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Showing papers presented at "Conference on Computability in Europe in 2007"


Journal ArticleDOI
01 Apr 2007
TL;DR: Call for papers for Special Issue of ACM Transactions on Multimedia Computing, Communications and Applications on Interactive Digital Television.
Abstract: Call for papers for Special Issue of ACM Transactions on Multimedia Computing, Communications and Applications on Interactive Digital Television

201 citations


Journal ArticleDOI
Kalle Jegers1
01 Jan 2007
TL;DR: An initial outline for a new model of pervasive player enjoyment is presented, that is, the pervasive gameflow model, which is described and discussed in terms of additions and elaborations to the general game flow model.
Abstract: Player enjoyment is perhaps the most important issue in successful game design, but so far it has not been addressed in the area of pervasive games. Departing from the general gameflow model, this article presents an initial outline for a new model of pervasive player enjoyment, that is, the pervasive gameflow model, which is described and discussed in terms of additions and elaborations to the general gameflow model. It is intended to serve as a departure point for further empirical studies on player enjoyment in pervasive games.

176 citations


Journal ArticleDOI
Eva Nieuwdorp1
01 Apr 2007
TL;DR: In the current nascent stage of research into pervasive gaming, in which attempts are being made to theorize this concept, an analytical overview of the use of the term pervasive is argued for to understand the different perspectives in which it is used.
Abstract: In this article I argue for a critical analysis and re-evaluation of the term pervasive. Due to ambiguous definitions in both the discourse on computing and gaming, this term has become theoretically entangled with others. This has led to multiple definitions, and also, in the case of gaming, to the neglect of the different perspectives in which a game can be pervasive. In the current nascent stage of research into pervasive gaming, in which attempts are being made to theorize this concept, I argue for an analytical overview of the use of the term pervasive in order to understand the different perspectives in which it is used.

124 citations


Journal ArticleDOI
01 Apr 2007
TL;DR: In this paper, the authors argue that a digital table is a conducive form factor for general co-located home gaming as it affords: (a) seating in collaboratively relevant positions that give all equal opportunity to reach into the surface and share a common view; (b) rich whole-handed gesture input usually seen only when handling physical objects; (c) the ability to monitor how others use space and access objects on the surface; and (d) communication with each other and interact on top of the surface via gestures and verbal utterance.
Abstract: There is a large disparity between the rich physical interfaces of co-located arcade games and the generic input devices seen in most home console systems. In this article we argue that a digital table is a conducive form factor for general co-located home gaming as it affords: (a) seating in collaboratively relevant positions that give all equal opportunity to reach into the surface and share a common view; (b) rich whole-handed gesture input usually seen only when handling physical objects; (c) the ability to monitor how others use space and access objects on the surface; and (d) the ability to communicate with each other and interact on top of the surface via gestures and verbal utterance. Our thesis is that multimodal gesture and speech input benefits collaborative interaction over such a digital table. To investigate this thesis, we designed a multimodal, multiplayer gaming environment that allows players to interact directly atop a digital table via speech and rich whole-hand gestures. We transform two commercial single-player computer games, representing a strategy and simulation game genre, to work within this setting.

118 citations


Journal ArticleDOI
01 Jan 2007
TL;DR: The technical realization of Epidemic Menace is explained, the evaluation methodologies used, and some evaluation results are explained, and experiences from two Epidemia Menace game events are described.
Abstract: Crossmedia games employ a wide variety of gaming interfaces based on stationary and mobile devices to facilitate different game experiences within a single game. This article presents the crossmedia game Epidemic Menace, introduces the game concept, and describes experiences from two Epidemic Menace game events. We also explain the technical realization of Epidemic Menace, the evaluation methodologies we used, and some evaluation results.

71 citations


Journal ArticleDOI
01 Apr 2007
TL;DR: This work presents some of the results of an MHP-based (Multimedia Home Platform) iTV field trial in Salzburg, Austria concerning usability issues and iTV usage, and discusses whether using iTV is an active or a passive user experience.
Abstract: How users experience interactive TV (iTV) is determined by various factors; usability is a major one. In industry, usability is often seen as the key element that determines acceptance of new technologies by users. We present some of the results of an MHP-based (Multimedia Home Platform) iTV field trial in Salzburg, Austria concerning usability issues and iTV usage.We give an introduction to the field trial and an overview of its methods to ensure usability of iTV services and to measure their use. We present results from a usability test, give design recommendations, and discuss whether using iTV is an active or a passive user experience.

