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Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
A homotopical algebra of graphs related to zeta series
 Homology, Homotopy and its Applications 10 (2008), 1–13. MR2506131 (2010f:18010). [C95] Crans, Sjoerd E. Quillen
"... Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equiv ..."
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Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two FreydKelly factorization systems based on Folding, Injecting, and Covering graph morphisms. 0. Introduction. In this paper we develop a notion of homotopy within graphs, and demonstrate its relevance to the study of zeta series and spectrum of a finite graph. We will work throughout with a particular category of graphs, described in Section 1 below. Our graphs will be directed and possibly infinite, with loops and multiple arcs allowed.
Applying category theory to improve the performance of a neural architecture
 Neurocomputing, (Article in
, 2009
"... NOTICE: this is the author’s version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been m ..."
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NOTICE: this is the author’s version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. The article is now in press. A recentlydeveloped mathematical semantic theory explains the relationship between knowledge and its representation in connectionist systems. The semantic theory is based upon category theory, the mathematical theory of structure. A product of its explanatory capability is a set of principles to guide the design of future neural architectures and enhancements to existing designs. We claim that this mathematical semantic approach to network design is an effective basis for advancing the state of the art. We offer two experiments to support this claim. One of these involves multispectral imaging using data from a satellite camera.
Monstrous moonshine and the classification of CFT
 Lectures given in Istanbul
, 1998
"... In these notes we give an introduction both to Monstrous Moonshine and to the classification of rational conformal field theories, using this as an excuse to explore several related structures and go on a little tour of modern math. We will discuss Lie algebras, modular functions, the finite simple ..."
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In these notes we give an introduction both to Monstrous Moonshine and to the classification of rational conformal field theories, using this as an excuse to explore several related structures and go on a little tour of modern math. We will discuss Lie algebras, modular functions, the finite simple group classification, vertex operator algebras, Fermat’s Last
Quantum picturalism
, 2009
"... Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to dis ..."
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Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’? We claim that the quantum mechanical formalism doesn’t support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is ‘lowlevel’. In this review we present steps towards a diagrammatic ‘highlevel ’ alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the nocloning theorem, and phenomena such as quantum teleportation. As a logic, it supports ‘automation’: it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required stepstone towards a deeper conceptual understanding of quantum theory, as well as its
Modification of the ART1 Architecture Based on Category Theoretic Design Principles
"... Abstract — Many studies have addressed the knowledge representation capability of neural networks. A recentlydeveloped mathematical semantic theory explains the relationship between knowledge and its representation in connectionist systems. The theory yields design principles for neural networks wh ..."
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Abstract — Many studies have addressed the knowledge representation capability of neural networks. A recentlydeveloped mathematical semantic theory explains the relationship between knowledge and its representation in connectionist systems. The theory yields design principles for neural networks whose behavioral repertoire expresses any desired capability that can be expressed logically. In this paper, we show how the design principle of limit formation can be applied to modify the ART1 architecture, yielding a discrimination capability that goes beyond vigilance. Simulations of this new design illustrate the increased discrimination ability it provides for multispectral image analysis. 0780390482/05/$20.00 ©2005 IEEE
CATEGORICAL ANALYSIS
, 2001
"... Abstract. We propose the categorification of algebraic analysis in terms of a specific 2category, called here the Leibniz 2category given by generators and relations which include the Leibnizlike relation (strict 3cell) among extended 2cells. The Leibniz 2category offers the ‘most general ’ no ..."
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Abstract. We propose the categorification of algebraic analysis in terms of a specific 2category, called here the Leibniz 2category given by generators and relations which include the Leibnizlike relation (strict 3cell) among extended 2cells. The Leibniz 2category offers the ‘most general ’ notion of a ‘(co)derivation’, as a strict 3cell, for a general (al co)gebra, not necessarily (co)associative, not necessarily (co)unital, nor necessarily (co)commutative. We outline a program in which every 2cell related to a partial (co)derivation (called a Leibniz strict 3cell) is translated into an appropriate 2cell related to a Cartan’slike (co)derivation (called a Cartan strict 3cell), and vice versa. We found also that a nonLeibniz component (2cell related to a Leibniz 3cell), responsible
Strategic Health Information Management and Forecast: The BirdWatching Approach
"... Abstract. To facilitate communication and the exchange of information between patients, nurses, lab technicians, health insurers, physicians, policy makers, and existing knowledgebased systems, a set of shared standard terminologies and controlled vocabularies are necessary. In modern health inform ..."
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Abstract. To facilitate communication and the exchange of information between patients, nurses, lab technicians, health insurers, physicians, policy makers, and existing knowledgebased systems, a set of shared standard terminologies and controlled vocabularies are necessary. In modern health information management systems, these vocabularies are defined within formal representations called ontologies, where terminologies are only meaningful once linked to a descriptive dataset. When the datasets and their conveyed knowledge are changed, the ontological structure is altered accordingly. Despite the importance of this topic, the problem of managing evolving ontological structures is inadequately addressed by available tools and algorithms, partly because handling ontological change is not a purely computational affair. In this paper, we propose a framework inspired by a social activity, birdwatching. Using this model, the evolving ontological structures can be monitored and analyzed based on their state at a given time. Moreover, patterns of changes can be derived and used to predict and approximate a system’s behavior based on potential future changes. Keywords: Change management, Biomedical ontologies, Multiagent system, Health information management.