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The Algebra of Open and Interconnected Systems

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The Algebra of Open and
Interconnected Systems

Brendan Fong

Hertford College
University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy in Computer Science

Trinity 2016

For all those who have prepared food so I could eat and created homes so I could live over the past four years. You too have laboured to produce this; I hope I have done your labours justice.

Abstract

Herein we develop category-theoretic tools for understanding networkstyle diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction

networks, and so on. The key feature is that the language is comprised

of a number of components with multiple (input/output) terminals, each

possibly labelled with some type, that may then be connected together along these terminals to form a larger network. The components form

hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor.

The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corela-

tions. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the ‘decorations’ specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.

Contents

Preface

v

I Mathematical Foundations

1

1 Hypergraph categories: the algebra of interconnection

1.1 The algebra of interconnection . . . . . . . . . . . . . . . . . . . . . . 1.2 Symmetric monoidal categories . . . . . . . . . . . . . . . . . . . . .
1.2.1 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 String diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

25679

1.3 Hypergraph categories . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Frobenius monoids . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 Hypergraph categories . . . . . . . . . . . . . . . . . . . . . . 12 1.3.3 Hypergraph categories are self-dual compact closed . . . . . . 13 1.3.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Example: cospan categories . . . . . . . . . . . . . . . . . . . . . . . 17

2 Decorated cospans: from closed systems to open

22

2.1 From closed systems to open . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Decorated cospan categories . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Composing decorated cospans . . . . . . . . . . . . . . . . . . 26 2.2.2 The hypergraph structure . . . . . . . . . . . . . . . . . . . . 30 2.2.3 A bicategory of decorated cospans . . . . . . . . . . . . . . . . 35
2.3 Functors between decorated cospan categories . . . . . . . . . . . . . 35 2.4 Decorations in Set are general . . . . . . . . . . . . . . . . . . . . . . 40 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 Labelled graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.2 Linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . 44

i

2.5.3 Electrical circuits . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Corelations: a tool for black boxing 48

3.1 The idea of black boxing . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Corelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Factorisation systems . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.2 Corelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.3 Categories of corelations . . . . . . . . . . . . . . . . . . . . . 58
3.3 Functors between corelation categories . . . . . . . . . . . . . . . . . 62 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Equivalence relations as corelations in Set . . . . . . . . . . . 67 3.4.2 Linear relations as corelations in Vect . . . . . . . . . . . . . . 68

4 Decorated corelations: black-boxed open systems

71

4.1 Black-boxed open systems . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Decorated corelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 Adjoining right adjoints . . . . . . . . . . . . . . . . . . . . . 75 4.2.2 Decorated corelations . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.3 Categories of decorated corelations . . . . . . . . . . . . . . . 78
4.3 Functors between decorated corelation categories . . . . . . . . . . . 80 4.4 All hypergraph categories are decorated corelation categories . . . . . 84
4.4.1 Representing a morphism with its name . . . . . . . . . . . . 85 4.4.2 A categorical equivalence . . . . . . . . . . . . . . . . . . . . . 89 4.4.3 Factorisations as decorations . . . . . . . . . . . . . . . . . . . 91
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.2 Two constructions for linear relations . . . . . . . . . . . . . . 97

II Applications

99

5 Signal flow diagrams

100

5.1 Behavioural control theory . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Linear time-invariant dynamical systems . . . . . . . . . . . . . . . . 106 5.3 Presentation of LTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3.2 Presentations of Mat k[s, s−1] and Cospan Mat k[s, s−1] . . . . 111 5.3.3 Corelations in Mat k[s, s−1] . . . . . . . . . . . . . . . . . . . . 114

ii

5.3.4 Presentation of Corel Mat k[s, s−1] . . . . . . . . . . . . . . . . 116
5.4 Operational semantics . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.5 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5.1 A categorical characterisation . . . . . . . . . . . . . . . . . . 122 5.5.2 Control and interconnection . . . . . . . . . . . . . . . . . . . 124 5.5.3 Comparison to matrix methods . . . . . . . . . . . . . . . . . 125

