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Towards a typed geometry of interaction
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.
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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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