One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[n] : n ∈ ω) approximating α and (b[n] : n ∈ ω) approximating β and a positive constant C such that for all n, C(α − a[n]) ≥ (β − b[n]). See [R. M. Solovay, Draft of a Paper (or Series of Papers) on Chaitin’s Work, manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1974, p. 215] and [G. (...) J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350–359]. We show that every recursively enumerable random real dominates all other recursively enumerable reals. We conclude that the recursively enumerable random reals are exactly the Ω-numbers [G. J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350–359]. Second, we show that the sets in a universal Martin-Lof test for randomness have random measure, and every recursively enumerable random number is the sum of the measures represented in a universal Martin-Lof test. (shrink)

We propose a lexical account of event nouns, in particular of deverbal nominalisations, whose meaning is related to the event expressed by their base verb. The literature on nominalisations often assumes that the semantics of the base verb completely defines the structure of action nominals. We argue that the information in the base verb is not sufficient to completely determine the semantics of action nominals. We exhibit some data from different languages, especially from Romance language, which show that nominalisations focus (...) on some aspects of the verb semantics. The selected aspects, however, seem to be idiosyncratic and do not automatically result from the internal structure of the verb nor from its interaction with the morphological suffix. We therefore propose a partially lexicalist approach view of deverbal nouns. It is made precise and computable by using the Montagovian generative lexicon, a type theoretical framework introduced by Bassac, Mery and Retoré in this journal in 2010. This extension of Montague semantics with a richer type system easily incorporates lexical phenomena like the semantics of action nominals in particular deverbals, including their polysemy and felicitous copredications. (shrink)

We propose a lexical account of event nouns, in particular of deverbal nominalisations, whose meaning is related to the event expressed by their base verb. The literature on nominalisations often assumes that the semantics of the base verb completely defines the structure of action nominals. We argue that the information in the base verb is not sufficient to completely determine the semantics of action nominals. We exhibit some data from different languages, especially from Romance language, which show that nominalisations focus (...) on some aspects of the verb semantics. The selected aspects, however, seem to be idiosyncratic and do not automatically result from the internal structure of the verb nor from its interaction with the morphological suffix. We therefore propose a partially lexicalist approach view of deverbal nouns. It is made precise and computable by using the Montagovian generative lexicon, a type theoretical framework introduced by Bassac, Mery and Retoré in this journal in 2010. This extension of Montague semantics with a richer type system easily incorporates lexical phenomena like the semantics of action nominals in particular deverbals, including their polysemy and felicitous copredications. (shrink)

The importance of non-codifying DNA polymorphism for the administration of justice is now well known. In Spain, however, this type of test has given rise to questions in recent years: Should consent be obtained before biological samples are taken from an individual for DNA analysis? Does society perceive these techniques and methods of analysis as being reliable? There appears to be lack of knowledge concerning the basic norms that regulate databases containing private or personal information and the protection that information (...) of this type must be given. This opinion survey and the subsequent analysis of the results in ethical terms may serve to reveal the criteria and the degree of information that society has with regard to DNA databases. In the study, 73.20% of the population surveyed was in favour of specific legislation for computer files in which DNA analysis results for forensic purposes are stored. (shrink)

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

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore how (...) the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (shrink)

In the first author's thesis [10], a sequential language, LRT, for real number computation is investigated. That thesis includes a proof that all polynomials are programmable, but that work comes short of giving a complete characterization of the expressive power of the language even for first-order functions. The technical problem is that LRT is non-deterministic. So a natural characterization of its expressive power should be in terms of relations rather than in terms of functions. In [2], Brattka examines a (...) formalization of recursive relations in the style of Kleene's recursive functions on the natural numbers. This paper is an expanded version of [13] which establishes the expressive power of LRTp, a variant of LRT, in terms of Brattka's recursive relations. Because Brattka already did the work of establishing the precise connection between his recursive relations and Type 2 Theory of Effectivity, we thus obtain a complete characterization of first-order definability in LRTp. (shrink)

With the innovation of computer, virtual clothing has also emerged. This research mainly discusses the real-time animation complex of interactive clothing design based on computer simulation. In the process of realizing virtual clothing, the sample interpolation synthesis method is used, and the human body sample library is constructed using the above two methods first, and then, the human body model is obtained by interpolation calculation according to the personalized parameters. Building a clothing model is particularly important for the effect (...) of trying on. The clothing that needs to be displayed can be scanned and then input into the computer to build the model. The model can be directly built in 3DMAX and other software and then its surface texture can be mapped, or the clothing model can be directly built. The 3D model in the 3ds file is loaded by the loop body nested switch branch selection structure. Correspondingly, the write-back operation of 3ds files is similar. Just follow the general structure of the 3ds file and write the root block, version information block, edit information block, key frame information block, etc. to a brand new file in sequence. The main reason for this article to perform the 3ds file write-back operation is that, after the clothing model is dynamically simulated through the dynamic principle, the deformed key animation frame needs to be saved as a 3ds file so that it can be further imported into the 3DSMAX software and generated by the renderer, form high-quality picture information, and finally get high-definition animation video. In the CPU-GPU hybrid method, modules such as force calculation, collision processing, and position update use the GPU method, while overstretching is processed by the CPU method, making the overall performance 10 times higher than the pure CPU method. This research helps to promote the development of 3D virtual clothing design. (shrink)

