Computable Real Research Articles

Computability in linear algebra

Many problems in Linear Algebra can be solved by Gaussian Elimination. This famous algorithm applies to an algebraic model of real number computation where operations + ,-, * ,/ and tests like, e.g., < and == are presumed exact. Implementations of algebraic algorithms on actual digital computers often lead to numerical instabilities, thus revealing a serious discrepancy between model and reality. A different model of real number computation dating back to Alan Turing considers real numbers as limits of rational approximations. In that widely believed to be more realistic notion of computability, we investigate problems from Linear Algebra. Our central results yield algorithms which in this sense • solve systems of linear equations A · x = b , • determine the spectral resolution of a symmetric matrix B, • and compute a linear subspace's dimension from its Euclidean distance function, provided the rank of A and the number of distinct eigenvalues of B are known. Without such restrictions, the first two problems are shown to be, in general, uncomputable.

A Two-Layer Approach to the Computability and Complexity of Real Numbers

We present a new approach to computability of real numbers in which real functions have type-1 representations, which also includes the ability to reason about the complexity of real numbers and functions. We discuss how this allows efficient implementations of exact real numbers and also present a new real number system that is based on it.

Real functions incrementally computable by finite automata

This is an investigation into exact real number computation using the incremental approach of Potts (Ph.D. Thesis, Department of Computing, Imperial College, 1998), Edalat and Potts (Electronic Notes in Computer Science, Vol. 6, Elsevier Science Publishers, Amsterdam, 2000), Nielsen and Kornerup (J. Universal Comput. Sci. 1(7) (1995) 527), and Vuillemin (IEEE Trans. on Comput. 39(8) (1990) 1087) where numbers are represented as infinite streams of digits, each of which is a Möbius transformation. The objective is to determine for each particular system of digits which functions R→ R can be computed by a finite transducer and ultimately to search for the most finitely expressible Möbius representations of real numbers. The main result is that locally such functions are either not continuously differentiable or equal to some Möbius transformation. This is proved using elementary properties of finite transition graphs and Möbius transformations. Applying the results to the standard signed-digit representations, we can classify functions that are finitely computable in such a representation and are continuously differentiable everywhere except for finitely many points. They are exactly those functions whose graph is a fractured line connecting finitely many points with rational coordinates.

On the hierarchy and extension of monotonically computable real numbers

A real number x is called h- monotonically computable ( h-mc for short), for some function h : N→ N , if there is a computable sequence ( x s ) of rational numbers converging to x such that h( n)| x− x n |⩾| x− x m | for all m> n. x is called ω- monotonically computable ( ω-mc) if it is h-mc for some computable function h. Thus, the class of ω-mc real numbers is an extension of the class of monotonically computable real numbers introduced in (Math. Logic Quart. 48(3) (2002) 459), where only constant functions h≡ c are considered and the corresponding real numbers are called c-monotonically computable. In (Math. Logic Quart. 48(3) (2002) 459) it is shown that the classes of c-mc real numbers form a proper hierarchy inside the class of weakly computable real numbers which is the arithmetical closure of the 1-mc real numbers. In this paper, we show that this hierarchy is dense, i.e., for any real numbers c 2> c 1⩾1, there is a c 2-mc real number which is not c 1-mc and there is also an ω-mc real number which is not c-mc for any c∈ R . Furthermore, we show that the class of all ω-mc real numbers is incomparable with the class of weakly computable real numbers.

On the Turing Degrees of Weakly Computable Real Numbers

The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree.

Computability of Real Numbers by Using a Given Class of Functions in the Set of the Natural Numbers

Given a class ℱ oft otal functions in the set oft he natural numbers, one could study the real numbers that have arbitrarily close rational approximations explicitly expressible by means of functions from ℱ. We do this for classes ℱsatisfying certain closedness conditions. The conditions in question are satisfied for example by the class of all recursive functions, by the class of the primitive recursive ones, by any of the Grzegorczyk classes ℰnwith n ≥ 2, by the class of all functions recursive in a given function and by the class of the functions primitive recursive in it, as well as by the class of all total functions in the set of the natural numbers.

Real number computation through Gray code embedding

We propose an embedding G of the unit open interval to the set {0,1} ⊥,1 ω of infinite sequences of {0,1} with at most one undefined element. This embedding is based on Gray code and it is a topological embedding with a natural topology on {0,1} ⊥,1 ω . We also define a machine called an indeterministic multihead Type 2 machine which input/output sequences in {0,1} ⊥,1 ω , and show that the computability notion induced on real functions through the embedding G is equivalent to the one induced by the signed digit representation and Type 2 machines. We also show that basic algorithms can be expressed naturally with respect to this embedding.

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)

The closure properties on real numbers under limits and computable operators

In effective analysis, various classes of real numbers are discussed. For example, the classes of computable, semi-computable, weakly computable, recursively approximable real numbers, etc. All these classes correspond to some kind of (weak) computability of the real numbers. In this paper we discuss mathematical closure properties of these classes under the limit, effective limit and computable function. Among others, we show that the class of weakly computable real numbers is not closed under effective limit and partial computable functions while the class of recursively approximable real numbers is closed under effective limit and partial computable functions.

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.

