Algebraic Characterizations of Computable Analysis Real Functions

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.

Computability

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.

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.

Lecture Notes in Computer Science, 1982

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].

2013

There have been several different approaches of investigating computation over the real numbers. Among such computable analysis seems to be the most amenable to physical realization and the most widely accepted model that can also act as a unifying framework. Computable analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. A representation based approach to the field was then developed by C. Kreitz and K. Weihrauch [1983]. Any typical representation is based on using the rationals, a countable dense subset of the reals with finite representation, to approximate the real numbers. The purpose of this article is to investigate the transition phenomena between rational computation and the completion of such computation (in some given representation) that induces a computability concept over the reals. This gives deeper insights into the nature of real computation (and generally computation over infinite objects) and how it conceptually differs from the corresponding foundational notion of discrete computation. We have studied both the computability and the complexity-theoretic transition phenomena. The main conclusion is the finding of a conceptual gap between rational and real computation manifested, for instance, by the existence of computable rational functions whose extensions to the reals are not computable and vice versa. This gap can be attributed to two main reasons: (1) continuity and smoothness of real functions play necessary roles in their computability and complexity-theoretic properties whereas the situation is the contrary in rational computation and (2) real computation is approximate and hence robust whereas rational computation is exact and rigid. Despite these negative results, if we relax our framework to include relative computation, then we can bridge the rational-real computation gap relative to ∆2 oracles of the arithmetical hierarchy. We have shown that ∆2 is a tight bound for the rational-real computational equivalence. That is, if a continuous function over the rationals is computable, then its extension to the reals is computable relative to a ∆2 oracle; and vice versa. This result can also be considered an extension of the Shoenfield's Limit Lemma from classical recursion theory to the computable analysis context.

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.

Bulletin of the American Mathematical Society, 1989

We present a model for computation over the reals or an arbitrary (ordered) ring R. In this general setting, we obtain universal machines, partial recursive functions, as well as JVP-complete problems. While our theory reflects the classical over Z (e.g., the computable functions are the recursive functions) it also reflects the special mathematical character of the underlying ring R (e.g., complements of Julia sets provide natural examples of R. E. undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.

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

2006

Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis.

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

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.

Proc. of the International Workshop on Foundational and Practical Aspects of Resource Analysis (FOPARA ‘09), 2010

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955] as an approach for investigating computation over the real numbers. It is based on enhancing the Turing machine model by introducing oracles that allow the machine to access finitary portions of the real infinite objects. Classes of computable real functions have been extensively studied as well as complexity-theoretic classes of functions restricted to compact domains. However, much less have been done regarding complexity of arbitrary real functions. In this article we give a definition of polynomial time computability of arbitrary real functions. Then we present two main applications based on that definition. The first one, which has already been published, concerns the relationships between polynomial time real computability and the corresponding notion over continuous rational functions. The second application, which is a new contribution to this article, concerns the construction of a function algebra that captures polynomial time real computability.