6 Passive linear networks

128

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Circuits of linear resistors . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.1 Circuits as labelled graphs . . . . . . . . . . . . . . . . . . . . 137 6.2.2 Ohm’s law, Kirchhoff’s laws, and the principle of minimum power139 6.2.3 A Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . 142 6.2.4 Equivalent circuits . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2.5 Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.3 Inductors and capacitors . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.3.1 The frequency domain and Ohm’s law revisited . . . . . . . . 151 6.3.2 The mechanical analogy . . . . . . . . . . . . . . . . . . . . . 154 6.3.3 Generalized Dirichlet forms . . . . . . . . . . . . . . . . . . . 155 6.3.4 A generalized minimizability result . . . . . . . . . . . . . . . 157
6.4 A compositional perspective . . . . . . . . . . . . . . . . . . . . . . . 160
6.4.1 The category of open circuits . . . . . . . . . . . . . . . . . . 161 6.4.2 An obstruction to black boxing . . . . . . . . . . . . . . . . . 163 6.4.3 The category of Dirichlet cospans . . . . . . . . . . . . . . . . 165 6.4.4 A category of behaviours . . . . . . . . . . . . . . . . . . . . . 168
6.5 Networks as Lagrangian relations . . . . . . . . . . . . . . . . . . . . 170
6.5.1 Symplectic vector spaces . . . . . . . . . . . . . . . . . . . . . 170 6.5.2 Lagrangian subspaces from quadratic forms . . . . . . . . . . 173 6.5.3 Lagrangian relations . . . . . . . . . . . . . . . . . . . . . . . 175 6.5.4 The symmetric monoidal category of Lagrangian relations . . 177
6.6 Ideal wires and corelations . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6.1 Ideal wires as corelations . . . . . . . . . . . . . . . . . . . . . 179 6.6.2 Potentials on corelations . . . . . . . . . . . . . . . . . . . . . 180 6.6.3 Currents on corelations . . . . . . . . . . . . . . . . . . . . . . 182 6.6.4 The symplectification functor . . . . . . . . . . . . . . . . . . 183 6.6.5 Lagrangian relations as decorated corelations . . . . . . . . . . 186

iii

6.7 The black box functor . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.7.2 Minimization via composition of relations . . . . . . . . . . . 191 6.7.3 Proof of functoriality . . . . . . . . . . . . . . . . . . . . . . . 192

7 Further directions

195

7.1 Bound colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.2 Open circuits and other systems . . . . . . . . . . . . . . . . . . . . . 198

Bibliography

200

iv

Preface

This is a thesis in the mathematical sciences, with emphasis on the mathematics. But before we get to the category theory, I want to say a few words about the scientific

tradition in which this thesis is situated.
Mathematics is the language of science. Twinned so intimately with physics, over the past centuries mathematics has become a superb—indeed, unreasonably effective—language for understanding planets moving in space, particles in a vacuum,

the structure of spacetime, and so on. Yet, while Wigner speaks of the unreasonable

effectiveness of mathematics in the natural sciences [Wig60], equally eminent mathematicians, not least Gelfand, speak of the unreasonable ineffectiveness of mathematics

in biology [Mon07] and related fields. Why such a difference?
A contrast between physics and biology is that while physical systems can often

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    SAT and SMT Solving in a Nutshell Erika Abrah´ am´ RWTH Aachen University, Germany LuFG Theory of Hybrid Systems February 27, 2020 Erika Abrah´ am´ - SAT and SMT solving 1 / 16 What is this talk about? Satisfiability problem The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. We focus on the automated solution of the satisfiability problem for first-order logic over arithmetic theories, especially using SAT and SMT solving. Erika Abrah´ am´ - SAT and SMT solving 2 / 16 CAS SAT SMT (propositional logic) (SAT modulo theories) Enumeration Computer algebra DP (resolution) systems [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62] Decision procedures NP-completeness [Cook’71] for combined theories CAD Conflict-directed [Shostak’79] [Nelson, Oppen’79] backjumping Partial CAD Virtual CDCL [GRASP’97] [zChaff’04] DPLL(T) substitution Watched literals Equalities and uninterpreted Clause learning/forgetting functions Variable ordering heuristics Bit-vectors Restarts Array theory Arithmetic Decision procedures for first-order logic over arithmetic theories in mathematical logic 1940 Computer architecture development 1960 1970 1980 2000 2010 Erika Abrah´ am´ - SAT and SMT solving 3 / 16 SAT SMT (propositional logic) (SAT modulo theories) Enumeration DP (resolution) [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62] Decision procedures NP-completeness [Cook’71] for combined theories Conflict-directed [Shostak’79] [Nelson, Oppen’79] backjumping CDCL [GRASP’97] [zChaff’04]
  • Programming for Computations – Python

