Weak computability and representation of reals

Archive for Mathematical Logic, 2008

The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First we provide an alternative proof of the result from Campagnolo, Moore and Costa [3], which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Secondly, we build on that result to extend a result of Bournez and Hainry [1], which provided a function algebra for the C 2 real elementary computable functions; our result does not require the restriction to C 2 functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the operation of a Turing Machine using a function algebra. We avoid this simulation, using a technique we call lifting, which allows us to lift the classic result regarding the elementary computable functions to a result on the reals. The two new techniques bring a different perspective to these problems, and furthermore appear more easily applicable to other problems of this sort.

Lecture Notes in Computer Science, 2011

▸ For any class F of total functions in N, we define what it means for a real function to be conditionally F-computable. This notion extends the notion of uniform F-computability of real functions introduced in the paper [SkWeGe 10].

Journal of Logic and Computation, 2010

The results on the subject of the talk are obtained by the authors and Ivan Georgiev during the period June 2008-July 2009. Outline 1 Introduction The class M 2 F-computability of real numbers 2 Proving M 2-computability by using appropriate partial sums M 2-computability of the number e M 2-computability of Liouville's number A partial generalization 3 Stronger tools for proving M 2-computability of real numbers M 2-computable real-valued function with natural arguments Logarithmically bounded summation M 2-computability of sums of series 4 Applications of the stronger tools M 2-computability of π A generalization Some other M 2-computable constants Preservation of M 2-computability by certain functions 5 Conclusion 6 References

Computability

Lecture Notes in Computer Science, 2003

Let h : N → Q be a computable function. A real number x is h-monotonically computable (h-mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the h-monotonic computability of semi-computable real numbers which are limits of monotone computable sequences of rational numbers. Especially, we show a sufficient and necessary condition for the function h such that the h-monotonic computability is simply equivalent to the normal computability.

MLQ, 2005

Let h : N → Q be a computable function. A real number x is called h-monotonically computable (h-mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x h-monotonically in the sense that h(n)|x − xn| ≥ |x − xm| for all n and m > n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h : N → (0, 1) Q , we show that the class h-MC coincides with the classes of computable and semi-computable real numbers if and only if i∈N (1 − h(i)) = ∞ and the sum i∈N (1 − h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then h-MC = SC no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h-MC = c-MC. Finally, if h is monotone and unbounded, then h-MC contains all ω-mc real numbers which are g-mc for some computable function g.

2011

Computable analysis is an approach to real continuous computation that is based on extending the normal Turing machine model. It was introduced by A. Turing 1936, A. Grzegorczyk 1955, and D. Lacombe 1955. Since the introduction of Moore's real recursion theory in 1996 several classes of computable analysis functions have been characterized by functions algebras. On the one hand these algebraic characterizations provide a unifying theoretical framework that interconnects computable analysis with other approaches to real computation such as the GPAC and Moore's recursion theory. On the other hand they provide machine-independent characterizations and hence a different perspective on computable analysis, a perspective that is more intuitive and natural especially from the vantage point of the mathematical analysis community. In this article we give an introduction to the field of computable analysis and a survey of the different algebraic characterizations of computable analysis classes starting from the elementary functions up to the total computable ones passing through the Grzegorczyk hierarchy. Unfortunately, not much work has been done in characterizing the sub elementary, in particular the lower complexity-theoretic classes. Some of the author's published work in that latter direction are presented in this article. This includes the introduction of a function algebra that is an extension of the Bellantoni-Cook class. The extended class can exactly characterize discrete polynomial time computation, however, can only partially characterize polynomial time real computation. Furthermore, there exists a gap between the computation concept over the rational numbers and the corresponding one over the reals. This difference is illustrated by the existence of computable rational functions whose extension to the reals are not computable and vice versa. Understanding this gap might help us extend the algebraic discrete complexity classes to the reals. This article surveys many of the major results in the area and their implications.

Theoretical Computer Science, 2005

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema.

MLQ, 2002

A real number x is called k-monotonically computable (k-mc), for constant k > 0, if there is a computable sequence (x n) n∈N of rational numbers which converges to x such that the convergence is k-monotonic in the sense that k • |x − x n | ≥ |x − x m | for any m > n and x is monotonically computable (mc) if it is k-mc for some k > 0. x is weakly computable if there is a computable sequence (x s) s∈N of rational numbers converging to x such that the sum s∈N |x s − x s+1 | is finite. In this paper we show that all mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.

Theoretical Computer Science, 2002

We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

Lecture Notes in Computer Science, 2004

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.

Electronic Notes in Theoretical Computer Science, 2007

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure-Dedekindcomplete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computability of real numbers. However, the primitive recursive (p.r., for short) versions of these representations can lead to different notions of p.r. real numbers. Several interesting results about p.r. real numbers can be found in literatures. In this paper we summarize the known results about the primitive recursiveness of real numbers for different representations as well as show some new relationships. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers.

Theoretical Computer Science, 2006

A real x is called h-bounded computable, for some function h : N → N, if there is a computable sequence (x s) of rational numbers which converges to x such that, for any n ∈ N, at most h(n) non-overlapping pairs of its members are separated by a distance larger than 2 −n. In this paper we discuss properties of h-bounded computable reals for various functions h. We will show a simple sufficient condition for a class of functions h such that the corresponding h-bounded computable reals form an algebraic field. A hierarchy theorem for h-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h-bounded computability for special functions h.

MLQ, 2007

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure-Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if "computable" is replaced by "primitive recursive" (p. r., for short), these definitions lead to a number of different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b-adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers.

Theoretical Computer Science, 2005

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk