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The denotational semantics approach to the semantics of programming languages understands the language constructions by assigning elements of mathematical structures to them. The structures form so-called categories of domains and the study of their closure properties is the subject of domain theory [Sco70,Sco82,Plo83a,GS90,AJ94].
Typically, categories of domains consist of suitably complete partially ordered sets together with continuous maps. But, what is a category of domains? Our aim in this thesis is to answer this question by axiomatising the categorical structure needed on a category so that it can be considered a category of domains. Criteria required from categories of domains can be of the most varied sort. For example, we could ask them to
have fixed-point operators for endomorphisms and endofunctors;
have a rich collection of type constructors: coproducts, products, exponentials, powerdomains, dependent types, polymorphic types, etc;
have a Stone dual providing a logic of observable properties [Abr87, Vic89,Zha91];
have only computable maps [Sco76,Smy77,McC84,Ros86,Pho90a].
The criteria adopted here will be quite modest but rich enough for the denotational semantics of deterministic programming languages. For us a category of domains will be a category with the structure necessary to support the interpretation of the metalanguage FPC (a type theory with sums, products, exponentials and recursive types). And our axiomatic approach will aim not only at clarifying the categorical structure needed on a category for doing domain theory but also at relating such mathematical criteria with computational criteria.
This thesis is an investigation into axiomatic categorical domain theory as needed for the denotational semantics of deterministic programming languages.
To provide a direct semantic treatment of non-terminating computations, we make partiality the core of our theory. Thus, we focus on categories of partial maps. We study representability of partial maps and show its equivalence with classifiability. We observe that, once partiality is taken as primitive, a notion of approximation may be derived. In fact, two notions of approximation, contextual approximation and specialisation, based on testing and observing partial maps are considered and shown to coincide. Further we characterise when the approximation relation between partial maps is domain-theoretic in the (technical) sense that the category of partial maps Cpo-enriches with respect to it.
Concerning the semantics of type constructors in categories of partial maps, we present a characterisation of colimits of diagrams of total maps; study order-enriched partial cartesian closure; and provide conditions to guarantee the existence of the limits needed to solve recursive type equations. Concerning the semantics of recursive types, we motivate the study of enriched algebraic compactness and make it the central concept when interpreting recursive types. We establish the fundamental property of algebraically compact categories, namely that recursive types on them admit canonical interpretations, and show that in algebraically compact categories recursive types reduce to inductive types. Special attention is paid to Cpo-algebraic compactness, leading to the identification of a 2-category of kinds with very strong closure properties.
This article discusses the nature of transcontextual references to time in the light of one composition, Kaija Saariaho's radiophonic work Stilleben.The term transcontextuality indicates that a sound has a dual meaning, which refers both to the musical context created by the composer and to its original, natural or cultural context. The concept of transcontextuality is particularly rewarding because a substantial part of Stilleben's sound material is drawn from an earlier composition by Saariaho, Lichtbogen. Thefocus of the study is in defining the ways the composition refers to time outside of the musical time of Stilleben itself. These references are divided into two categories: (i) references to the musical time of Lichtbogen, and (ii) references to the passage of time in a non-musical context. The first category leads to a music analytical approach where dominance/subordination relations of the time structures of the two compositions in question are studied. It is concluded that the temporal structure of Stilleben is defined by the temporality of Lichtbogen onvarious structural levels. The second category gives rise to reasoning with respect to narrative interpretation: environmental sounds convey stories and thus refer directly to the passage of time.
The theme for this issue of Organised Sound is the Time Domain - a fundamental concern for composers, artists and engineers working with the musical application of technology. The range of material encompassed by this theme is vast. In fact many would say that it is impossible to divorce any aspect of musical work from its existence within the time domain. Consequently, time is likely to be the focus of future articles in Organised Sound.
The design and construction of an organised sound space to support information representations in the human–computer interface is described. The design of the sound space is guided by four principles which match perceptual structure with data structure to improve natural comprehension of an auditory display. These principles – completeness, comprehensibility, consistency, and cohesiveness – have been generalised from the use of colourdisplays in scientific visualisation. The choice of perceptual parameters to represent different types of data is informed by the body of psychoacoustic literature. The raw material for the construction of the sound space is the McGill University Master Samples (MUMS) palette of musical instrument samples. This is an important choice because this reference resource enables reproduction and confirmation of the results. The construction was carried out in four stages – the ‘pedestal’, the ‘skin’, the ‘skeleton’ and the ‘flesh’. The pedestal consists of eight equally discriminable timbres organised in a circle by perceptual similarity. The skin is the boundary of variation in the space, defining the limits of dynamic range for pitch and brightness at each timbre. The skeleton characterises the internal behaviour of the space at a number of perceptually measured points. The flesh is a continuous medium moulded to the skeleton and skin, realised by a three-dimensional (3D) regularised linear spline interpolation. The concrete realisation of the sound space can be investigated through a user interface, called the GamutExplorer. Colour visualisations of slices and wireframe views of the 3D space can be chosen, and sounds can be picked with a mouse.
Physical modelling of musical instruments is an exciting new paradigm in digital sound synthesis. The basic idea is to imitate the sound production mechanism of an acoustic musical instrument using a computer program. The sound produced by such a model will automatically resemble that of the real instrument, if the model has been devised in a proper way. In this article we review the history and present techniques of physical modelling. It appears that the many seemingly very different modelling methods try to achieve the same result: to simulate the solutions of the wave equation in a simplified manner. We concentrate on the digital waveguide modelling technique which has gained much popularity among both researchers and engineers in the music technology industry. The benefits and drawbacks of the new technology are considered, and concurrent research topics are discussed. The physical modelling approach offers many new applications, especially in the fields of multimedia and virtual reality.
The purpose of Discovery Strategy is to define compositional methods which can lead first-time listeners to the musical substance of an electroacoustic work. Discovery Strategy provides a steering mechanism which directs first-time listeners towards identifying features that lie beneath the surface of the music. A key component of this mechanism is the skilful integration of familiar sounds into more abstract contexts. The implementation of the strategy rests on a consideration of how different listeners perceive music through time, and how their perception changes over repeated listenings. The tape work Undercurrents (Field 1994) follows these principles, demonstrating that it is not necessary to compromise structural sophistication while pursuing Discovery Strategy methods.
The use of video as a performance medium, concentrating on interactive digital video used synchronously with realtime interactive audio, is outlined. Specifically, the author's experiments with realtime video in Macromind Director and in Opcode's Max are explored, working with motion-sensing systems in performance environments. A number of experimental pieces using sonic and visual environments driven by realtime motion-sensing performance systems are described, using Max to re-map incoming sensing data and using Macromind Director to control MIDI sequences, digital sound files, and digital video in realtime. Some of the compositional challenges presented by realtime systems are also explored, particularly the compositional issues arising from the addition of video into the (real)time domain. The author's current work in the development of a CD-ROM published in 1996 is also explored. The CD-ROM attempts to provide a genuine degree of interaction significantly more sophisticated than the usual point-and-click navigation, allowing the viewer a degree of creativity in his or her interaction with the material.
Many software packages for computer music encourage the composer to take either a time domain approach or a frequency domain approach. This paper examines the possibilities afforded by recent software developments of working at the intersection of these two domains. It investigates the relationship between the FOF algorithm, originally used in the CHANT program, and more traditional approaches to granular synthesis, and considers how they can be combined. The author's compositions are used as illustrations of these techniques. The significance of using the FOF algorithm in granulating sound files is explained (FOG). Methods of using and controlling the FOG unit-generator are explained. Compositional and aesthetic issues arising from working with sound at this ambiguous intersection are investigated.
Figures B.I to B.8 show the test environments for the exploration experiments. Two figures are given for each environment. The first diagram shows the walls and objects in the environment. It also shows the positions and orientations from which exploration experiments were started. The second diagram for shows the ‘ideal’ free-space map which would result from complete knowledge of the objects in the environment.
Experience with human control of the exploration process suggested that map quality could be increased rapidly in the early stages of exploration by heading into open regions of space instead of staying close to one of the walls (Section 15.4). The ‘Longest Lines’ strategy described in this chapter was motivated by this observation. The essential idea is to perform a full sensor scan and head in the direction of the longest reading. As many steps as possible are then taken in that direction until an obstacle is encountered. The algorithm then continues by heading in the direction of the longest reading from this new position.
This strategy shares with wall-following the fact that it is totally reactive. Navigational decisions are made solely on the basis of the latest sensor readings.
Section 16.2 gives the details of the implementation and Section 16.3 compares the results to those of Wall-Following and Supervised Wall-Following. Section 16.4 summarises the experimental results and considers the strengths and weaknesses of the strategy.
16.2 Implementation
The strategy, as described in the previous section, is straightforward. The only slight complication is the problem of multiple reflections. Wall-following used the shortest range readings from each viewpoint; multiple reflections were not a problem because they typically cause long range readings. On the other hand, the ‘Longest Lines’ strategy is particularly interested in the long readings. It is therefore necessary to acknowledge the likelihood of multiple reflections and to compensate for them.