DOCSLIB.ORG

A Relation-Algebraic Approach to Graph Structure Transformation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Relation-Algebraic Approach to Graph Structure Transformation A Relation-Algebraic Approach to Graph Structure Transformation WOLFRAM KAHL u0 PhiL PhiR ChiL Xi Chi PsiL Psi Bericht Nr. 2002-03 June 2002 Universität der Bundeswehr München Fakultät für . INFORMATIK Werner-Heisenberg-Weg 39 • 85577 Neubiberg • Germany A Relation-Algebraic Approach to Graph Structure Transformation Wolfram Kahl1 Abstract Graph transformation is a rapidly expanding ¯eld of research, motivated by a wide range of applications. Such transformations can be speci¯ed at di®erent levels of abstraction. On one end, there are \programmed" graph transformation systems with very ¯ne control over manip- ulation of individual graph items. Di®erent flavours of rule-based graph transformation systems match their patterns in more generic ways and therefore interact with the graph structure at slightly higher levels. A rather dominant approach to graph transformation uses the abstractions of cate- gory theory to de¯ne matching and replacement almost on a black-box level, using easily understandable basic category-theoretic concepts, like pushouts and pullbacks. However, some details cannot be covered on this level, and most authors refrain from resorting to the much more advanced category-theoretic tools of topos theory that are available for graphs, too | topos theory is not felt to be an appropriate language for specifying graph transformation. In this thesis we show that the language of relations can be used very naturally to cover all the problems of the categoric approach to graph transformation. Although much of this follows from the well-known fact that every graph-structure category is a topos, very little of this power has been exploited before, and even surveys of the categoric approach to graph transformation state essential conditions outside the category-theoretic framework. One achievement is therefore the capability to provide descriptions of all graph trans- formation e®ects on a suitable level of abstraction from the concrete choice of graph structures. Another important result is the de¯nition of a graph rewriting approach where rela- tional matchings can match rule parameters to arbitrary subgraphs, which then can be copied or deleted by rewriting. At the same time, the rules are still as intuitive as in the double-pushout approach, and there is no need to use complicated encodings as in the pullback approaches. In short: A natural way to achieve a double-pushout-like rewriting concept that incorpo- rates some kind of \graph variable" matching and replication is to amalgamate pushouts and pullbacks, and the relation-algebraic approach o®ers appropriate abstractions that allow to formalise this in a fully component-free yet intuitively accessible manner. 1 Current e-mail address: [email protected]. This habilitation thesis was completed during the author's appointment at the Institute for Software Technology of UniversitÄat der Bundeswehr MuncÄ hen. Acknowledgements Although my interest in graphs and structures may have been \hard-wired" in my brain, it was Gunther Schmidt who led me on the one hand to applications of graph transformation in software development, and on the other hand to relation-algebraic formalisations in general, and to their applications to graphs in particular. Besides guiding me with his deep understanding of the calculus of relations as a valuable tool for formalisation and reasoning, he also alerted me to the importance of being aware of the axiomatic basis underlying one's work, and refraining from assuming more than necessary for the task at hand. Another important influence has been Yasuo Kawahara, who introduced me to Dede- kind categories and to working in weaker axiomatisations of relation categories, and whose pioneering work on \pushout-complements in toposes" provided an important stepping stone for the current thesis. I wish to thank many friends and colleagues for discussions on more or less directly related topics that considerably helped to shape my ideas, let me only mention Andrea Corradini, Jules Desharnais, Hitoshi Furusawa, Fabio Gadducci, H¶el`ene Jacquet, Bernhard MÄoller, John Pfaltz, and Michael Winter. Finally, I am in¯nitely grateful to my wife Liping and daughter Cynthia for their patience, understanding and support during the genesis of this thesis. Contents 1 Introduction 1 1.1 Graph Grammars and Graph Transformation . 1 1.2 The Categoric Approaches to Graph Transformation . 3 1.2.1 Pushouts . 4 1.2.2 More About Pushouts in Graphs . 7 1.2.3 The Double-Pushout Approach . 7 1.2.4 The Gluing Condition . 9 1.2.5 Restricting Derivations . 12 1.2.6 The Single-Pushout Approach . 13 1.2.7 Shortcomings of the Pushout Approaches . 15 1.2.8 The Pullback Approach . 15 1.3 Graph Transformation with Relational Matching . 21 1.4 Relation Algebras and Generalisations . 24 1.5 Contrasting the Relational and the Categoric Approaches . 26 1.6 Overview . 27 1.7 Eddi . 29 2 Graph Structures and Their Parts 30 2.1 Preliminaries: Sets, Lattices . 31 2.2 Signatures . 34 2.3 Algebras and Subalgebras . 