46 citations


Journal ArticleDOI
01 Jul 2007
TL;DR: It is concluded that continuous treatment of the two-dimensional emotion space provides a metric for comparing and assessing emotion ratings, and that spatial interpolation in this space provided a viable method for predicting the effect of music on perceived emotion of film.
Abstract: This article presents quantitative and visual methods for the analysis of the effect of music on emotion perceived in film. We discover strong, visible, and quantifiable trends in the effect of music on perceived emotion of film. We perform studies using both selected classical and composed music segments annotated with diverse emotions, paired with ambiguous film clips. We collect and analyze viewers' ratings of stress, activity, and dominance in the silent film, and film with various music soundtracks. The results are mapped onto a three-dimensional emotion space for visual and quantitative analysis. Aggregate scatter plots and center of mass results are presented using this emotion space. We find that the center of mass of the perceived emotion of film and music combined is consistently situated on a path between the center of mass of the musicalone response and that of the film-alone response. Regression analysis based on this trajectory observation results in coefficients with high R2 values (R2 = 0.675 for stress, R2 = 0.817 for activity, and R2 = 0.813 for dominance) for the first study with classical music selections, and lower R2 values (R2 = 0.199 for stress, R2 = 0.405 for activity, and R2 = 0.660 for dominance) for the second study with composed music. We conclude that continuous treatment of the two-dimensional emotion space provides a metric for comparing and assessing emotion ratings, and that spatial interpolation in this space provides a viable method for predicting the effect of music on perceived emotion of film.

44 citations


Journal ArticleDOI
01 Apr 2007
TL;DR: The findings suggest that interactive TV applications provide support for education and entertainment for children and young people, as well as continuous education for all.
Abstract: Despite the criticism concerning the value of TV content, research reveals several worthwhile aspects -- one of them is the opportunity to learn. In this article we explore the characteristics of interactive TV applications that facilitate education and interactive entertainment. In doing so we analyze research methods and empirical results from experimental and field studies. The findings suggest that interactive TV applications provide support for education and entertainment for children and young people, as well as continuous education for all. In particular, interactive TV is especially suitable for (1) informal learning and (2) for engaging and motivating its audience. We conclude with an agenda for future interactive TV research in entertainment and education.

42 citations


Journal ArticleDOI
01 Apr 2007
TL;DR: It is found that hard of hearing viewers were significantly more positive about this style of captioning than deaf viewers and that some viewers believed that these augmentations were useful and enhanced their viewing experience.
Abstract: Television and film have become important equalization mechanisms for the dissemination and distribution of cultural materials. Closed captioning has allowed people who are deaf and hard of hearing to be included as audience members. However, some of the audio information such as music, sound effects, and speech prosody are not generally provided for in captioning. To include some of this information in closed captions, we generated graphical representations of the emotive information that is normally represented with nondialog sound. Eleven deaf and hard of hearing viewers watched two different video clips containing static and dynamic enhanced captions and compared them with conventional closed captions of the same clips. These viewers then provided verbal and written feedback regarding positive and negative aspects of the various captions. We found that hard of hearing viewers were significantly more positive about this style of captioning than deaf viewers and that some viewers believed that these augmentations were useful and enhanced their viewing experience.

39 citations


Journal ArticleDOI
01 Jul 2007
TL;DR: The VoodooIO Gaming Kit is presented, a real-time adaptable gaming controller that allows players to define and actively reconfigure their gaming space, making it appropriate to their personal preference and gaming needs.
Abstract: Existing gaming controllers are limited in their end-user configurability. As a complement to current game control technology, we present the VoodooIO Gaming Kit, a real-time adaptable gaming controller. We introduce the concept of appropriable gaming devices, which allow players to define and actively reconfigure their gaming space, making it appropriate to their personal preference and gaming needs. The technology and its conceived usage are illustrated through its application to two commercially available computer games, as well as through the results of a formal user study.

39 citations


Journal ArticleDOI
01 Apr 2007
TL;DR: Analysis of how young people respond to television and new media and how media is used in their everyday lives, in their social relations, and in building an identity suggests a shift is about to happen in their relationship to the media.
Abstract: With the ever-changing and ubiquitous media environment as a backdrop, this article analyzes how young people respond to television and new media and how media is used in their everyday lives, in their social relations, and in building an identity. The analysis is based on findings from a user study of 10 to15 year-olds in Norway. The respondents were recruited among active Internet users, and since they were early adopters of new technology, they can be considered a vanguard. Starting with a broad outline of some essential earlier studies on young people's use of media in Europe and in the United States, the results of this study are presented and ideas for further development are discussed. Media, and in particular visual and social media, play an increasingly important role in young people's lives. But a shift is about to happen in their relationship to the media; from being an audience and users to becoming participants and creators as well. This article is a contribution to the previous rather poor research on these ongoing changes.

