A Simple Characterization of the Computability of Real Functions

Domains and Processes, 2001

There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch's Type Two Theory of Effectivity or on domain-theoretic approaches, in which case the partial objects appearing during computations are made explicit. A further, more analysis-oriented line of research is based on Grzegorczyk's work. All these approaches are claimed to be equivalent, but not in all cases proofs have been given. In this paper it is shown that a real number as well as a real-valued function are computable in Weihrauch's sense if and only if they are definable in Escardó's functional language Real PCF, an extension of the language PCF by a new ground type for (total and partial) real numbers. This is exactly the case if the number is a computable element in the continuous domain of all compact real intervals and/or the function has a computable extension to this domain. For defining the semantics of the language Real PCF a full subcategory of the category of bounded-complete ω-continuous directed-complete partial orders is introduced and it is defined when a domain in this category is effectively given. The subcategory of effectively given domains contains the interval domain and is Cartesian closed. * The paper mainly contains results from the second author's diploma thesis [18] written under the supervision of the first author.

Computability

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.

Proc. of the Thirty-Second Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2017), 2017

—We investigate interrelationships among different notions from mathematical analysis, effective topology, and classical computability theory. Our main object of study is the class of computable functions defined over a bounded domain with the boundary being a left-c.e. number. We investigate necessary and sufficient conditions under which such function can be computably extended. It turns out that this depends on the behavior of the function near the boundary as well as on the class of left-c.e. numbers to which the boundary belongs, that is, how it can be constructed. Of particular interest a class of functions is investigated: sawtooth functions constructed from computable enumerations of c.e. sets.

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].

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.

Springer Proceedings in Mathematics & StatistiAdvances in Mathematical Logic: Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Kobe, Japan, September 2018, Selected, Revised Contributionscs., 2021

We investigate a sort of a unifying theory of computability of real functions, continuous or discontinuous, called here "irrational-based" (IB-) computability. All the examples which are computable in our various theories presented previously are IB-computable. The basic requirements of computability, the sequential computability and the effective continuity, are defined relative to computable irrational real sequences. The family of IB-computable functions is closed under IB-effective convergence. In order to certify the fruitfulness of IB-computability, quite a number of examples are presented.

Theoretical Computer Science, 2000

In this paper we extend computability theory to the spaces of continuous, upper semi-continuous and lower semi-continuous real functions. We apply the framework of TTE, Type-2 Theory of E ectivity, where not only computable elements but also computable functions on the spaces can be considered. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on di erent intuitive concepts of "e ectivity" and prove their equivalence. Computability of several operations on the function spaces is investigated, among others limits, mappings to open sets, images of compact sets and preimages of open sets, maximum and minimum values. The positive results usually show computability in all arguments, negative results usually express discontinuity. Several of the problems have computable but not extensional solutions. Since computable functions map computable elements to computable elements, many previously known results on computability are obtained as simple corollaries.

Proc. of the 5th International Workshop on De- velopments in Computational Models 2009, 2009

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real functions, characterizes it, and compares it with the corresponding notion over rational functions. Assuming continuity, the main conclusion is that there is a conceptual difference between polynomial time computation over the rationals and the reals manifested by the fact that there are polynomial time computable rational functions whose extensions to the reals are not polynomial time computable and vice versa.

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.