36 2.4 Subalgebras in Graph Structures . 41 2.5 Partial Algebras and Weak Partial Subalgebras . 44 2.6 Pseudo-Complements . 46 2.7 Semi-Complements . 50 2.8 Naijve Graph Rewriting . 56 2.9 Discreteness in Graph Structures . 57 2.10 Coregular Parts and Base Elements . 63 3 Allegories of §-Algebras 66 3.1 Preliminaries: Categories and Allegories . 66 3.2 Abstract §-Algebras and Relational Homomorphisms . 73 3.3 Constructions in §-AlgD . 77 4 Dedekind Categories of Graph Structures 84 4.1 Preliminaries: Distributive Allegories and Dedekind Categories . 84 4.2 Joins in General §-Algebra Allegories . 88 4.3 Relational Homomorphisms Between Graph Structures . 90 4.4 Pseudo- and Semi-Complements in §-GSD . 93 4.5 Constructions in §-GSD . 94 4.6 Discrete Relations . 95 iii iv CONTENTS 5 Categoric Rewriting in a Relational Setting 98 5.1 Pullbacks . 99 5.2 Transitive and Difunctional Closures . 102 5.3 Pushouts . 103 5.4 Pushout Complements . 107 5.5 Pullback Complements . 116 5.6 Pushouts of Partial Functions . 120 5.7 Summary . 129 6 Relational Rewriting in Dedekind Categories 131 6.1 Gluing Setup . 131 6.2 Amalgamating Pushouts and Pullbacks to Pullouts . 134 6.3 Pullout Complements . 138 6.4 Pullout Rewriting . 140 6.5 The Weak Pullout Construction . 143 7 Conclusion and Outlook 151 Bibliography 153 A Proofs of Auxiliary Properties 165 A.1 Allegory Properties . 165 A.2 Partial Identities . 167 A.3 Dedekind Category Properties . 169 A.4 Semi-Complements and Partial Identities in Dedekind Categories . 170 A.5 Symmetric Quotients . 172 B Proofs for Relational Rewriting, Chapter 6 173 B.1 Correctness of the Glued Tabulation Construction . 173 B.2 Correctness of the Pullout Complement Construction . 179 B.3 Monomorphy of Weak Pullouts . 184 B.4 Correctness of the Direct Result Construction . 190 B.5 The Straight Host Construction Yields Weak Pullout Complements . 199 Index 201 Chapter 1 Introduction We start this introductory chapter with an overview over graph transformation, followed by a more detailed presentation of the categoric approaches to graph transformation. In Sect. 1.3 we sketch the problem of relational matching and shortly sketch our solution. All this serves as introduction into the problem area of the present thesis. The solutions rely on axiomatisations of categories of relations, and we present some historical background for this in Sect. 1.4. Although categories of relations are still cat- egories, the style of relational formalisations is frequently very di®erent from the style of category-theoretic formalisations; we discuss this e®ect in Sect. 1.5. Finally, we present an overview over the remainder of this thesis in Sect. 1.6. 1.1 Graph Grammars and Graph Transformation Research on graph grammars started in the late 1960s [PR69, Sch70, EPS73] and has since evolved into a very active area, as documented by a gradually increasing number of conference series devoted to the topic [CER78, ENR82, ENRR87, FK87, EKR91, CR93b, SE93, C+96, EEKR00, ET00]. Given the fact that graphs are a useful means for modelling many di®erent kinds of complex systems, graph grammars and graph replacement was, from the beginning, strongly motivated by applications in computer science, such as networks, and also in biology, such as modelling the growth of plants [LC79, PL90]. Important areas of applications currently include applied software systems and systems modelling [Jon90, Jon91, Jon92, EB94], many aspects of parallelism, distributed systems and synchronisation mechanisms [BFH85, Kre87, KW87, Sch90], implementation of term rewriting and operational semantics of functional (and logic) programming [Pad82, HP91, CR93a, PvE93, SPvE93, Kah96, Plu99, BS99], operational semantics of object oriented languages [WG96, WG99], and many aspects of visual programming languages [BMST99]. A survey of the foundations of graph transformation is contained in the ¯rst volume [Roz97] of a three-volume handbook. The second volume [EEKR99] concentrates on ap- plications, languages, and tools, and there already has been a ¯rst international workshop on \Applications of Graph Transformations with Industrial Relevance" [NSM00]. In recent years, there has been a shift of attention, accompanied by a shift of terminology, from \graph grammars" to \graph transformation", from \productions" to \(transfor- mation) rules", as problems of describing, generating, and parsing classes of graphs (as \graph languages") are nowadays overshadowed by problems of using graph transformation to specify or describe complex system behaviour and step-wise adaptation or re¯nement of systems or abstract entities represented by graphs. Accordingly, we shall use the following nomenclature for the essential items involved in an individual graph transformation step: 1 2 1. Introduction ² A rule usually contains at least a left-hand side and a right-hand side, which usually both are graphs, and some indication of how instances of the right-hand side are going to replace matched instances of the left-hand side. ² Rule application to some application graph requires identi¯cation of a redex in the application graph, usually via some kind of matching from the rule's left-hand side to the application graph.
