if for every increasing computable function r : N → N the sequence ( d ( a r ( n + 1 ) , a r ( n ) ) ) n converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ( a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

Computability of sets in Euclidean space

Abstract We consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝ n and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of computable set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set.

Σ-Definability in Hereditarily Finite Superstructures and Computable Analysis

We construct a computable real function not Σ-definable in hereditarily finite superstructures over the extensions with decidable theory of the reals.

Computability and complexity of Julia sets: a review

Since A. M. Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1][10][13]. Some are of interest in nonlinear and statistical physics while others are extensions of the mathematical theory of computation. In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3]. 1 Computability and complexity Chaos and fractals have been studied from the viewpoint of computability in physics [6][12][1][2]. Investigation has focused on the nature of complexity arising from simple nonlinear equations. Unpredictability in chaotic attractors and final state sensitivity in fractal basins are discussed in terms of computability and complexity in the theory of computation. We summarize the leading results of a recently developed computability theory for Julia sets by Braverman and Yampolsky [3]. First, we introduce the classical notions of computability introduced by Turing [14], and computable real functions introduced by Pour-El [13]. Turing computability is defined by rather a physical model of human computation, called a Turing machine, which is an automaton which consisting of finite internal states and a head to read/write symbols on an external tape. The length of the tape is not restricted, but the number of alphabets is finite. It can manipulate individual symbols on a tape, according to a transition diagram, which tells the machine what action to undertake depending on the current internal state ∗s133038@math.sci.hokudai.ac.jp †ysato@math.sci.hokudai.ac.jp ‡zin@math.sci.hokudai.ac.jp

Relative computability and uniform continuity of relations

A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based, computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f :[0,1]→ℝ is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multi- valued functions f :[0,1]⇒ℝ (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y ∈ f ( x ), new ways arise of (linearly) ordering quantifiers — yet none turns out as satisfactory. We are thus led to a concept of uniform continuity based on the Henkin quantifier ; and prove it necessary for relative computability of compact real relations. In fact iterating this condition yields a strict hierarchy of notions each necessary — and the ω-th level also sufficient — for relative computability.

Conditional computability of real functions with respect to a class of operators

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question.

On the Kolmogorov complexity of continuous real functions

Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects—such as rational numbers—used to approximate them.This idea will be taken further here by extending the definition to continuous functions over real numbers, based on the fact that every continuous real function can be represented as the limit of a sequence of finitely-representable enclosures, such as polynomials with rational coefficients.Based on this definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that ‘almost every’ continuous real function has such a high-growth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well.

On the Weak Computability of Continuous Real Functions

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

Plottable Real Number Functions and the Computable Graph Theorem

The Graph Theorem of classical recursion theory states that a total function on the natural numbers is computable if and only if its graph is recursive. It is known that this result can be generalized to real number functions where it has an important practical interpretation: the total computable real number functions are precisely those which can be effectively plotted with any given resolution. We generalize the Graph Theorem to appropriate partial real number functions and even further to functions defined on certain computable metric spaces. Besides the nonuniform version of the Graph Theorem which logically relates computability properties of the function and computability properties of its graph, we also discuss the uniform version: given a program of a function, can we algorithmically derive a description of its graph? And, vice versa, given a description of the graph, can we derive a program of the function? While the passage from functions to graphs is always computable, the inverse direction from graphs to functions is problematic, and it turns out that the answers to the uniform and the nonuniform questions do not coincide. We prove that in both cases certain topological and computational properties (such as compactness or effective local connectedness) are sufficient for a positive answer, and we provide counterexamples which show that the corresponding properties are not superfluous. Additionally, we briefly discuss the special situation of the linear case.

Hierarchies of function classes defined by the first-value operator

The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within Δ 0 2 . The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class Δ B 2 . Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.

Real Hypercomputation and Continuity

By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.

Revising Type-2 Computation and Degrees of Discontinuity

By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS2000), Brattka (MLQ2005), and Ziegler (ToCS2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the jump of a representation: both a TTE–counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield.We also consider Markov and Banach/Mazur oracle–computation of discontinuous functions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.

The basic feasible functionals in computable analysis

We give a correspondence between two notions of complexity for real functions: poly-time computability according to Ko and a notion that arises naturally when one considers the application of Mehlhorn's class of the basic feasible functionals to computable analysis. We show that both notions define the same set of polynomial-time computable real functions.