Book ChapterDOI
18 Jun 2007
TL;DR: The operations are classified with respect to effective measurability in the Borel hierarchy and it turns out that most operations can be located in the first three levels of the hierarchy, or they are not even Borel measurable at all.
Abstract: We study the Borel complexity of topological operations on closed subsets of computable metric spaces. The investigated operations include set theoretic operations such as union and intersection, but also typical topological operations such as the closure of the complement, the closure of the interior, the boundary and the derivative of a set. These operations are studied with respect to different computability structures on the hyperspace of closed subsets. These structures include positive or negative information on the represented closed subsets. Topologically, they correspond to the lower or upper Fell topology, respectively, and the induced computability concepts generalize the classical notions of recursively enumerable (r.e.) or co-r.e. subsets, respectively. The operations are classified with respect to effective measurability in the Borel hierarchy and it turns out that most operations can be located in the first three levels of the hierarchy, or they are not even Borel measurable at all. In some cases the effective Borel measurability depends on further properties of the underlying metric spaces, such as effective local compactness and effective local connectedness.

Book ChapterDOI
18 Jun 2007
TL;DR: An algorithm that decides whether or not a regular language coincides with its primitive hairpin root is presented and a cubic time algorithm for computing the common ancestors of two given words is discussed.
Abstract: We define the hairpin reduction as the inverse operation of a formal operation on words and languages suggested by DNA biochemistry, namely the hairpin completion, introduced in [3] We settle the closure properties of some classes in the Chomsky hierarchy as well as some complexity classes under the non-iterated version of the hairpin reduction, in comparison with the hairpin completion Then an algorithm that decides whether or not a regular language coincides with its primitive hairpin root is presented Finally, we discuss a cubic time algorithm for computing the common ancestors of two given words This algorithm may be used also for computing the closest or farthest primitive hairpin root of a given word

Book ChapterDOI
18 Jun 2007
TL;DR: This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete SIERpinski triangle and nothing else, and proves that the standard discrete Siersiak triangle cannotstrictly self-assemble in the Tile Assembly Model.
Abstract: Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assemblyof discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannotstrictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of Winfree counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2005).

Book ChapterDOI
18 Jun 2007
TL;DR: This article describes liquid computing as a new framework for analyzing those types of computations that are usually carried out in biological organisms - either in the brain, or in the form of gene regulation within a single cell.
Abstract: This review addresses structural differences between that type of computation on which computability theory and computational complexity theory have focused so far, and those computations that are usually carried out in biological organisms (either in the brain, or in the form of gene regulation within a single cell). These differences concern the role of time, the way in which the input is presented, the way in which an algorithm is implemented, and in the end also the definition of what a computation is. This article describes liquid computing as a new framework for analyzing those types of computations that are usually carried out in biological organisms.

Journal ArticleDOI
01 Jan 2007
TL;DR: A system integrating highly interactive narrative structures in a real-time 3D environment based on a theoretical foundation of narrative and drama, an interactive drama engine (IDE) has been implemented that comprises an action calculus system, a text-generation system, and a behavior engine as well as an innovative adaptive user interface.
Abstract: Interactive fiction and adventure video games are narrative genres which provide the player with the option of acting as the main character of the story. However these genres do not fully match the expectations of their authors and readers because the player cannot deeply affect the storyline. This article describes a system integrating highly interactive narrative structures in a real-time 3D environment. Based on a theoretical foundation of narrative and drama, an interactive drama engine (IDE) has been implemented. It comprises an action calculus system, a text-generation system, a behavior engine, an animation engine as well as an innovative adaptive user interface. The IDE is demonstrated with two scenarios.

Journal ArticleDOI
Michael J. Darnell1
01 Apr 2007
TL;DR: The results show that overall the most common way people interacted with TV was in avoiding advertisements and finding new programming to watch when the current show ended.
Abstract: To understand how people really interact with digital TV and digital video recorder (DVR) systems, 10 people were recruited, 6 with DVRs and 4 with non-DVR digital cable or satellite TV systems. Video-recording equipment was set up in their homes to record their TV viewing. The person viewing as well as the TV content being viewed were recorded as the person watched TV. Afterwards, the recordings were reviewed and analyzed by the experimenter together with the viewer, who then described his or her behavior. The results show that overall the most common way people interacted with TV was in avoiding advertisements and finding new programming to watch when the current show ended. These results can be used to find ways to improve how users experience TV systems.