Load more

Recommended publications
  • Homomorphism and Isomorphism Theorems Generalized from a Relational Perspective
    Homomorphism and Isomorphism Theorems Generalized from a Relational Perspective Gunther Schmidt Institute for Software Technology, Department of Computing Science Universit¨at der Bundeswehr M¨unchen, 85577 Neubiberg, Germany [email protected] Abstract. The homomorphism and isomorphism theorems traditionally taught to students in a group theory or linear algebra lecture are by no means theorems of group theory. They are for a long time seen as general concepts of universal algebra. This article goes even further and identifies them as relational properties which to study does not even require the concept of an algebra. In addition it is shown how the homomorphism and isomorphism theorems generalize to not necessarily algebraic and thus relational structures. Keywords: homomorphism theorem, isomorphism theorem, relation al- gebra, congruence, multi-covering. 1 Introduction Relation algebra has received increasing interest during the last years. Many areas have been reconsidered from the relational point of view, which often pro- vided additional insight. Here, the classical homomorphism and isomorphism theorems (see [1], e.g.) are reviewed from a relational perspective, thereby sim- plifying and slightly generalizing them. The paper is organized as follows. First we recall the notion of a heterogeneous relation algebra and some of the very basic rules governing work with relations. With these, function and equivalence properties may be formulated concisely. The relational concept of homomorphism is defined as well as the concept of a congruence which is related with the concept of a multi-covering, which have connections with topology, complex analysis, and with the equivalence problem for flow-chart programs. We deal with the relationship between mappings and equivalence relations.
    [Show full text]
  • Thomas Ströhlein's Endgame Tables: a 50Th Anniversary
    Thomas Ströhlein's Endgame Tables: a 50th Anniversary Article Supplemental Material The Festschrift on the 40th Anniversary of the Munich Faculty of Informatics Haworth, G. (2020) Thomas Ströhlein's Endgame Tables: a 50th Anniversary. ICGA Journal, 42 (2-3). pp. 165-170. ISSN 1389-6911 doi: https://doi.org/10.3233/ICG-200151 Available at http://centaur.reading.ac.uk/90000/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . Published version at: https://content.iospress.com/articles/icga-journal/icg200151 To link to this article DOI: http://dx.doi.org/10.3233/ICG-200151 Publisher: The International Computer Games Association All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online 40 Jahre Informatik in Munchen:¨ 1967 – 2007 Festschrift Herausgegeben von Friedrich L. Bauer unter Mitwirkung von Helmut Angstl, Uwe Baumgarten, Rudolf Bayer, Hedwig Berghofer, Arndt Bode, Wilfried Brauer, Stephan Braun, Manfred Broy, Roland Bulirsch, Hans-Joachim Bungartz, Herbert Ehler, Jurgen¨ Eickel, Ursula Eschbach, Anton Gerold, Rupert Gnatz, Ulrich Guntzer,¨ Hellmuth Haag, Winfried Hahn (†), Heinz-Gerd Hegering, Ursula Hill-Samelson, Peter Hubwieser, Eike Jessen, Fred Kroger,¨ Hans Kuß, Klaus Lagally, Hans Langmaack, Heinrich Mayer, Ernst Mayr, Gerhard Muller,¨ Heinrich Noth,¨ Manfred Paul, Ulrich Peters, Hartmut Petzold, Walter Proebster, Bernd Radig, Angelika Reiser, Werner Rub,¨ Gerd Sapper, Gunther Schmidt, Fred B.