Monotonically Computable Real Numbers

A real number x is called semi-computable if it is a limit of an increasing or decreasing computable sequence (x n ), n∈ℕ of ratio­nal numbers. In this case, a later member of the sequence is always a better approximation to x in the sense that |x − x n | ≥ |x − x m |, if n ≤ m. As a natural generalization, we call a real number x k-monotone computable (k-mc, for short), for any real number k >0,, if there is a computable sequence (xn),n∈ℕ of rational numbers which converges to x k-monotonically in the sense that k · |x − x n | ≥ |x − x m | for any n ≤ m and x is monotonically computable (mc, for short) if it is k-mc for some k >0. Various properties of k-mc real numbers are discussed in this pa­per. Among others we show that a real number is computable if it is k-mc for some k <1; the 1-mc real numbers are just the semi-computable real numbers and the set of all mc real numbers is contained properly in the set of weakly computable real numbers, where x is weakly computable if there are semi-computable real numbers y, z such that x = y + z. Furthermore, we show also an infinite hierarchy of mc real numbers.

Can partial indexings be totalized?

AbstractIn examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topologicalT0-spaces and study the problem in which cases the canonical numberings of such spaces can be totalized,i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded into a totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets.

The topological entropy of iterated piecewise affine maps is uncomputable

We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.

Minkowski Geometric Algebra of Complex Sets

A geometric algebra of point sets in the complex plane is proposed, based on two fundamental operations: Minkowski sums and products. Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention. Many interesting applications, interpretations, and connections arise from the geometric algebra based on these operations. Minkowski products with lines and circles are intimately related to problems of wavefront reflection or refraction in geometrical optics. The Minkowski algebra is also the natural extension, to complex numbers, of interval-arithmetic methods for monitoring propagation of errors or uncertainties in real-number computations. The Minkowski sums and products offer basic 'shape operators' for applications such as computer-aided design and mathematical morphology, and may also prove useful in other contexts where complex variables play a fundamental role – Fourier analysis, conformal mapping, stability of control systems, etc.

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (xs)s∈N of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers.

Preface: Volume 24

The usual implementation of real numbers in today's computers as floating point numbers has the well-known deficiency that most numbers can only be represented up to some fixed accuracy. Consequently, even the basic arithmetic operations cannot be performed exactly, leading to the ubiquitous round-off errors. This is a serious problem in all disciplines where high accuracy calculations are required. One of the ultimate goals of the theoretical and practical research in this area is to overcome these problems by improving the present implementations and algorithms or providing alternatives.This volume contains the proceedings of the Workshop on Real Number Computation which was held in June 19-20, 1998, in Indianapolis, Indiana, USA. The workshop preceded the IEEE Symposium on Logic in Computer Science (LICS).The meeting aimed to present an introduction to the interdisciplinary area of Real Number Computation. The subject is understood in a broad sense and covers various different fields like Recursion Theory, Interval Analysis, Computer Arithmetic, Semantics of Programming Languages, Number Theory, and Numerical Analysis. The workshop was meant to provide researchers from these different communities an opportunity to meet and exchange ideas.More than 30 participants from eight countries made this workshop a succesful event.Contributions to the proceedings were invited after the workshop from all participants and were sent to referees. Comments and suggestions were forwarded to the authors who thus had a chance to amend their texts. The final versions were accepted by the editors. We would like to take this opportunity to thank all referees for their time and cooperation, and Michael Mislove for his invitation to publish the proceedings in the ENTCS series.23rd August 1999Abbas Edalat, David Matula and Philipp Sünderhauf, Guest Editors

Type Two Theory of Effectivity and Real PCF

There have been many suggestions for what should be a computable real number or function. Some of these exhibited pathological properties [16]. Presently, research concentrates on domain theoretic approaches [4, 9] based on an idea of Scott [15], an application of Weihrauch's Type Two Theory of Effectivity [19, 20, 21] and an approach introduced by Pour-El and Richards for the more general case of Banach spaces [13].All these approaches are claimed to be equivalent, but only in a few cases proofs have been given [3].In this paper we show that a real number as well as a function on the reals are computable in Weihrauch's approach if and only if they are definable in Real PCF [5,6], an extension of the functional language PCF [12] by new ground type representing intervals which approximate real numbers.

Microcomputer based Harmonic Estimation of Thyristor Converter Waveform using Recursive Discrete Fourier Transform

The paper presents an application of recursive discrete Fourier transform (RDFT) method involving real number computations for estimating the harmonic magnitudes in the waveforms of a 6 pulse, full wave thyristor bridge circuit. The signal is modelled in state variable form as the output of a dynamic system. A dead-beat spectral observer in discrete time, operating at the Nyquist sampling rate estimates the harmonic amplitude components recursively since it performs one step per-sample updating. The estimates rapidly converge to yield the transform coefficient after exactly the requisite number of samples decided by Nyquist rate, even in the presence of numerical truncation errors. A microcomputer based instrumentation scheme for on-line implementation of the RDFT is suggested.

Lp‐Computability

AbstractIn this paper we investigate conditions for Lp‐computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ⊂ ℝq, we show that an Lp function f ∈ Lp(I) is LP‐computable if and only if (i) f is sequentially computable as a linear functional and (ii) the Lp‐modulus function of f is effectively continuous at the origin of ℝq.

A Real Number Structure that is Effectively Categorical

AbstractOn 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 basebby 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.