Journal ArticleDOI
01 Jan 2007
TL;DR: An initial set of criteria is proposed that can be used to evaluate how well a given planning algorithm fits IS systems and an analysis using available planning systems is presented.
Abstract: Interactive storytelling (IS) is an important research topic in the broader area of interactive digital entertainment. Research in IS focuses in the search for techniques that allow the creation of systems that can generate stories that change as a consequence of a player's actions. There have been numerous articles attempting to reach this goal by employing a planning algorithm as part of the solution, but they typically do not discuss in detail why a given planning algorithm was used. Given this reality, this article makes two main contributions. First, it proposes an initial set of criteria that can be used to evaluate how well a given planning algorithm fits IS systems. Second, on the basis of these criteria, it presents an analysis using available planning systems.

Book ChapterDOI
18 Jun 2007
TL;DR: It is shown that viruses can be classified as solutions of fixed point equations which are obtained from different versions of Kleene's second recursion theorem.
Abstract: We study computer virology from an abstract point of view. Viruses and worms are self-replicating programs, whose constructions are essentially based on Kleene's second recursion theorem. We show that we can classify viruses as solutions of fixed point equations which are obtained from different versions of Kleene's second recursion theorem. This lead us to consider four classes of viruses which various polymorphic features. We propose to use virus distribution in order to deal with mutations. Topics covered.Computability theoretic aspects of programs, computer virology.

Book ChapterDOI
18 Jun 2007
TL;DR: Much improved FPT algorithms for a large number of graph problems, for input of bounded max leaf number, are described, based on the polynomial-time extremal structure theory associated to the parametermax leaf number.
Abstract: In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidthinfluences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf numberof a connected graph Gis the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring or Hamilton Path or Minimum Dominating Set for graphs of bounded max leaf number? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input of bounded max leaf number, based on the polynomial-time extremal structure theory associated to the parameter max leaf number. (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach.

Book ChapterDOI
18 Jun 2007
TL;DR: It is proved that signal machines and the infinite-dimension linear BSS machines can simulate one another and be able to carry out any Turing-computation.
Abstract: geometrical computation naturally arises as acontinuous counterpart of cellular automata. It relies on signals(dimensionless points) traveling at constant speed in a continuousspace in continuous time. When signals collide, they are replacedby new signals according to some collision rules. This simpledynamics relies on real numbers with exact precision and is alreadyknown to be able to carry out any (discrete) Turing-computation.The Blum, Shub and Small (BSS) model is famous for computing overℝ (considered here as a ℝ unlimited register machine)by performing algebraic computations. We prove that signal machines (set of signals and correspondingrules) and the infinite-dimension linear (multiplications are onlyby constants) BSS machines can simulate one another.

Book ChapterDOI
18 Jun 2007
TL;DR: Two complexity-theoretic formulations of Bennett's logical depth are introduced: finite-state depth and polynomial-time depth; it is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth law holds, implying that deep sequences cannot be created easily from shallow sequences.
Abstract: This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depthand polynomial-time depth. It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth lawholds, implying that deep sequences cannot be created easily from shallow sequences. Furthermore, the E analogue of the halting language is shown to be polynomial-time deep, by proving a more general result: every language to which a nonnegligible subset of E can be reduced in uniform exponential time is polynomial-time deep.

Book ChapterDOI
18 Jun 2007
TL;DR: The L-derivative, however, is shown to be upper semi continuous with respect to the weak* topology, a result which is not known to hold for the Clarke gradient on infinite dimensional Banach spaces.
Abstract: We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. The values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. This is also the case for the Clarke generalised gradient. The L-derivative, however, is shown to be upper semi continuous with respect to the weak* topology, a result which is not known to hold for the Clarke gradient on infinite dimensional Banach spaces. We also formulate the notion of primitive maps dual to the L-derivative, an extension of Fundamental Theorem of Calculus for the L-derivative and a domain for computation of real-valued functions on a Banach space with a corresponding computability theory.

Book ChapterDOI
Eric Allender1
18 Jun 2007
TL;DR: This survey talk will focus on some of the remaining open questions dealing with graph reachability problems, and particular attention will be paid to planar directed graphs.
Abstract: There has been a great deal of progress in the fifteen years that have elapsed since Wigderson published his survey on the complexity of the graph connectivity problem [Wig92]. Most significantly, Reingold solved the longstanding question of the complexity of the s-tconnectivity problem in undirected graphs, showing that this is complete for logspace (L) [Rei05]. This survey talk will focus on some of the remaining open questions dealing with graph reachability problems. Particular attention will be paid to these topics: Reachability in planar directed graphs (and more generally, in graphs of low genus) [ADR05, BTV07]. Reachability in different classes of grid graphs [ABC+06]. Reachability in mangroves [AL98].