    [Show full text]
  • Moduli Spaces and Invariant Theory
    MODULI SPACES AND INVARIANT THEORY JENIA TEVELEV CONTENTS §1. Syllabus 3 §1.1. Prerequisites 3 §1.2. Course description 3 §1.3. Course grading and expectations 4 §1.4. Tentative topics 4 §1.5. Textbooks 4 References 4 §2. Geometry of lines 5 §2.1. Grassmannian as a complex manifold. 5 §2.2. Moduli space or a parameter space? 7 §2.3. Stiefel coordinates. 8 §2.4. Complete system of (semi-)invariants. 8 §2.5. Plücker coordinates. 9 §2.6. First Fundamental Theorem 10 §2.7. Equations of the Grassmannian 11 §2.8. Homogeneous ideal 13 §2.9. Hilbert polynomial 15 §2.10. Enumerative geometry 17 §2.11. Transversality. 19 §2.12. Homework 1 21 §3. Fine moduli spaces 23 §3.1. Categories 23 §3.2. Functors 25 §3.3. Equivalence of Categories 26 §3.4. Representable Functors 28 §3.5. Natural Transformations 28 §3.6. Yoneda’s Lemma 29 §3.7. Grassmannian as a fine moduli space 31 §4. Algebraic curves and Riemann surfaces 37 §4.1. Elliptic and Abelian integrals 37 §4.2. Finitely generated fields of transcendence degree 1 38 §4.3. Analytic approach 40 §4.4. Genus and meromorphic forms 41 §4.5. Divisors and linear equivalence 42 §4.6. Branched covers and Riemann–Hurwitz formula 43 §4.7. Riemann–Roch formula 45 §4.8. Linear systems 45 §5. Moduli of elliptic curves 47 1 2 JENIA TEVELEV §5.1. Curves of genus 1. 47 §5.2. J-invariant 50 §5.3. Monstrous Moonshine 52 §5.4. Families of elliptic curves 53 §5.5. The j-line is a coarse moduli space 54 §5.6.
    [Show full text]
  • Introduction to Abstract Algebra (Math 113)
    Introduction to Abstract Algebra (Math 113) Alexander Paulin, with edits by David Corwin FOR FALL 2019 MATH 113 002 ONLY Contents 1 Introduction 4 1.1 What is Algebra? . 4 1.2 Sets . 6 1.3 Functions . 9 1.4 Equivalence Relations . 12 2 The Structure of + and × on Z 15 2.1 Basic Observations . 15 2.2 Factorization and the Fundamental Theorem of Arithmetic . 17 2.3 Congruences . 20 3 Groups 23 1 3.1 Basic Definitions . 23 3.1.1 Cayley Tables for Binary Operations and Groups . 28 3.2 Subgroups, Cosets and Lagrange's Theorem . 30 3.3 Generating Sets for Groups . 35 3.4 Permutation Groups and Finite Symmetric Groups . 40 3.4.1 Active vs. Passive Notation for Permutations . 40 3.4.2 The Symmetric Group Sym3 . 43 3.4.3 Symmetric Groups in General . 44 3.5 Group Actions . 52 3.5.1 The Orbit-Stabiliser Theorem . 55 3.5.2 Centralizers and Conjugacy Classes . 59 3.5.3 Sylow's Theorem . 66 3.6 Symmetry of Sets with Extra Structure . 68 3.7 Normal Subgroups and Isomorphism Theorems . 73 3.8 Direct Products and Direct Sums . 83 3.9 Finitely Generated Abelian Groups . 85 3.10 Finite Abelian Groups . 90 3.11 The Classification of Finite Groups (Proofs Omitted) . 95 4 Rings, Ideals, and Homomorphisms 100 2 4.1 Basic Definitions . 100 4.2 Ideals, Quotient Rings and the First Isomorphism Theorem for Rings . 105 4.3 Properties of Elements of Rings . 109 4.4 Polynomial Rings . 112 4.5 Ring Extensions . 115 4.6 Field of Fractions .
