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Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by S ..."
Abstract

Cited by 44 (10 self)
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...
Heterotic Computing
"... Abstract. Nonclassical computation has tended to consider only single computational models: neural, analog, quantum, etc. However, combined computational models can both have more computational power, and more natural programming approaches, than such ‘pure ’ models alone. Here we outline a propose ..."
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Cited by 1 (1 self)
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Abstract. Nonclassical computation has tended to consider only single computational models: neural, analog, quantum, etc. However, combined computational models can both have more computational power, and more natural programming approaches, than such ‘pure ’ models alone. Here we outline a proposed new approach, which we term heterotic computing 4. We discuss how this might be incorporated in an accessible refinementbased computational framework for combining diverse computational models, and describe a range of physical exemplars (combinations of classical discrete, quantum discrete, classical analog, and quantum analog) that could be used to demonstrate the capability. 1
A Framework for Heterotic Computing
"... Computational devices combining two or more different parts, one controlling the operation of the other, for example, derive their power from the interaction, in addition to the capabilities of the parts. Nonclassical computation has tended to consider only single computational models: neural, anal ..."
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Computational devices combining two or more different parts, one controlling the operation of the other, for example, derive their power from the interaction, in addition to the capabilities of the parts. Nonclassical computation has tended to consider only single computational models: neural, analog, quantum, chemical, biological, neglecting to account for the contribution from the experimental controls. In this position paper, we propose a framework suitable for analysing combined computational models, from abstract theory to practical programming tools. Focusing on the simplest example of one system controlled by another through a sequence of operations in which only one system is active at a time, the output from one system becomes the input to the other for the next step, and vice versa. We outline the categorical machinery required for handling diverse computational systems in such combinations, with their interactions explicitly accounted for. Drawing on prior work in refinement and retrenchment, we suggest an appropriate framework for developing programming tools from the categorical framework. We place this work in the context of two contrasting concepts of “efficiency”: theoretical comparisons to determine the relative computational power do not always reflect the practical comparison of real resources for a finitesized computational task, especially when the inputs include (approximations of) real numbers. Finally we outline the limitations of our simple model, and identify some of the extensions that will be required to treat more complex interacting computational systems. 1