Computational complexity of real functions

AbstractRecursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects. We investigate the computational complexity of real functions using the methods of recursive function theory. Partial recursive real functions are defined and their domains are characterized as the recursively open sets. We define the time complexity of recursive real continuous functions and show that the time complexity and the modulus of uniform continuity of a function are closely related. We study the complexity of the roots and the differentiability of polynomial time computable real functions. In particular, a polynomial time computable real function may have a root of arbitrarily high complexity and may be nowhere differentiable. The concepts of the space complexity and nondeterministic computation are used to study the complexity of the integrals and the maximum values of real functions. These problems are shown to be related to the “P=?NP” and the “P=?PSPACE” questions

Continuous-time computation with restricted integration capabilities

AbstractRecursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals, linear differential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results in this paper, combined with earlier results, suggest that there is a close connection between analog complexity classes and subrecursive classes, at least in the region between FLINSPACE and the primitive recursive functions

Elementary recursive complexity results in real algebraic geometry (Women in Mathematics)

I shall discuss two important results in real algebraic geometry - quantifier elimination, proving that the projection of a semi-algebraic set is semi-algebraic - Hilbert 17th problem, proving that a non negative polynomial is always a sum of squares of rational functions from the point of view of effectivity and complexity. The two problems look at first sight totally un related at all but it turns out that modern computer algebra techniques play a key role in proving elementary recursive complexity results for both these problems

Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions

Accepted for publication in International Journal of Unconventional ComputingInternational audienceRecursive analysis is the most classical approach to model and discuss computations over the real numbers.Recently, it has been shown that computability classes of functions in the sense of recursive analysis can be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of R-recursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomial-time computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Informational aspects of the haptic stimulation by the light for correction of the human functional state

Introduction. The study of the laws and principles of information processes in the biological systems of the human body in extreme forms of its activities and the development of the theory of medical information systems of such appointment, taking into account the status and trends of convergence of society, ecosystems and technology become very relevant. This state of affairs makes it possible to affirm that it is an actual scientific and applied problem of radical change of the existing paradigm of designing information systems. The purpose of the article is to specify the informational aspects of low intensity, haptic stimulation by the light, which is essential for correction of the functional state of an organism of the human being, who works in extreme conditions, to develop and study such methods and systems. Methods. Analyses of requirements, functions and systems for designing synthesis of information technologies and the control biotechnical system of correction of the functional state of an organism of the human, who works in extreme conditions. The theoretical and experimental dependences between the stimulation energy of light emission diode (LED) and the energy are transferred through the layered bio media design. Mathematical modelling and computational simulation. Comparison of these real and model data. Results. The base aspects requirements, functions and systems for designing synthesis of information technologies and the control biotechnical system of correction of the functional state of an organism of the human, who works in extreme conditions, low intensity, haptic stimulation by the light are defined. The methods for determining of intensity I0 of light emission diode, recursive expression , and formula for coefficient Cm , where M — quantity of bio media layers were developed. The bridges, which connects Maxwell’s phenomenological theory with the atomistic theory of matter and optics, were used. Computer simulation studies have confirmed the specification of requirements, functional and structural schemas of biotechnical system. Conclusions. Thanking to specification of requirements possibility-using recursive determining of the light flux intensity after every bio media layer was got. Under the effect of recurstion low computation complexity was caused. Information technology means (for automation optimal control) of the human state under external influences on the organism was developed. Further study to confirm statistical significance in representative samples of observations was opened

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page