    [Show full text]
  • Developing and Benchmarking Native Linux Applications on Android
    Developing and Benchmarking Native Linux Applications on Android Leonid Batyuk, Aubrey-Derrick Schmidt, Hans-Gunther Schmidt, Ahmet Camtepe, and Sahin Albayrak Technische Universit¨at Berlin, 10587 Berlin, Germany {aubrey.schmidt,leonid.batyuk,hans-gunther.schmidt,ahmet.camtepe, sahin.albayrak}@dai-labor.de http://www.dai-labor.de Abstract. Smartphones get increasingly popular where more and more smartphone platforms emerge. Special attention was gained by the open source platform Android which was presented by the Open Handset Al- liance (OHA) hosting members like Google, Motorola, and HTC. An- droid uses a Linux kernel and a stripped-down userland with a custom Java VM set on top. The resulting system joins the advantages of both environments, while third-parties are intended to develop only Java ap- plications at the moment. In this work, we present the benefit of using native applications in Android. Android includes a fully functional Linux, and using it for heavy computational tasks when developing applications can bring in substantional performance increase. We present how to develop native applications and software components, as well as how to let Linux appli- cations and components communicate with Java programs. Additionally, we present performance measurements of native and Java applications executing identical tasks. The results show that native C applications can be up to 30 times as fast as an identical algorithm running in Dalvik VM. Java applications can become a speed-up of up to 10 times if utilizing JNI. Keywords: software, performance, smartphones, android, C, Java. 1 Introduction With the growing customer interest in smartphones the number of available platforms increases steadily.
    [Show full text]
  • Impulse Response Sequences and Construction of Number Sequence Identities
    1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.8.2 47 6 23 11 Impulse Response Sequences and Construction of Number Sequence Identities Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington, IL 61702-2900 USA [email protected] Abstract In this paper, we investigate impulse response sequences over the integers by pre- senting their generating functions and expressions. We also establish some of the corre- sponding identities. In addition, we give the relationship between an impulse response sequence and all linear recurring sequences satisfying the same linear recurrence rela- tion, which can be used to transfer the identities among different sequences. Finally, we discuss some applications of impulse response sequences to the structure of Stirling numbers of the second kind, the Wythoff array, and the Boustrophedon transform. 1 Introduction Many number and polynomial sequences can be defined, characterized, evaluated, and clas- sified by linear recurrence relations with certain orders. A number sequence an is called a sequence of order r if it satisfies the linear recurrence relation of order r { } r an = pjan j, n r, (1) − ≥ j=1 X for some constants pj, j = 1, 2,...,r, pr = 0, and initial conditions aj (j = 0, 1,...,r 1). Linear recurrence relations with constant6 coefficients are important in subjects including− 1 combinatorics, pseudo-random number generation, circuit design, and cryptography, and they have been studied extensively. To construct an explicit formula of the general term of a number sequence of order r, one may use a generating function, a characteristic equation, or a matrix method (See Comtet [4], Niven, Zuckerman, and Montgomery [12], Strang [15], Wilf [16], etc.) Let Ar be the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1) with coefficient set E = p ,p ,...,p .
    [Show full text]
  • Introduction to Algebraic Theory of Linear Systems of Differential
    Introduction to algebraic theory of linear systems of differential equations Claude Sabbah Introduction In this set of notes we shall study properties of linear differential equations of a complex variable with coefficients in either the ring convergent (or formal) power series or in the polynomial ring. The point of view that we have chosen is intentionally algebraic. This explains why we shall consider the D-module associated with such equations. In order to solve an equation (or a system) we shall begin by finding a “normal form” for the corresponding D-module, that is by expressing it as a direct sum of elementary D-modules. It is then easy to solve the corresponding equation (in a similar way, one can solve a system of linear equations on a vector space by first putting the matrix in a simple form, e.g. triangular form or (better) Jordan canonical form). This lecture is supposed to be an introduction to D-module theory. That is why we have tried to set out this classical subject in a way that can be generalized in the case of many variables. For instance we have introduced the notion of holonomy (which is not difficult in dimension one), we have emphazised the connection between D-modules and meromorphic connections. Because the notion of a normal form does not extend easily in higher dimension, we have also stressed upon the notions of nearby and vanishing cycles. The first chapter consists of a local algebraic study of D-modules. The coefficient ring is either C{x} (convergent power series) or C[[x]] (formal power series).
