Jason Morton – an Approach to Computational Category Theory
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A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience. -
Knowledge Representation in Bicategories of Relations
Knowledge Representation in Bicategories of Relations Evan Patterson Department of Statistics, Stanford University Abstract We introduce the relational ontology log, or relational olog, a knowledge representation system based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a recent categorical framework for knowledge representation. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. In this paper, we investigate relational ologs both for their own sake and to gain insight into the relationship between the algebraic and logical approaches to knowledge representation. On a practical level, we show by example that relational ologs have a friendly and intuitive—yet fully precise—graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. Although we make extensive use of categorical language, this paper is designed to be self-contained and has considerable expository content. The only prerequisites are knowledge of first-order logic and the rudiments of category theory. 1. Introduction arXiv:1706.00526v2 [cs.AI] 1 Nov 2017 The representation of human knowledge in computable form is among the oldest and most fundamental problems of artificial intelligence. Several recent trends are stimulating continued research in the field of knowledge representation (KR). -
The ACT Pipeline from Abstraction to Application
Addressing hypercomplexity: The ACT pipeline from abstraction to application David I. Spivak (Joint with Brendan Fong, Paolo Perrone, Evan Patterson) Department of Mathematics Massachusetts Institute of Technology 0 / 25 Introduction Outline 1 Introduction Why? How? What? Plan of the talk 2 Some new applications 3 Catlab.jl 4 Conclusion 0 / 25 Introduction Why? What's wrong? As problem complexity increases, good organization becomes crucial. (Courtesy of Spencer Breiner, NIST) 1 / 25 Introduction Why? The problem is hypercomplexity The world is becoming more complex. We need to deal with more and more diverse sorts of interaction. Rather than closed dynamical systems, everything is open. The state of one system affects those of systems in its environment. Examples are everywhere. Designing anything requires input from more diverse stakeholders. Software is developed by larger teams, touching a common codebase. Scientists need to use experimental data obtained by their peers. (But the data structure and assumptions don't simply line up.) We need to handle this new complexity; traditional methods are faltering. 2 / 25 Introduction How? How might we deal with hypercomplexity? Organize the problem spaces effectively, as objects in their own right. Find commonalities in the various problems we work on. Consider the ways we build complex problems out of simple ones. Think of problem spaces and solutions as mathematical entities. With that in hand, how do you solve a hypercomplex problem? Migrating subproblems to groups who can solve them. Take the returned solutions and fit them together. Ensure in advance that these solutions will compose. Do this according the mathematical problem-space structure. -
Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. -
Category Theory Cheat Sheet
Musa Al-hassy https://github.com/alhassy/CatsCheatSheet July 5, 2019 Even when morphisms are functions, the objects need not be sets: Sometimes the objects . are operations —with an appropriate definition of typing for the functions. The categories Reference Sheet for Elementary Category Theory of F -algebras are an example of this. “Gluing” Morphisms Together Categories Traditional function application is replaced by the more generic concept of functional A category C consists of a collection of “objects” Obj C, a collection of “(homo)morphisms” composition suggested by morphism-arrow chaining: HomC(a; b) for any a; b : Obj C —also denoted “a !C b”—, an operation Id associating Whenever we have two morphisms such that the target type of one of them, say g : B A a morphism Ida : Hom(a; a) to each object a, and a dependently-typed “composition” is the same as the source type of the other, say f : C B then “f after g”, their composite operation _ ◦ _ : 8fABC : Objg ! Hom(B; C) ! Hom(A; B) ! Hom(A; C) that is morphism, f ◦ g : C A can be defined. It “glues” f and g together, “sequentially”: required to be associative with Id as identity. It is convenient to define a pair of operations src; tgt from morphisms to objects as follows: f g f◦g C − B − A ) C − A composition-Type f : X !C Y ≡ src f = X ^ tgt f = Y src; tgt-Definition Composition is the basis for gluing morphisms together to build more complex morphisms. Instead of HomC we can instead assume primitive a ternary relation _ : _ !C _ and However, not every two morphisms can be glued together by composition. -
Logic and Categories As Tools for Building Theories