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10, 3, 7] that every real of strictly positive effective Hausdorff (...) dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one). (shrink)

In [8], the third author defined a reducibility$\le _w^{\rm{*}}$that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals${\cal R}$lies strictly above certain related structures. In the present paper, we show that$\left \equiv _w^{\rm{*}}{\cal R}$. More generally, for the weak-looking structure${\cal R}$ℚconsisting of the real numbers with just the ordering and constants naming the rationals, allo-minimal expansions of${\cal R}$ℚare equivalent to${\cal R}$. Using this, we (...) show that for any analytic functionf,$\left \equiv _w^{\rm{*}}{\cal R}$. $is noto-minimal.). (shrink)

In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.

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. (shrink)

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that equation image. Note that equation image is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate equation image by μ. It is known that whenever A so presents α then A ≤wttα, (...) where ≤wtt denotes weak truth table reducibility, and that the wtt-degrees of presentations form an ideal ℐ in the computably enumerable wtt-degrees. We prove that any such ideal is equation image, and conversely that if ℐ is any nonempty equation image ideal in the computably enumerable wtt-degrees then there is a computable enumerable real α such that ℐ = ℐ. We also prove a kind of Rice Theorem for these ideals, namely that if the index set of such a equation image ideal is not empty or equal to ω then it is equation image-complete. (shrink)

Recent new paradigms of computation, based on biological and physical models, address in a radically new way questions of efficiency and challenge assumptions ...

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in (...) question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question. (shrink)

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅' of the probability that the output be in some set.

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \not\leq_T Y}$$\end{document}. A real X is relatively simple and above if there is a real Y (...) We prove that every nonempty \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above. (shrink)

Let h : ℕ → ℚ be a computable function. A real number x is called h-monotonically computable if there is a computable sequence of rational numbers which converges to x h-monotonically in the sense that h|x – xn| ≥ |x – xm| for all n andm > 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 → ℚ, we show that the class (...) h-MC coincides with the classes of computable and semi-computable real numbers if and only if Σi∈ℕ) = ∞and the sum Σi∈ℕ) is a computable real number, respectively. On the other hand, if h ≥ 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. (shrink)

In the age of digitization, the world seems to be reducible to a digital computer. However, mathematically, modern quantum field theories do not only depend on discrete, but also continuous concepts. Ancient debates in natural philosophy on atomism versus the continuum are deeply involved in modern research on digital and computational physics. This example underlines that modern physics, in the tradition of Newton’s Principia Mathematica Philosophiae Naturalis, is a further development of natural philosophy with the rigorous methods of mathematics, measuring, (...) and computing. We consider fundamental concepts of natural philosophy with mathematical and computational methods and ask for their ontological and epistemic status. The following article refers to the author’s book, “The Digital and the Real World. Computational Foundations of Mathematics, Science, Technology, and Philosophy.”. (shrink)

1. One theory or many? In 2004 a very interesting and readable article by Lenore Blum, entitled “Computing over the reals: Where Turing meets Newton,” appeared in the Notices of the American Mathematical Society. It explained a basic model of computation over the reals due to Blum, Michael Shub and Steve Smale (1989), subsequently exposited at length in their influential book, Complexity and Real Computation (1997), coauthored with Felipe Cucker. The ‘Turing’ in the title of Blum’s article refers (...) of course to Alan Turing, famous for his explication of the notion of mechanical computation on discrete data such as the integers. The ‘Newton’ there has to do to with the well known numerical method due to Isaac Newton for approximating the zeros of continuous functions under suitable conditions that is taken to be a paradigm of scientific computing. Finally, the meaning of “Turing meets Newton” in the title of Blum’s article has another, more particular aspect: in connection with the problem of controlling errors in the numerical solution of linear equations and inversion of matrices,Turing (1948) defined a notion of condition for the relation of changes in the output of a computation due to small changes in the input that is analogous to Newton’s definition of the derivative. The thrust of Blum’s 2004 article was that the BSS model of computation on the reals is the appropriate foundation for scientific computing in general. By way of response, two years later another very interesting and equally readable article appeared in the Notices, this time by Mark Braverman and Stephen Cook (2006) entitled “Computing over the reals: Foundations for scientific computing,” in which the authors argued that the requisite foundation is provided by a quite different “bit computation” model, that is in fact prima facie incompatible with the BSS model. The bit computation model goes back to ideas due to Stefan Banach and Stanislaw Mazur in the latter part of the 1930s, but the first publication was not made until Mazur (1963).. (shrink)

Scientists of many disciplines use theoretical models to explain and predict the dynamics of the world. They often have to rely on digital computer simulations to draw predictions fromthe model. But to deliver phenomenologically adequate results, simulations deviate from the assumptions of the theoretical model. Therefore the role of simulations in scientific explanation demands itself an explanation. This paper analyzes the relation between real-world system, theoretical model, and simulation. It is argued that simulations do not explain processes in the (...) real world directly. The way in which simulations help explaining real-world processes is conceived as indirect, mediated by the theoretical model. Simulacra are characterized further, and turn out to be a priori measurable. This gives a clue to a better understanding of the epistemic role of computer simulations in scientific research. (shrink)

The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence (...) and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic. (shrink)

Interpretation is the process whereby a hearer reasons to an interpretation of a speaker's discourse. The hearer normally adopts a credulous attitude to the discourse, at least for the purposes of interpreting it. That is to say the hearer tries to accommodate the truth of all the speaker's utterances in deriving an intended model. We present a nonmonotonic logical model of this process which defines unique minimal preferred models and efficiently simulates a kind of closed-world reasoning of particular interest for (...) human cognition. Byrne's “suppression” data (Byrne, 1989) are used to illustrate how variants on this logic can capture and motivate subtly different interpretative stances which different subjects adopt, thus indicating where more fine-grained empirical data are required to understand what subjects are doing in this task. We then show that this logical competence model can be implemented in spreading activation network models. A one pass process interprets the textual input by constructing a network which then computes minimal preferred models for (3-valued) valuations of the set of propositions of the text. The neural implementation distinguishes easy forward reasoning from more complex backward reasoning in a way that may be useful in explaining directionality in human reasoning. (shrink)

Interpretation is the process whereby a hearer reasons to an interpretation of a speaker's discourse. The hearer normally adopts a credulous attitude to the discourse, at least for the purposes of interpreting it. That is to say the hearer tries to accommodate the truth of all the speaker's utterances in deriving an intended model. We present a nonmonotonic logical model of this process which defines unique minimal preferred models and efficiently simulates a kind of closed-world reasoning of particular interest for (...) human cognition. Byrne's “suppression” data (Byrne, 1989) are used to illustrate how variants on this logic can capture and motivate subtly different interpretative stances which different subjects adopt, thus indicating where more fine-grained empirical data are required to understand what subjects are doing in this task. We then show that this logical competence model can be implemented in spreading activation network models. A one pass process interprets the textual input by constructing a network which then computes minimal preferred models for (3-valued) valuations of the set of propositions of the text. The neural implementation distinguishes easy forward reasoning from more complex backward reasoning in a way that may be useful in explaining directionality in human reasoning. (shrink)

We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any (...) Borel quotient of the reals reduces to it. (shrink)

Recording and manipulating neuronal ensemble activity is a key requirement in advanced neuromodulatory and behavior studies. Devices capable of both recording and manipulating neuronal activity brain-computer interfaces should ideally operate un-tethered and allow chronic longitudinal manipulations in the freely moving animal. In this study, we designed a new intracortical BCI feasible of telemetric recording and stimulating local gray and white matter of visual neural circuit after irradiation exposure. To increase the translational reliance, we put forward a Göttingen minipig model. The (...) animal was stereotactically irradiated at the level of the visual cortex upon defining the target by a fused cerebral MRI and CT scan. A fully implantable neural telemetry system consisting of a 64 channel intracortical multielectrode array, a telemetry capsule, and an inductive rechargeable battery was then implanted into the visual cortex to record and manipulate local field potentials, and multi-unit activity. We achieved a 3-month stability of the functionality of the un-tethered BCI in terms of telemetric radio-communication, inductive battery charging, and device biocompatibility for 3 months. Finally, we could reliably record the local signature of sub- and suprathreshold neuronal activity in the visual cortex with high bandwidth without complications. The ability to wireless induction charging combined with the entirely implantable design, the rather high recording bandwidth, and the ability to record and stimulate simultaneously put forward a wireless BCI capable of long-term un-tethered real-time communication for causal preclinical circuit-based closed-loop interventions. (shrink)

We consider two classical computability notions for functions mapping all computable real numbers to computable real numbers. It is clear that any function that is computable in the sense of Markov, i.e., computable with respect to a standard Gödel numbering of the computable real numbers, is computable in the sense of Banach and Mazur, i.e., it maps any computable sequence of real numbers to a computable sequence of real numbers. We show that the converse is (...) not true. This solves a long-standing open problem posed by Kushner. (shrink)

The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.