    [Show full text]
  • Publications by Gunther Schmidt
    Publications by Gunther Schmidt 2020 | R¨uckblick auf die Anf¨angeder M¨unchner Informatik | Dokumente, Belege, Ver¨offent- lichungen und Erinnerungen von fr¨uhund lange Beteiligten. Die blaue Stunde der Infor- matik. Springer-Vieweg, 2020. viii+200 Seiten, ISBN 978-3-658-28754-2, ISBN 978-3-658- 28755-9 (eBook). https://doi.org/10.1007/978-3-658-28755-9 2019 | with Rudolf Berghammer and Michael Winter. Cryptomorphic topological structures: A computational, relation-algebraic approach. Journal of Logical and Algebraic Methods in Programming 102 (2019), 17{45. https://doi.org/10.1016/j.jlamp.2018. 09.004. 2018 | with Michael Winter. Relational Topology, volume 2208 of Lecture Notes in Mathe- matics. Springer-Verlag, 2018. https://doi.org/10.1007/978-3-319-74451-3 2017 | Partiality III: Observability and dispersion, 2017. In preparation. 2015 | Arranging binary relations nicely | A Guide. Technical Report 2015-01, Fakult¨atf¨ur Informatik, Universit¨atder Bundeswehr M¨unchen, December 2015. https://titurel. org/HandBookPrinted.pdf | A relational view on stochastics. The Festschrift on the occasion of the 60th birthday of Jos´eN. Oliveira, 2015. 2014 | with Michael Winter. Relational Topology. Technical Report 2014-03, Fakult¨atf¨ur Informatik, Universit¨atder Bundeswehr M¨unchen, 76 pages, Nov. 2014. | with Lothar Schmitz. Prof. Friedrich L. Bauer und seine Verbindungen zur Fakult¨at f¨urInformatik | Der Vater der deutschen Informatik. Hochschulkurier der Universit¨atder Bundeswehr, 50, 2014. 10{11. | with Michael Winter. Relational Mathematics Continued. Tech. Rep. 2014- 01, Fakult¨atf¨urInformatik, Universit¨atder Bundeswehr M¨unchen, 45 pages, Apr. 2014. http://arxiv.org/abs/1403.6957.
    [Show full text]
  • A Proposal for a Multilevel Relational Reference Language
    A Proposal for a Multilevel Relational Reference Language Gunther Schmidt Institute for Software Technology, Department of Computing Science Federal Armed Forces University Munich, 85577 Neubiberg e-Mail: [email protected] Abstract A highly expressive multilevel relational reference language is proposed that covers most possibilities to use relations in practical applications. The language is designed to describe work in a heterogeneous setting. It originated from a Haskell-based system announced in [Sch02], fore- runners of which were [HBS94,Hat97]. This language is intended to serve a variety of purposes. First, it shall allow to formulate all of the problems that have so far been tackled us- ing relational methods providing full syntax- and type-control. Trans- formation of relational terms and formulae in the broadest sense shall be possible as well as interpretation in many forms. In the most simple way, boolean matrices will serve as an interpretation, but also non- representable models as with the Rath-system may be used. Proofs of relational formulae in the style of Ralf or in Rasiowa-Sikorski style are aimed at. Cooperation and communication around this research was partly spon- sored by the European Cost Action 274: Tarski (Theory and Appli- cations of Relational Structures as Knowledge Instruments), which is gratefully acknowledged. 1Introduction When an engineer is about to design an artefact and has to apply Linear Algebra methods (such as solving systems of linear equations or determining eigenvalues and eigenvectors), he will approach the respective computing center and most certainly get the necessary software. When the matrices considered become boolean matrices, i.e., relations, the situation changes dramatically.
    [Show full text]
  • Mathematical Objects Arising from Equivalence Relations, and Their Implementation in Quine’S NF
    Mathematical Objects arising from Equivalence Relations, and their Implementation in Quine's NF Thomas Forster Department of Pure Mathematics and Mathematical Statistics, University of Cambridge January 4, 2014 ABSTRACT Many mathematical objects arise from equivalence classes, and invite implementation as those classes. Set existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories (e.g. NF and Church's CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for \low" sets, and thus|e.g.|a set of all (low) ordinals. However that set has an ordinal in turn, which is not a member of the set constructed, so no set of all ordinals is obtained thereby. This \recurrence problem" is discussed. This essay can be seen as a non-technical companion to [6] and is, like it, a precursor to|and will be subsumed into|the longer book-length treatment of these matters that I have promised myself I will write. 1 Introduction I would like to start off by insisting on some terminology: what we try to do to cardinals, ordinals (and other mathematical entities) when we come to set theory is not to define them but to implement them. We don't need to define cardinals: we know perfectly well what a cardinal is: a cardinal is that thing that two sets have in common (i.e., to which they are related in the same way) precisely when they are equinumerous. If we wish to tell a story in which every- thing is a set then we have to have ways of implementing these mathematical objects from outside set theory as sets.