Logic and Categories As Tools For Building Theories Samson Abramsky Oxford University Computing Laboratory 1 Introduction My aim in this short article is to provide an impression of some of the ideas emerging at the interface of logic and computer science, in a form which I hope will be accessible to philosophers. Why is this even a good idea? Because there has been a huge interaction of logic and computer science over the past half-century which has not only played an important r^olein shaping Computer Science, but has also greatly broadened the scope and enriched the content of logic itself.1 This huge effect of Computer Science on Logic over the past five decades has several aspects: new ways of using logic, new attitudes to logic, new questions and methods. These lead to new perspectives on the question: What logic is | and should be! Our main concern is with method and attitude rather than matter; nevertheless, we shall base the general points we wish to make on a case study: Category theory. Many other examples could have been used to illustrate our theme, but this will serve to illustrate some of the points we wish to make. 2 Category Theory Category theory is a vast subject. It has enormous potential for any serious version of `formal philosophy' | and yet this has hardly been realized. We shall begin with introduction to some basic elements of category theory, focussing on the fascinating conceptual issues which arise even at the most elementary level of the subject, and then discuss some its consequences and philosophical ramifications. -
Categorical Databases
Categorical databases David I. Spivak [email protected] Mathematics Department Massachusetts Institute of Technology Presented on 2014/02/28 at Oracle David I. Spivak (MIT) Categorical databases Presented on 2014/02/28 1 / 1 Introduction Purpose of the talk Purpose of the talk There is an fundamental connection between databases and categories. Category theory can simplify how we think about and use databases. We can clearly see all the working parts and how they fit together. Powerful theorems can be brought to bear on classical DB problems. David I. Spivak (MIT) Categorical databases Presented on 2014/02/28 2 / 1 Introduction The pros and cons of relational databases The pros and cons of relational databases Relational databases are reliable, scalable, and popular. They are provably reliable to the extent that they strictly adhere to the underlying mathematics. Make a distinction between the system you know and love, vs. the relational model, as a mathematical foundation for this system. David I. Spivak (MIT) Categorical databases Presented on 2014/02/28 3 / 1 Introduction The pros and cons of relational databases You're not really using the relational model. Current implementations have departed from the strict relational formalism: Tables may not be relational (duplicates, e.g from a query). Nulls (and labeled nulls) are commonly used. The theory of relations (150 years old) is not adequate to mathematically describe modern DBMS. The relational model does not offer guidance for schema mappings and data migration. Databases have been intuitively moving toward what's best described with a more modern mathematical foundation. David I. -
SHEAVES, TOPOSES, LANGUAGES Acceleration Based on A
Chapter 7 Logic of behavior: Sheaves, toposes, and internal languages 7.1 How can we prove our machine is safe? Imagine you are trying to design a system of interacting components. You wouldn’t be doing this if you didn’t have a goal in mind: you want the system to do something, to behave in a certain way. In other words, you want to restrict its possibilities to a smaller set: you want the car to remain on the road, you want the temperature to remain in a particular range, you want the bridge to be safe for trucks to pass. Out of all the possibilities, your system should only permit some. Since your system is made of components that interact in specified ways, the possible behavior of the whole—in any environment—is determined by the possible behaviors of each of its components in their local environments, together with the precise way in which they interact.1 In this chapter, we will discuss a logic wherein one can describe general types of behavior that occur over time, and prove properties of a larger-scale system from the properties and interaction patterns of its components. For example, suppose we want an autonomous vehicle to maintain a distance of some safe R from other objects. To do so, several components must interact: a 2 sensor that approximates the real distance by an internal variable S0, a controller that uses S0 to decide what action A to take, and a motor that moves the vehicle with an 1 The well-known concept of emergence is not about possibilities, it is about prediction. -
UNIVERSITY of CALIFORNIA RIVERSIDE the Grothendieck
UNIVERSITY OF CALIFORNIA RIVERSIDE The Grothendieck Construction in Categorical Network Theory A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Joseph Patrick Moeller December 2020 Dissertation Committee: Dr. John C. Baez, Chairperson Dr. Wee Liang Gan Dr. Carl Mautner Copyright by Joseph Patrick Moeller 2020 The Dissertation of Joseph Patrick Moeller is approved: Committee Chairperson University of California, Riverside Acknowledgments First of all, I owe all of my achievements to my wife, Paola. I couldn't have gotten here without my parents: Daniel, Andrea, Tonie, Maria, and Luis, or my siblings: Danielle, Anthony, Samantha, David, and Luis. I would like to thank my advisor, John Baez, for his support, dedication, and his unique and brilliant style of advising. I could not have become the researcher I am under another's instruction. I would also like to thank Christina Vasilakopoulou, whose kindness, energy, and expertise cultivated a deeper appreciation of category theory in me. My expe- rience was also greatly enriched by my academic siblings: Daniel Cicala, Kenny Courser, Brandon Coya, Jason Erbele, Jade Master, Franciscus Rebro, and Christian Williams, and by my cohort: Justin Davis, Ethan Kowalenko, Derek Lowenberg, Michel Manrique, and Michael Pierce. I would like to thank the UCR math department. Professors from whom I learned a ton of algebra, topology, and category theory include Julie Bergner, Vyjayanthi Chari, Wee-Liang Gan, Jos´eGonzalez, Jacob Greenstein, Carl Mautner, Reinhard Schultz, and Steffano Vidussi. Special thanks goes to the department chair Yat-Sun Poon, as well as Margarita Roman, Randy Morgan, and James Marberry, and many others who keep the whole thing together. -
Mathematics and the Brain: a Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness
entropy Review Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness 1,2,3, , 4,5,6,7, 8, Georg Northoff * y, Naotsugu Tsuchiya y and Hayato Saigo y 1 Mental Health Centre, Zhejiang University School of Medicine, Hangzhou 310058, China 2 Institute of Mental Health Research, University of Ottawa, Ottawa, ON K1Z 7K4 Canada 3 Centre for Cognition and Brain Disorders, Hangzhou Normal University, Hangzhou 310036, China 4 School of Psychological Sciences, Faculty of Medicine, Nursing and Health Sciences, Monash University, Melbourne, Victoria 3800, Australia; [email protected] 5 Turner Institute for Brain and Mental Health, Monash University, Melbourne, Victoria 3800, Australia 6 Advanced Telecommunication Research, Soraku-gun, Kyoto 619-0288, Japan 7 Center for Information and Neural Networks (CiNet), National Institute of Information and Communications Technology (NICT), Suita, Osaka 565-0871, Japan 8 Nagahama Institute of Bio-Science and Technology, Nagahama 526-0829, Japan; [email protected] * Correspondence: georg.northoff@theroyal.ca All authors contributed equally to the paper as it was a conjoint and equally distributed work between all y three authors. Received: 18 July 2019; Accepted: 9 October 2019; Published: 17 December 2019 Abstract: Consciousness is a central issue in neuroscience, however, we still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates. In this review, we provide a novel mathematical framework of category theory (CT), in which we can define and study the sameness between different domains of phenomena such as consciousness and its neural substrates. CT was designed and developed to deal with the relationships between various domains of phenomena. -
Category Theory for Dummies (I)
Category Theory for Dummies (I) James Cheney Programming Languages Discussion Group March 12, 2004 1 Not quite everything you’ve ever wanted to know... • You keep hearing about category theory. • Cool-sounding papers by brilliant researchers (e.g. Wadler’s “Theorems for free!”) • But it’s scary and incomprehensible. • And Category Theory is not even taught here. • Goal of this series: Familarity with basic ideas, not expertise 2 Outline • Categories: Why are they interesting? • Categories: What are they? Examples. • Some familiar properties expressed categorically • Some basic categorical constructions 3 Category theory • An abstract theory of “structured things” and “structure preserving function-like things”. • Independent of the concrete representation of the things and functions. • An alternative foundation for mathematics? (Lawvere) • Closely connected with computation, types and logic. • Forbiddingly complex notation for even simple ideas. 4 A mathematician’s eye view of the world Algebra Topology Logic Groups, Rings,... Topological Spaces Formulas Homomorphisms Continuous Functions Implication Set Theory 5 A category theorist’s eye view of the world Category Theory Objects Arrows Algebra Topology Logic Groups, Rings,... Topological Spaces Formulas Homomorphisms Continuous Functions Implication 6 My view (not authoritative): • Category theory helps organize thought about a collection of related things • and identify patterns that recur over and over. • It may suggest interesting ways of looking at them • but does not necessarily -
Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms
Article Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms Allen D. Parks 1,* and David J. Marchette 2 1 Electromagnetic and Sensor Systems Department, Naval Surface Warfare Center Dahlgren Division, Dahlgren, VA 22448, USA 2 Strategic and Computing Systems Department, Naval Surface Warfare Center Dahlgren Division, Dahlgren, VA 22448, USA; [email protected] * Correspondence: [email protected] Received: 25 October 2018; Accepted: 11 December 2018; Published: 24 January 2019 Abstract: The Müller-Wichards model (MW) is an algebraic method that quantitatively estimates the performance of sequential and/or parallel computer applications. Because of category theory’s expressive power and mathematical precision, a category theoretic reformulation of MW, i.e., CMW, is presented in this paper. The CMW is effectively numerically equivalent to MW and can be used to estimate the performance of any system that can be represented as numerical sequences of arithmetic, data movement, and delay processes. The CMW fundamental symmetry group is introduced and CMW’s category theoretic formalism is used to facilitate the identification of associated model invariants. The formalism also yields a natural approach to dividing systems into subsystems in a manner that preserves performance. Closed form models are developed and studied statistically, and special case closed form models are used to abstractly quantify the effect of parallelization upon processing time vs. loading, as well as to establish a system performance stationary action principle. Keywords: system performance modelling; categorification; performance functor; symmetries; invariants; effect of parallelization; system performance stationary action principle 1. Introduction In the late 1980s, D.