Effectively open real functions

A function f is continuous iff the pre-image f - 1 [ V ] of any open set V is open again. Dual to this topological property, f is called open iff the image f [ U ] of any open set U is open again. Several classical open mapping theorems in analysis provide a variety of sufficient conditions for openness. By the main theorem of recursive analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping V ↦ f - 1 [ V ] being effective: given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f - 1 [ V ] . Analogously, effective openness requires the mapping U ↦ f [ U ] on open real subsets to be effective. The present work combines real analysis with algebraic topology and Tarski's quantifier elimination to effectivize classical open mapping theorems and to establish several rich classes of real functions as effectively open.

Type-2 Computability and Moore's Recursive Functions

Regarding effectivity of functions on the reals, there have been several proposed models of analog, continuous-time computation, as opposed to the digital, discrete nature of the type-2 computability. We study one of them, Moore's real (primitive) recursive functions, whose definition mimics the classical characterization of recursive functions on N by the closure properties. We show that the class of type-2 computable real functions falls between Moore's classes of primitive recursive and recursive functions.

Hierarchies of Function Classes Defined by the First-Value Operator: (Extended Abstract)

The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point.In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within Δ20. The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class Δ2B. Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.

Effective Jordan Decomposition

The Jordan Decomposition Theorem says that any real function of bounded variation can be expressed as a difference of two increasing functions. This paper investigates the effective version of Jordan decomposition and discusses the properties of variation of computable real functions. First we show that the effective version of Jordan decomposition does not hold in general. Then we give a sufficient and necessary condition for computable real functions of bounded variation which have an effective Jordan decomposition. Applying this condition, we construct a computable real function which has a computable modulus of absolute continuity (hence of bounded variation) but is not effectively Jordan decomposable. Finally, we prove a version of this result restricted to polynomial time which answers negatively an open question of Ko in [8].

Index sets for computable differential equations

AbstractIndex sets are used to measure the complexity of properties associated with the differentiability of real functions and the existence of solutions to certain classic differential equations. The new notion of a locally computable real function is introduced and provides several examples of Σ04 complete sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Closure Properties of Real Number Classes under CBV Functions

CBV functions are computable real functions of bounded variation. In this paper we investigate the basic properties of CBV functions. We are especially interested in the question of whether a real number class is closed under CBV functions. The real number classes considered here include the classes of computable (EC), semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and recursively approximable (RA) real numbers. We show that the classes EC, RA and DBC are closed under CBV functions but SC and WC are not. Furthermore, WC$ is not even closed under computable monotone functions and, finally, the image sets of $\wc$ under computable monotone functions and CBV functions are different.

Preface: Volume 66, Issue 1

This part of the volume contains the papers accepted for presentation at the fifth workshop on Computability and Complexity in Analysis, CCA 2002, which took place on July 12-13, 2002 in Málaga, Spain. It was co-located with ICALP 2002, the 29th International Colloquium on Automata, Languages, and Programming. Both events were hosted by the Computer Science Department of the University of Málaga, Spain.The workshop is concerned with the theory of computability and complexity over the real numbers which is built on the Turing machine model. This theory was initiated by Turing, Grzegorczyk, Lacombe, Banach and Mazur, and after a long period of slow development has seen rapid growth in recent years. Newcomers can find introductions in the books by Pour-El/Richards, Ko, and Weihrauch.Until recently, most work in computability and complexity concentrated on problems over discrete structures, and many computer scientists do not know much about computable real functions. Maybe, for this reason problems including real number computation have been neglected, avoided or even ignored by the majority of computer scientists.Although still many even basic questions are unsettled, meanwhile a rich toolkit has been developed for numerous applications of computer science requiring real number computation like image processing, computational geometry, dynamical systems, hybrid systems and, last but not least, numerical mathematics and scientific computation.These proceedings contain 18 papers on different areas of computable analysis which range from recursion theoretic aspects to exact geometric computations. The workshop has been enriched by a demonstration of systems for exact real computation.We would like to thank the authors for their contributions, the program committee and the additional referees for their careful work and the local organizers for their assistance.June, 2002 Vasco Brattka Matthias Schröder Klaus WeihrauchProgram Committee•Peter Hertling (Hagen, Germany)•Ker-I Ko (Stony Brook, USA)•Marian Pour-El (Minnesota, USA)•Ludwig Staiger (Halle-Wittenberg, Germany)•John V. Tucker (Swansea, Wales)•Klaus Weihrauch, chair (Hagen, Germany)•Mariko Yasugi (Kyoto Sangyo, Japan)•Xizhong Zheng (Cottbus, Germany)•Ning Zhong (Cincinnati, USA)Organizing Committee•Vasco Brattka (Hagen, Germany)•Matthias Schröder (Hagen, Germany)Demonstration•Norbert Müller (Trier, Germany)