    [Show full text]
  • Generalising Some Graph Decomposition Algorithms Binh-Minh Bui-Xuan, Michel Habib, Vincent Limouzy, Fabien De Montgolfier
    Homogeneity vs. Adjacency: generalising some graph decomposition algorithms Binh-Minh Bui-Xuan, Michel Habib, Vincent Limouzy, Fabien de Montgolfier To cite this version: Binh-Minh Bui-Xuan, Michel Habib, Vincent Limouzy, Fabien de Montgolfier. Homogeneity vs. Ad- jacency: generalising some graph decomposition algorithms. WG: Graph-Theoretic Concepts in Com- puter Science, Jun 2006, Bergen, Norway. pp.278-288, 10.1007/11917496_25. hal-00020188 HAL Id: hal-00020188 https://hal.archives-ouvertes.fr/hal-00020188 Submitted on 13 Mar 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Homogeneity vs. Adjacency: generalising some graph decomposition algorithms B.-M. Bui Xuan1, M. Habib2, V. Limouzy2, and F. de Montgolfier2 1 LIRMM, Universit´eMontpellier 2, France. [email protected] 2 LIAFA, Universit´eParis 7, France. {habib,limouzy,fm}@liafa.jussieu.fr Abstract. In this paper, a new general decomposition theory inspired from modular graph decompo- sition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are still efficient. This theory not only unifies the usual modular decomposition generalisations such as modular de- composition of directed graphs or decomposition of 2-structures, but also star cutsets and bimodular decomposition.
    [Show full text]
  • Springer Book Archives Seite 823 P-Adic Numbers 1997 1984
    Springer Book Archives p-adic Numbers An Introduction Fernando Quadros Gouvea 1997 P-adic Numbers, p-adic Analysis, and Zeta- Functions Neal Koblitz 1984 Paartherapie und Paarsynthese Lernmodell Liebe Michael Cöllen 1997 Deanna J. Stouder; Peter A. Bisson; Robert J. Pacific Salmon And Their Ecosystems Status and future options Naiman 1997 Package Electrical Modeling, Thermal Modeling, and Processing for GaAs Wireless Applications Dean L. Monthei 1999 Packaging in the Envirnment Geoffrey M. Levy 1995 Packaging in the Environment Geoffrey M. Levy 1992 Packaging Pharmaceutical and Healthcare Products Frank A. Paine; H. Lockhart 1995 Packaging User's Handbook Frank A. Paine 1990 Pädiatrie upgrade 2002 Weiter- und Fortbildung B. Koletzko; D. Reinhardt; S. Stöckler-Ipsiroglu 2002 Pädiatrische Kardiologie Thomas Borth-Bruhns; Andrea Eichler 2004 Erkrankungen des Herzens bei Neugeborenen, Säuglingen, Kindern und Pädiatrische Kardiologie Heranwachsenden Jürgen Apitz 2002 Pädiatrische Nephrologie K. Schärer; O. Mehls 2002 Paediatric Emergencies Thomas Lissauer 1982 Paediatric Endocrinology in Clinical Practice A. Aynsley-Green 1984 Paediatric Neoplasia An Atlas and Text S. Variend 1993 Paediatrics N.D. Barnes; N.R.C. Roberton 1982 Proceedings of the First Convention of the Pain - A Medical and Anthropological Academia Eurasiana Neurochirurgia, Bonn, Challenge September 25-28, 1985 Jean Brihaye; Fritz Loew; H.W. Pia 1987 Pain and Neurogenic Inflammation S.D. Brain; P.K. Moore 1999 Nayef E. Saadé; Suhayl J. Jabbur; A. Vania Pain and Neuroimmune Interactions Apkarian 2000 J.M. Greep; H.A.J. Lemmens; D.B. Roos; H.C. Pain in Shoulder and Arm An Integrated View Urschel 1979 Pain Management and Anesthesiology M.A. Ashburn; P.G.
    [Show full text]