Journal ArticleDOI
01 Apr 2007
TL;DR: This article presents a new concept in t-learning, called entercation experiences, where TV programs are used as a hook to engage users in education, and develops an ontology for learning content based on the ADL SCORM (Sharable Content Object Reference Model) standard.
Abstract: IDTV (interactive digital TV) opens new learning opportunities where new forms of education are needed. In order to gain the participation of viewers, and overcome their typical passivity, it is essential to combine education and entertainment. In this article we present a new concept in t-learning, called entercation experiences, where we use TV programs as a hook to engage users in education. The key element in this approach is an ontology for learning content based on the ADL SCORM (Sharable Content Object Reference Model) standard. We have developed this ontology so that we can choose the appropriate elements to construct entercation experiences, and thus provide learning personalization.

Book ChapterDOI
18 Jun 2007
TL;DR: It is proved that the long-standing conjecture [Surjectivity $\Rightarrow$ Density of thePeriodic Orbits (DPO)] is equivalent to the conjecture [TopologicalMixing $\ rightarrow $ DPO].
Abstract: We consider the family of all the Cellular Automata (CA) sharingthe same local rule but have different memory. This family containsalso all the CA with memory m≤ 0 (one-sided CA) whichcan act both on Aℤand on Aℕ. We study several set theoretical andtopological properties for these classes. In particular weinvestigate if the properties of a given CA are preserved when weconsider the CA obtained by changing the memory of the original one(shifting operation). Furthermore we focus our attention to theone-sided CA acting on Aℕstarting fromthe one-sided CA acting on Aℤand havingthe same local rule (lifting operation). As a particularconsequence of these investigations, we prove that thelong-standing conjecture [Surjectivity $\Rightarrow$ Density of thePeriodic Orbits (DPO)] is equivalent to the conjecture [TopologicalMixing $\Rightarrow$ DPO].

Journal ArticleDOI
01 Apr 2007
TL;DR: It is concluded that variable text influences visual attention, memory, and cognitive workload more than pictorial information does; this conclusion supports the limited capacity model.
Abstract: This article explores how viewers respond to and remember extra information shown on television. Its purpose is to investigate the effects of typical print content on TV viewers' visual attention, message processing ability, and cognitive workload when print content appears simultaneously with video. Test subjects looked at various video clips simultaneously with print, while variable text and graphics changed the structural complexity of the message. Data of psychophysiological measurements such as eye-tracking and skin-conductivity were obtained during the test; memory and cognitive workload measurements were self-reported afterwards. The results show that visual attention was equally balanced between video and print content for lengthy texts, whereas short texts with an increasing number of images caused subjects to look longer and more often at the video. Both video and text recognition were better on lengthy texts. Subjective cognitive workload data shows that lengthy texts resulted in higher ratings for psychological stress than short ones did. We conclude that variable text influences visual attention, memory, and cognitive workload more than pictorial information does; this conclusion supports the limited capacity model.

Book ChapterDOI
18 Jun 2007
TL;DR: This work surveys some work concerned with small universal Turing machines, cellular automata, and other simple models of computation and finds that Rule 110, a well-known elementary cellular automaton, is also efficiently universal.
Abstract: We survey some work concerned with small universal Turing machines, cellular automata, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. As a related result we also find that Rule 110, a well-known elementary cellular automaton, is also efficiently universal. We also mention some new universal program-size results, including new small universal Turing machines and new semi-weakly universal Turing machines. We then discuss some ideas for future work arising out of these, and other, results.

Book ChapterDOI
18 Jun 2007
TL;DR: A jump inversion theorem for the degree spectra is presented and a structure is constructed as a Marker's extension of such that the jump spectrum of is exactly the degree spectrum of and the degree Spectrum of is a subset of thejump spectrum of.
Abstract: A jump inversion theorem for the degree spectra is presented. For a structure which degree spectrum is a subset of the jump spectrum of a structure , a structure is constructed as a Marker's extension of such that the jump spectrum of is exactly the degree spectrum of and the degree spectrum of is a subset of the degree spectrum of .

Book ChapterDOI
18 Jun 2007
TL;DR: The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930's working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit.
Abstract: The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930's working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alonewho achieved the characterization, in the opinion of Godel. We also explore Turing's oracle machine and its analogous properties in analysis.