In the wake of the Corrupted Blood incident, which afflicted the massively multiplayer online computer role-playing game World of Warcraft in 2005, it has been suggested that both, the incident itself and massively multiplayer online computer role-playing games in general, can be utilised to inform and assist real-world epidemic and public health research. In this paper, I engage critically with these claims.

"The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...) of an algorithm as central to AI, whereas it is argued here that for the purpose of explaining mental capabilities the architecture of an intelligent system is more important than the concept of an algorithm, using the premise that what makes something intelligent is not what it does but how it does it. What needs to be explained is also unclear: Penrose thinks we all know what consciousness is and claims that the ability to judge Go "del's formula to be true depends on it. He also suggests that quantum phenomena underly consciousness. This is rebutted by arguing that our existing concept of "consciousness" is too vague and muddled to be of use in science. This and related concepts will gradually be replaced by a more powerful theory-based taxonomy of types of mental states and processes. The central argument offered by Penrose against the strong AI thesis depends on a tempting but unjustified interpretation of Goedel's incompleteness theorem. Some critics are shown to have missed the point of his argument. A stronger criticism is mounted, and the relevance of mathematical Platonism analysed. Architectural requirements for intelligence are discussed and differences between serial and parallel implementations analysed. (shrink)

In order to benefit from the advantages of localist coding, neural models that feature winner-take-all representations at the top level of a network hierarchy must still solve the computational problems inherent in distributed representations at the lower levels.

This commentary suggests: (1) experimentalists must expand their subject pools beyond university students; (2) the pollution created by deception would not be a problem if experimentalists fully used non-student subjects; (3) one-shot games remain important and repeated games should not ignore social learning; (4) economists need to take better control of context; and (5) using computers in experiments creates potential problems.

Virtual reality constitutes an alternative, effective, and increasingly utilized treatment option for people suffering from psychiatric and neurological illnesses. However, the currently available VR simulations provide a predetermined simulative framework that does not take into account the unique personality traits of each individual; this could result in inaccurate, extreme, or unpredictable responses driven by patients who may be overly exposed and in an abrupt manner to the predetermined stimuli, or result in indifferent, almost non-existing, reactions when the stimuli do not (...) affect the patients adequately and thus stronger stimuli are recommended. In this study, we present a VR system that can recognize the individual differences and readjust the VR scenarios during the simulation according to the treatment aims. To investigate and present this dynamically adaptive VR system we employ an Anxiety Disorder condition as a case study, namely arachnophobia. This system consists of distinct anxiety states, aiming to dynamically modify the VR environment in such a way that it can keep the individual within a controlled, and appropriate for the therapy needs, anxiety state, which will be called “desired states” for the study. This happens by adjusting the VR stimulus, in real-time, according to the electrophysiological responses of each individual. These electrophysiological responses are collected by an external electrodermal activity biosensor that serves as a tracker of physiological changes. Thirty-six diagnosed arachnophobic individuals participated in a one-session trial. Participants were divided into two groups, the Experimental Group which was exposed to the proposed real-time adaptive virtual simulation, and the Control Group which was exposed to a pre-recorded static virtual simulation as proposed in the literature. These results demonstrate the proposed system’s ability to continuously construct an updated and adapted virtual environment that keeps the users within the appropriately chosen state for approximately twice the time compared to the pre-recorded static virtual simulation. Thus, such a system can increase the efficiency of VR stimulations for the treatment of central nervous system dysfunctions, as it provides numerically more controlled sessions without unexpected variations. (shrink)

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 et (...) al. (J Complex 18:977–1000, 2002), which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Second, we build on that result to extend a result of Bournez and Hainry (Theor Comput Sci 348(2–3):130–147, 2005), which provided a function algebra for the ${\mathcal{C}}^2$ real elementary computable functions; our result does not require the restriction to ${\mathcal{C}}^2$ 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. (shrink)

A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the 0'-computable reals. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real x we assign a degree structure, the structure of all possible ways available to approximate x. So the main criterion for (...) such classification is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results. (shrink)

This paper questions the notion of creativity found in certain artworks produced with A.I. technologies. The artistic examples concerned are: The Giver of names by David Rokeby, Oscar by Catherine Ikam and Louis Fléri, and Emotion Vending Machine by Maurice Benayoun. These artworks were selected because they stand out for their autonomous behavior in front of the human public. In this context, creativity is revealed as a consequence of the functional autonomy, which is very typical of these pieces of art. (...) The intention is to establish a relationship between the notion of autonomy and that of creativity, reviewing their meanings and applications in philosophy and in the field of life science. (shrink)