    Programming for Computations – Python

    15 Svein Linge · Hans Petter Langtangen Programming for Computations – Python Editorial Board T. J.Barth M.Griebel D.E.Keyes R.M.Nieminen D.Roose T.Schlick Texts in Computational 15 Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick More information about this series at http://www.springer.com/series/5151 Svein Linge Hans Petter Langtangen Programming for Computations – Python A Gentle Introduction to Numerical Simulations with Python Svein Linge Hans Petter Langtangen Department of Process, Energy and Simula Research Laboratory Environmental Technology Lysaker, Norway University College of Southeast Norway Porsgrunn, Norway On leave from: Department of Informatics University of Oslo Oslo, Norway ISSN 1611-0994 Texts in Computational Science and Engineering ISBN 978-3-319-32427-2 ISBN 978-3-319-32428-9 (eBook) DOI 10.1007/978-3-319-32428-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2016945368 Mathematic Subject Classification (2010): 26-01, 34A05, 34A30, 34A34, 39-01, 40-01, 65D15, 65D25, 65D30, 68-01, 68N01, 68N19, 68N30, 70-01, 92D25, 97-04, 97U50 © The Editor(s) (if applicable) and the Author(s) 2016 This book is published open access. Open Access This book is distributed under the terms of the Creative Commons Attribution-Non- Commercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated.
  • Modeling and Analysis of Hybrid Systems

    Modeling and Analysis of Hybrid Systems

    Building Bridges between Symbolic Computation and Satisfiability Checking Erika Abrah´ am´ RWTH Aachen University, Germany in cooperation with Florian Corzilius, Gereon Kremer, Stefan Schupp and others ISSAC’15, 7 July 2015 Photo: Prior Park, Bath / flickr Liam Gladdy What is this talk about? Satisfiability problem The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. We focus on the automated solution of the satisfiability problem for first-order logic over arithmetic theories, especially on similarities and differences in symbolic computation and SAT and SMT solving. Erika Abrah´ am´ - SMT solving and Symbolic Computation 2 / 39 CAS SAT SMT (propositional logic) (SAT modulo theories) Enumeration Computer algebra DP (resolution) systems [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62] Decision procedures NP-completeness [Cook’71] for combined theories CAD Conflict-directed [Shostak’79] [Nelson, Oppen’79] backjumping Partial CAD Virtual CDCL [GRASP’97] [zChaff’04] DPLL(T) substitution Watched literals Equalities and uninterpreted Clause learning/forgetting functions Variable ordering heuristics Bit-vectors Restarts Array theory Arithmetic Decision procedures for first-order logic over arithmetic theories in mathematical logic 1940 Computer architecture development 1960 1970 1980 2000 2010 Erika Abrah´ am´ - SMT solving and Symbolic Computation 3 / 39 SAT SMT (propositional logic) (SAT modulo theories) Enumeration DP (resolution) [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62]
  • Easy-To-Use Chinese MTEX Suite Hongbin Ma

    Easy-To-Use Chinese MTEX Suite Hongbin Ma

    The PracTEX Journal, 2012, No. 1 Article revision 2012/06/25 Easy-to-use Chinese MTEX Suite Hongbin Ma Email cnedu.bit.mathmhb@ Address School of Automation, Beijing Institute of Technology, Beijing 100081, P. R. China 1 Motivation of Developing MTEX As the main developer of Chinese MTEX Suite , or simply MTEX [1], I started to fall in love with TEX[2] and LATEX[3] in 2002 when I was still a graduate student major- ing in mathematics and cybernetics at the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. At that time, recommended by some se- nior students, I started to use Chinese CTEX Suite , or simply CTEX[4], which was maintained by Dr. Lingyun Wu[5], a researcher in our academy, and is roughly a collection of pre-configured MiKTEX system[6] packaged with other tools such as customized WinEdt[7] for Chinese TEXers. CTEX brings significant benefits to China TEX users and it helps much to popularize the use of LATEX in China, es- pecially in the educational and academic areas with large requirement on mathe- matics typesetting. Furthermore, at that time, CTEX provides one way to typeset Chinese documents easily with LATEX and CCT [8](Chinese-Typesetting-System ), which was initially developed by another researcher Prof. Linbo Zhang[9] in our academy since 1998 for the purpose of typesetting Chinese with LATEX. Besides CCT system, another system called TY (Tian-Yuan ) system [10] was invented by a group in Eastern China Normal University so as to overcome the difficulties of typesetting Chinese with LATEX using different idea.
  • Robust Computer Algebra, Theorem Proving, and Oracle AI

