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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
Bicompleteness in the Category of Partial Graphs with Total Homomorphisms
"... Category Theory is becoming an useful tool to formalize abstract concepts making easy to construct proofs and investigate properties while graphs are commonly used to model systems. Partiality is a important mathematical concept used in Mathematics and Computer Science. In this paper we define a cat ..."
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Category Theory is becoming an useful tool to formalize abstract concepts making easy to construct proofs and investigate properties while graphs are commonly used to model systems. Partiality is a important mathematical concept used in Mathematics and Computer Science. In this paper we define a category where objects are partial graphs whose arcs may have source and/or target nodes undefined and morphisms are total homomorphisms of
*autonomous categories, Unique decomposition categories.
"... We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of ..."
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We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of Abramsky’s GoI situations–ones based on Unique Decomposition Categories (UDC’s)–exactly captures Girard’s functional analytic models in his first GoI paper, including Girard’s original Execution formula in Hilbert spaces, his notions of orthogonality, types, datum, algorithm, etc. Here we associate to a UDCbased GoI Situation a denotational model (a ∗autonomous category (without units) with additional exponential structure). We then relate this model to some of the standard GoI models via a fullyfaithful embedding into a doublegluing category, thus connecting up GoI with earlier Full Completeness
A Categorical Model for the Geometry of Interaction Abstract
"... We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via ..."
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We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.
Towards a Typed Geometry of Interaction Abstract
"... Girard’s Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cutelimination. We introduce a typed version of GoI, called Multiobject GoI (MGoI) for multiplicative linear logic without units in categories which include previous (untyped) GoI models, as well ..."
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Girard’s Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cutelimination. We introduce a typed version of GoI, called Multiobject GoI (MGoI) for multiplicative linear logic without units in categories which include previous (untyped) GoI models, as well as models not possible in the original untyped version. The development of MGoI depends on a new theory of partial traces and trace classes, as well as an abstract notion of orthogonality (related to work of Hyland and Schalk.) We develop Girard’s original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of Cut Elimination. We prove Soundness and Completeness Theorems for the MGoI interpretation in partially traced categories with an orthogonality. Moreover, as an application of our MGoI interpretation, we prove a completeness theorem for the original untyped GoI interpretation of MLL in a traced unique decomposition category.
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
CTCS 2004 Preliminary Version From Geometry of Interaction to Denotational Semantics
"... This is a preliminary version. The final version will be published inElectronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs ..."
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This is a preliminary version. The final version will be published inElectronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs