DOCSLIB.ORG

FORMAL LOGICAL METHODS for SYSTEM SECURITY and CORRECTNESS NATO Science for Peace and Security Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
FORMAL LOGICAL METHODS for SYSTEM SECURITY and CORRECTNESS NATO Science for Peace and Security Series FORMAL LOGICAL METHODS FOR SYSTEM SECURITY AND CORRECTNESS NATO Science for Peace and Security Series This Series presents the results of scientific meetings supported under the NATO Programme: Science for Peace and Security (SPS). The NATO SPS Programme supports meetings in the following Key Priority areas: (1) Defence Against Terrorism; (2) Countering other Threats to Security and (3) NATO, Partner and Mediterranean Dialogue Country Priorities. The types of meeting supported are generally “Advanced Study Institutes” and “Advanced Research Workshops”. The NATO SPS Series collects together the results of these meetings. The meetings are co-organized by scientists from NATO countries and scientists from NATO’s “Partner” or “Mediterranean Dialogue” countries. The observations and recommendations made at the meetings, as well as the contents of the volumes in the Series, reflect those of participants and contributors only; they should not necessarily be regarded as reflecting NATO views or policy. Advanced Study Institutes (ASI) are high-level tutorial courses to convey the latest developments in a subject to an advanced-level audience. Advanced Research Workshops (ARW) are expert meetings where an intense but informal exchange of views at the frontiers of a subject aims at identifying directions for future action. Following a transformation of the programme in 2006 the Series has been re-named and re- organised. Recent volumes on topics not related to security, which result from meetings supported under the programme earlier, may be found in the NATO Science Series. The Series is published by IOS Press, Amsterdam, and Springer Science and Business Media, Dordrecht, in conjunction with the NATO Public Diplomacy Division. Sub-Series A. Chemistry and Biology Springer Science and Business Media B. Physics and Biophysics Springer Science and Business Media C. Environmental Security Springer Science and Business Media D. Information and Communication Security IOS Press E. Human and Societal Dynamics IOS Press http://www.nato.int/science http://www.springer.com http://www.iospress.nl Sub-Series D: Information and Communication Security – Vol. 14 ISSN 1874-6268 Formal Logical Methods for System Security and Correctness Edited by Orna Grumberg Technion, Israel Tobias Nipkow Technische Universität München, Germany and Christian Pfaller Technische Universität München, Germany Amsterdam • Berlin • Oxford • Tokyo • Washington, DC Published in cooperation with NATO Public Diplomacy Division Proceedings of the NATO Advanced Study Institute on Formal Logical Methods for System Security and Correctness Marktoberdorf, Germany 31 July–12 August 2007 © 2008 IOS Press. All rights reserved. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-58603-843-4 Library of Congress Control Number: 2008922610 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Distributor in the USA and Canada Gazelle Books Services Ltd. IOS Press, Inc. White Cross Mills 4502 Rachael Manor Drive Hightown Fairfax, VA 22032 Lancaster LA1 4XS USA United Kingdom fax: +1 703 323 3668 fax: +44 1524 63232 e-mail: [email protected] e-mail: [email protected] LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS Formal Logical Methods for System Security and Correctness v O. Grumberg et al. (Eds.) IOS Press, 2008 © 2008 IOS Press. All rights reserved. Preface The objective of our Summer School 2007 on Formal Logical Methods for System Security and Correctness was to present the state-of-the-art in the field of proof technol- ogy in connection with secure and correct software. The lecturers have shown that meth- ods of correct-by-construction program and process synthesis allow a high level pro- gramming method more amenable to security and reliability analysis and guarantees. By providing the necessary theoretical background and presenting corresponding applica- tion oriented concepts, the objective was an in-depth presentation of such methods cov- ering both theoretical foundations and industrial practice. In detail the following courses were given: GILLES BARTHE lectured on Verification Methods for Software Security and Cor- rectness. The objective of the lectures was to present static enforcement mechanisms to ensure reliability and security of mobile code. First, he introduced a type based verifier for ensuring information flow policies and a verification condition generator for Java bytecode programs. He also described how these mechanisms have been certified using the proof assistant Coq. Second, he related these two enforcement mechanisms to their counterparts for Java programs. ROBERT CONSTABLE’s lectures Logical Foundations of Computer Security were concerned with developing correct-by-construction security protocols for distributed sys- tems on communication networks. He used computational type theory to express logi- cally sound cryptographic services and established them by machine generated formal proofs. In his course Building a Software Model-Checker JAVIER ESPARZA introduced jMoped, a tool for the analysis of Java programs. He then explained the theory and algo- rithms behind the tool. In jMoped is assumed that variables have a finite range. He started by considering the computational complexity of verifying different classes of programs satisfying this constraint. After choosing a reasonable class of programs, he introduced a model-checking algorithm based on pushdown automata and then addressed the problem of data. He presented an approach to this problem based on BDDs and counterexample- based abstraction refinement with interpolants. With Automatic Refinement and Vacuity Detection for Symbolic Trajectory Evalu- ation ORNA GRUMBERG presented a powerful model checking technique called Sym- bolic Trajectory Evaluation (STE), which is particularly suitable for hardware. STE is applied to a circuit M, described as a graph over nodes (gates and latches). The specifi- cation consists of assertions in a restricted temporal language. The assertions are of the form A =⇒ C, where the antecedent A expresses constraints on nodes n at different times t, and the consequent C expresses requirements that should hold on such nodes (n, t). Abstraction in STE is derived from the specification by initializing all inputs not appearing in A to the X (“” unknown ) value. A refinement amounts to changing the as- sertion in order to present node values more accurately. A symbolic simulation and the specific type of abstraction, used in STE, was described. We proposed a technique for automatic refinement of assertions in STE, in case the model checking results in X.In vi this course the notion of hidden vacuity for STE was defined and several methods for detecting it was suggested. JOHN HARRISON lectured on Automated and Interactive Theorem Proving.Hecov- ered a range of topics from Boolean satisfiability checking (SAT), several approaches to first-order automated theorem proving, special methods for equations, decision pro- cedures for important special theories, and interactive proofs. He gave some suitable references. MARTIN HOFMANN gave a series of lectures on Correctness of Effect-based Pro- gram Transformations in which a type system was considered capable of tracking read- ing, writing and allocation in a higher-order language with dynamically allocated refer- ences. He gave a denotational semantics to this type system which allowed us to validate a number of effect-dependent program equivalences in the sense of observational equiv- alence. On the way we learned popular techniques such as parametrised logical relations, regions, admissible relations, etc which belong to the toolbox of researchers in principles of programming languages. Abstract and Concrete Models of Recursion was the course of MARTIN HYLAND. Systems of information flow are fundamental in computing systems generally and secu- rity protocols in particular. One key issue is feedback (or recursion) and we developed an approach based on the notion of trace. He covered applications to fixed point theory, automata theory and topics in the theory of processes. In his course Security Analysis of Network Protocols JOHN MITCHELL provided an introduction to network protocols that have security requirements. He covered a variety of contemporary security protocols and gave students information needed to carry out case studies using automated tools and formal techniques. The first lectures surveyed protocols and their properties, including secrecy, authentication, key establishment, and fairness. The second part covered standard formal models and tools used in security protocol analysis, and described their advantages and limitations. With his lectures on The Engineering Challenges of Trustworthy Computing GREG MORRISETT talked about a range of language, compiler, and verification techniques that can be used to address safety and security issues in systems software today. Some of the techniques, such as software fault isolation, are aimed at legacy software and provide relatively weak but important guarantees, and come with significant overhead. Other techniques, such as proof-carrying code, offer the potential of fine-grained protection with low overhead, but introduce significant verification challenges. The focus of
Load more

Recommended publications
  • Bialgebras in Rel
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Electronic Notes in Theoretical Computer Science 265 (2010) 337–350 www.elsevier.com/locate/entcs Bialgebras in Rel Masahito Hasegawa1,2 Research Institute for Mathematical Sciences Kyoto University Kyoto, Japan Abstract We study bialgebras in the compact closed category Rel of sets and binary relations. We show that various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras in Rel.In particular, for any group G we derive a ribbon category of crossed G-sets as the category of modules of a Hopf algebra in Rel which is obtained by the quantum double construction. This category of crossed G-sets serves as a model of the braided variant of propositional linear logic. Keywords: monoidal categories, bialgebras and Hopf algebras, linear logic 1 Introduction For last two decades it has been shown that there are plenty of important exam- ples of traced monoidal categories [21] and ribbon categories (tortile monoidal cat- egories) [32,33] in mathematics and theoretical computer science. In mathematics, most interesting ribbon categories are those of representations of quantum groups (quasi-triangular Hopf algebras)[9,23] in the category of finite-dimensional vector spaces. In many of them, we have non-symmetric braidings [20]: in terms of the graphical presentation [19,31], the braid c = is distinguished from its inverse c−1 = , and this is the key property for providing non-trivial invariants (or de- notational semantics) of knots, tangles and so on [12,23,33,35] as well as solutions of the quantum Yang-Baxter equation [9,23].
    [Show full text]
  • On the Completeness of the Traced Monoidal Category Axioms in (Rel,+)∗
    Acta Cybernetica 23 (2017) 327–347. On the Completeness of the Traced Monoidal Category Axioms in (Rel,+)∗ Mikl´os Barthaa To the memory of my friend and former colleague Zolt´an Esik´ Abstract It is shown that the traced monoidal category of finite sets and relations with coproduct as tensor is complete for the extension of the traced sym- metric monoidal axioms by two simple axioms, which capture the additive nature of trace in this category. The result is derived from a theorem saying that already the structure of finite partial injections as a traced monoidal category is complete for the given axioms. In practical terms this means that if two biaccessible flowchart schemes are not isomorphic, then there exists an interpretation of the schemes by partial injections which distinguishes them. Keywords: monoidal categories, trace, iteration, feedback, identities in cat- egories, equational completeness 1 Introduction It was in March, 2012 that Zolt´an visited me at the Memorial University in St. John’s, Newfoundland. I thought we would find out a research topic together, but he arrived with a specific problem in mind. He knew about the result found by Hasegawa, Hofmann, and Plotkin [12] a few years earlier on the completeness of the category of finite dimensional vector spaces for the traced monoidal category axioms, and wanted to see if there is a similar completeness statement true for the matrix iteration theory of finite sets and relations as a traced monoidal category. Unfortunately, during the short time we spent together, we could not find the solu- tion, and after he had left I started to work on some other problems.
    [Show full text]
  • A Survey of Graphical Languages for Monoidal Categories
    5 Traced categories 32 5.1 Righttracedcategories . 33 5.2 Planartracedcategories. 34 5.3 Sphericaltracedcategories . 35 A survey of graphical languages for monoidal 5.4 Spacialtracedcategories . 36 categories 5.5 Braidedtracedcategories . 36 5.6 Balancedtracedcategories . 39 5.7 Symmetrictracedcategories . 40 Peter Selinger 6 Products, coproducts, and biproducts 41 Dalhousie University 6.1 Products................................. 41 6.2 Coproducts ............................... 43 6.3 Biproducts................................ 44 Abstract 6.4 Traced product, coproduct, and biproduct categories . ........ 45 This article is intended as a reference guide to various notions of monoidal cat- 6.5 Uniformityandregulartrees . 47 egories and their associated string diagrams. It is hoped that this will be useful not 6.6 Cartesiancenter............................. 48 just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of 7 Dagger categories 48 topological notions, and have omitted most proofs. Nevertheless, the exposition 7.1 Daggermonoidalcategories . 50 is sufficiently detailed to make it clear what is presently known, and to serve as 7.2 Otherprogressivedaggermonoidalnotions . ... 50 a starting place for more in-depth study. Where possible, we provide pointers to 7.3 Daggerpivotalcategories. 51 more rigorous treatments in the literature. Where we include results that have only 7.4 Otherdaggerpivotalnotions . 54 been proved in special cases, we indicate this in the form of caveats. 7.5 Daggertracedcategories . 55 7.6 Daggerbiproducts............................ 56 Contents 8 Bicategories 57 1 Introduction 2 9 Beyond a single tensor product 58 2 Categories 5 10 Summary 59 3 Monoidal categories 8 3.1 (Planar)monoidalcategories . 9 1 Introduction 3.2 Spacialmonoidalcategories .
    [Show full text]
  • A Survey of Graphical Languages for Monoidal Categories
    A survey of graphical languages for monoidal categories Peter Selinger Dalhousie University Abstract This article is intended as a reference guide to various notions of monoidal cat- egories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats. Contents 1 Introduction 2 2 Categories 5 3 Monoidal categories 8 3.1 (Planar)monoidalcategories . 9 3.2 Spacialmonoidalcategories . 14 3.3 Braidedmonoidalcategories . 14 3.4 Balancedmonoidalcategories . 16 arXiv:0908.3347v1 [math.CT] 23 Aug 2009 3.5 Symmetricmonoidalcategories . 17 4 Autonomous categories 18 4.1 (Planar)autonomouscategories. 18 4.2 (Planar)pivotalcategories . 22 4.3 Sphericalpivotalcategories. 25 4.4 Spacialpivotalcategories . 25 4.5 Braidedautonomouscategories. 26 4.6 Braidedpivotalcategories. 27 4.7 Tortilecategories ............................ 29 4.8 Compactclosedcategories . 30 1 5 Traced categories 32 5.1 Righttracedcategories . 33 5.2 Planartracedcategories. 34 5.3 Sphericaltracedcategories . 35 5.4 Spacialtracedcategories . 36 5.5 Braidedtracedcategories . 36 5.6 Balancedtracedcategories . 39 5.7 Symmetrictracedcategories . 40 6 Products, coproducts, and biproducts 41 6.1 Products................................. 41 6.2 Coproducts ............................... 43 6.3 Biproducts................................ 44 6.4 Traced product, coproduct, and biproduct categories .
    [Show full text]
  • Some Aspects of Categories in Computer Science
    This is a technical rep ort A mo died version of this rep ort will b e published in Handbook of Algebra Vol edited by M Hazewinkel Some Asp ects of Categories in Computer Science P J Scott Dept of Mathematics University of Ottawa Ottawa Ontario CANADA Sept Contents Intro duction Categories Lamb da Calculi and FormulasasTyp es Cartesian Closed Categories Simply Typ ed Lamb da Calculi FormulasasTyp es the fundamental equivalence Some Datatyp es Polymorphism Polymorphic lamb da calculi What is a Mo del of System F The Untyp ed World Mo dels and Denotational Semantics CMonoids and Categorical Combinators Church vs Curry Typing Logical Relations and Logical Permutations Logical Relations and Syntax Example ReductionFree Normalization Categorical Normal Forms P category theory and normalization algorithms Example PCF PCF Adequacy Parametricity Dinaturality Reynolds Parametricity
    [Show full text]
  • Traced Monoidal Categories As Algebraic Structures in Prof
    Traced monoidal categories as algebraic structures in Prof Nick Hu Jamie Vicary University of Oxford University of Cambridge [email protected] [email protected] Abstract We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal cat- egories as algebraic structures in Prof, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced ∗-autonomous categories, prov- ing a new equivalence result between the left ⊗-trace and the right -trace, and describing a new condition under which traced ∗-autonomous categories become` autonomous. 1 Introduction One way to interpret category theory is as a theory of systems and processes, whereby monoidal structure naturally lends itself to enable processes to be juxtaposed in parallel. Following this analogy, the presence of a trace structure embodies the notion of feedback: some output of a process is directly fed back in to one of its inputs. For instance, if we think of processes as programs, then feedback is some kind of recursion [11]. This becomes clearer still when we consider how tracing is depicted in the standard graphical calculus [16, § 5], as follows: A X A f f X (1) B X B Many important algebraic structures which are typically defined as sets-with-structure, like mon- oids, groups, or rings, may be described abstractly as algebraic structure, which when interpreted in Set yield the original definition.
    [Show full text]
  • U Ottawa L'universite Canadienne Canada's University FACULTE DES ETUDES SUPERIEURES ^=1 FACULTY of GRADUATE and ET POSTOCTORALES U Ottawa POSDOCTORAL STUDIES
    nm u Ottawa L'Universite canadienne Canada's university FACULTE DES ETUDES SUPERIEURES ^=1 FACULTY OF GRADUATE AND ET POSTOCTORALES U Ottawa POSDOCTORAL STUDIES L'Universite canadienne Canada's university Octavio Malherbe ^_^^-^^^„„„^______ Ph.D. (Mathematics) _.__...„___„„ Department of Mathematics and Statistics Categorical Models of Computation: Partially Traced Categories and Presheaf Models of Quantum Computation TITRE DE LA THESE / TITLE OF THESIS Philip Scott „„„„„„„„_^ Peter Selinger Richard Bulte Pieter Hofstra Robin Cockett Benjamin Steinberg University of Calgary Gary W. Slater Le Doyen de la Faculte des etudes superieures et postdoctorales / Dean of the Faculty of Graduate and Postdoctoral Studies CATEGORICAL MODELS OF COMPUTATION: PARTIALLY TRACED CATEGORIES AND PRESHEAF MODELS OF QUANTUM COMPUTATION By Octavio Malherbe, B.Sc, M.Sc. Thesis submitted to the Faculty of Graduate and Postdoctoral Studies University of Ottawa in partial fulfillment of the requirements for the PhD degree in the awa-Carleton Institute for Graduate Studies and Research in Mathematics and Statistics © Octavio Malherbe, Ottawa, Canada, 2010 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-73903-7 Our file Notre reference ISBN: 978-0-494-73903-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license
    [Show full text]
  • The Uniformity Principle on Traced Monoidal Categories†
    Publ. RIMS, Kyoto Univ. 40 (2004), 991–1014 The Uniformity Principle on Traced Monoidal Categories† By Masahito Hasegawa∗ Abstract The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin’s principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits. §1. Introduction Traced monoidal categories were introduced by Joyal, Street and Verity [16] as the categorical structure for cyclic phenomena arising from various areas of mathematics, most notably knot theory [25]. They are (balanced) monoidal categories [15] enriched with a trace, which is a natural generalization of the traditional notion of trace in linear algebra, and can be regarded as an operator to create a “loop”. For example, the braid closure operation — e.g. the con- struction of the trefoil knot from a braid depicted below — can be considered as the trace operator on the category of tangles. ⇒ = Communicated by R. Nakajima. Received December 16, 2003. 2000 Mathematics Subject Classification(s): 18D10, 19D23 †This article is an invited contribution to a special issue of Publications of RIMS com- memorating the fortieth anniversary of the founding of the Research Institute for Math- ematical Sciences. ∗Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan; PRESTO, Japan Science and Technology Agency, Japan c 2004 Research Institute for Mathematical Sciences, Kyoto University.
    [Show full text]
  • Arxiv:0910.2931V1 [Quant-Ph] 15 Oct 2009 Fslne 2] Esosta H Rmwr Fcmltl Oiiem Positive Completely of Framework the That Shows He [25]
    Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories Samson Abramsky Oxford University Computing Laboratory Wolfson Building, Parks Road, Oxford OX1 3QD, U.K. http://web.comlab.ox.ac.uk/oucl/work/samson.abramsky/ Abstract. We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in’ to the free construction. 1 Introduction In this preliminary section, we will discuss the background and motivation for the technical results in the main body of the paper, in a fairly wide-ranging fashion. The technical material itself should be essentially self-contained, from the level of a basic familiarity with monoidal categories (for which see e.g. [20]). 1.1 Background In recent work [4,5], the present author and Bob Coecke have developed a cat- egorical axiomatics for Quantum Mechanics, as a foundation for high-level ap- arXiv:0910.2931v1 [quant-ph] 15 Oct 2009 proaches to quantum informatics: type systems, logics, and languages for quan- tum programming and quantum protocol specification.
    [Show full text]
  • Tracing the Man in the Middle in Monoidal Categories Dusko Pavlovic
    Tracing the Man in the Middle in Monoidal Categories Dusko Pavlovic To cite this version: Dusko Pavlovic. Tracing the Man in the Middle in Monoidal Categories. 11th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Mar 2012, Tallinn, Estonia. pp.191-217, 10.1007/978-3-642-32784-1_11. hal-01539882 HAL Id: hal-01539882 https://hal.inria.fr/hal-01539882 Submitted on 15 Jun 2017 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. Distributed under a Creative Commons Attribution| 4.0 International License Tracing the Man in the Middle in Monoidal Categories Dusko Pavlovic Email: [email protected] Royal Holloway, University of London, and University of Twente Abstract. Man-in-the-Middle (MM) is not only a ubiquitous attack pattern in security, but also an important paradigm of network compu- tation and economics. Recognizing ongoing MM-attacks is an important security task; modeling MM-interactions is an interesting task for seman- tics of computation. Traced monoidal categories are a natural framework for MM-modelling, as the trace structure provides a tool to hide what happens in the middle. An effective analysis of what has been traced out seems to require an additional property of traces, called normality.
    [Show full text]
  • Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories
    Finite Dimensional Vector Spaces are Complete for Traced Symmetric Monoidal Categories Masahito Hasegawa1, Martin Hofmann2, and Gordon Plotkin3 1 RIMS, Kyoto University, [email protected] 2 LMU M¨unchen, Institut f¨urInformatik, [email protected] 3 LFCS, University of Edinburgh, [email protected] Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a sig- nature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists a strong traced functor into FinVectk which distinguishes them. There- fore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk. 1 Introduction This paper is affectionately dedicated to Professor B. Trakhtenbrot on the occa- sion of his 85th birthday. Cyclic networks of various kinds occur in computer sci- ence, and other fields, and have long been of interest to Professor Trakhtenbrot: see, e.g., [15, 9, 16, 8]. In this paper they arise in connection with Joyal, Street and Verity’s traced monoidal categories [6]. These categories were introduced to provide a categorical structure for cyclic phenomena arising in various areas of mathematics, in particular knot theory [17]; they are (balanced) monoidal cate- gories [5] enriched with a trace, a natural generalization of the traditional notion of trace in linear algebra that can be thought of as a ‘loop’ operator. In computer science, specialized versions of traced monoidal categories nat- urally arise as recursion/feedback operators as well as cyclic data structures.
    [Show full text]
  • Rewriting Graphically with Cartesian Traced Categories
    Rewriting with Cartesian Traced Monoidal Categories George Kaye and Dan R. Ghica July 8, 2021 Motivation. Constructors, architects, and engineers have always enjoyed using blueprints, diagrams, and other kinds of graphical representations of their designs. These are often more than simply illustrations aiding the under- standing of a formal specification: they are the specification itself. By contrast, in mathematics, diagrams have not been traditionally considered first-class citizens, although they are often used to help the reader visualise a construc- tion or a proof. However, the development of new formal diagrammatic languages for a variety of systems such as quantum communication and computation [8], computational linguistics [9] and signal-flow graphs [2,3], proved that diagrams can be used not just to aid understanding of proofs, but to formulate them as well. This approach has multifaceted advantages, from enabling the use of graph-theoretical techniques to aid reasoning [13] to making the teaching of algebraic concepts to younger students less intimidating [12]. String diagrams [25] are becoming the established mathematical language of diagrammatic reasoning, whereby equal terms are usually interpreted as isomorphic (or isotopic) diagrams. While the ‘only connectivity matters’ mantra is enough for reasoning about structural properties, properties which have computational content require a rewriting of the diagram. To make this possible, diagrams must be represented as combinatorial objects, such as graphs or hypergraphs, which have enough structure; the framework of adhesive categories [24] is often used to ensure that graph rewriting is always well-defined. These diagrammatic languages build on an infrastructure of (usually symmetric and strict) monoidal cate- gories [18], and in particular compact closed categories [20].
    [Show full text]