    Robust Computer Algebra, Theorem Proving, and Oracle AI

    Robust Computer Algebra, Theorem Proving, and Oracle AI Gopal P. Sarma∗ Nick J. Hayyz School of Medicine, Emory University, Atlanta, GA USA Vicarious FPC, San Francisco, CA USA Abstract In the context of superintelligent AI systems, the term “oracle” has two meanings. One refers to modular systems queried for domain-specific tasks. Another usage, referring to a class of systems which may be useful for addressing the value alignment and AI control problems, is a superintelligent AI system that only answers questions. The aim of this manuscript is to survey contemporary research problems related to oracles which align with long-term research goals of AI safety. We examine existing question answering systems and argue that their high degree of architectural heterogeneity makes them poor candidates for rigorous analysis as oracles. On the other hand, we identify computer algebra systems (CASs) as being primitive examples of domain-specific oracles for mathematics and argue that efforts to integrate computer algebra systems with theorem provers, systems which have largely been developed independent of one another, provide a concrete set of problems related to the notion of provable safety that has emerged in the AI safety community. We review approaches to interfacing CASs with theorem provers, describe well-defined architectural deficiencies that have been identified with CASs, and suggest possible lines of research and practical software projects for scientists interested in AI safety. I. Introduction other hand, issues of privacy and surveillance, access and inequality, or economics and policy are Recently, significant public attention has been also of utmost importance and are distinct from drawn to the consequences of achieving human- the specific technical challenges posed by most level artificial intelligence.
  • On the Atkinson Formula for the Ζ Function

    On the Atkinson Formula for the Ζ Function

    On the Atkinson formula for the ζ function Daniele Dona, Sebastian Zuniga Alterman June 23, 2021 Abstract Thanks to Atkinson (1938), we know the first two terms of the asymptotic for- mula for the square mean integral value of the Riemann zeta function ζ on the criti- cal line. Following both his work and the approach of Titchmarsh (1986), we present an explicit version of the Atkinson formula, improving on a recent bound by Simoniˇc 1 3 (2020). Moreover, we extend the Atkinson formula to the range ℜ(s) ∈ 4 , 4 , giv- ing an explicit bound for the square mean integral value of ζ and improving on a bound by Helfgott and the authors (2019). We use mostly classical tools, such as the approximate functional equation and the explicit convexity bounds of the zeta function given by Backlund (1918). 1 Introduction The search for meaningful bounds for ζ(s) in the range 0 < (s) < 1 has spanned more ℜ than a century. The classical conjecture on L∞ bounds, called the Lindel¨of hypothesis, 1 ε states that ζ 2 + it = oε( t ) for any ε > 0; by Hadamard’s three-line theorem and | | ε the functional equation of ζ, this implies in particular that ζ(τ + it) = oε( t ) for 1 | | | | 1 2 τ+ε 1 2 <τ< 1 and ζ(τ + it) = oε( t − ) for 0 <τ< 2 . | 1 | τ | | − +ε Bounds of order t 2 are called convexity bounds, and bounds with even lower exponent are called subconvexity| | bounds. The current best bound is due to Bourgain [6], 13 1 84 +ε who showed that ζ 2 + it = oε( t ).
  • Satisfiability Checking and Symbolic Computation

    Satisfiability Checking and Symbolic Computation

    Satisfiability Checking and Symbolic Computation E. Abrah´am´ 1, J. Abbott11, B. Becker2, A.M. Bigatti3, M. Brain10, B. Buchberger4, A. Cimatti5, J.H. Davenport6, M. England7, P. Fontaine8, S. Forrest9, A. Griggio5, D. Kroening10, W.M. Seiler11 and T. Sturm12 1RWTH Aachen University, Aachen, Germany; 2Albert-Ludwigs-Universit¨at, Freiburg, Germany; 3Universit`adegli studi di Genova, Italy; 4Johannes Kepler Universit¨at, Linz, Austria; 5Fondazione Bruno Kessler, Trento, Italy; 6University of Bath, Bath, U.K.; 7Coventry University, Coventry, U.K.; 8LORIA, Inria, Universit´ede Lorraine, Nancy, France; 9Maplesoft Europe Ltd; 10University of Oxford, Oxford, U.K.; 11Universit¨at Kassel, Kassel, Germany; 12CNRS, LORIA, Nancy, France and Max-Planck-Institut f¨ur Informatik, Saarbr¨ucken, Germany. Abstract Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the two communities are currently only weakly con- nected. We introduce a new project SC2 to build a joint community in this area, supported by a newly accepted EU (H2020-FETOPEN-CSA) project of the same name. We aim to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identi- fying common challenges, and developing a common roadmap. This abstract and accompanying poster describes the motivation and aims for the project, and reports on the first activities. 1 Introduction We describe a new project to bring together the communities of Symbolic Computation and Satisfiability Checking into a new joint community, SC2. Both communities have long histories, as illustrated by the tool development timeline in Figure 1, but traditionally they do not interact much even though they are now individually addressing similar problems in non-linear algebra.
  • Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 T I

    Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 T I

    INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation.