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1968 results in Distributed, Networked and Mobile Computing

Page 46 of 99

  • 15 - Graphical Networks

  • from Part III - Multihop Networks
  • Abbas El Gamal, Young-Han Kim, University of California, San Diego
  • Book:
    Network Information Theory
    Published online:
    05 June 2012
    Print publication:
    08 December 2011, pp 363-381

  • Summary

    So far we have studied single-hop networks in which each node is either a sender or a receiver. In this chapter, we begin the discussion of multihop networks, where some nodes can act as both senders and receivers and hence communication can be performed over multiple rounds. We consider the limits on communication of independent messages over networks modeled by a weighted directed acyclic graph. This network model represents, for example, a wired network or a wireless mesh network operated in time or frequency division, where the nodes may be servers, handsets, sensors, base stations, or routers. The edges in the graph represent point-to-point communication links that use channel coding to achieve close to error-free communication at rates below their respective capacities. We assume that each node wishes to communicate a message to other nodes over this graphical network. The nodes can also act as relays to help other nodes communicate their messages. What is the capacity region of this network?

    Although communication over such a graphical network is not hampered by noise or interference, the conditions on optimal information flow are not known in general. The difficulty arises in determining the optimal relaying strategies when several messages are to be sent to different destination nodes.

    We first consider the graphical multicast network, where a source node wishes to communicate a message to a set of destination nodes. We establish the cutset upper bound on the capacity and show that it is achievable error-free via routing when there is only one destination, leading to the celebrated max-flow min-cut theorem. When there are multiple destinations, routing alone cannot achieve the capacity, however. We show that the cutset bound is still achievable, but using more sophisticated coding at the relays. The proof of this result involves linear network coding in which the relays perform simple linear operations over a finite field.

    We then consider graphical networks with multiple independent messages. We show that the cutset bound is tight when the messages are to be sent to the same set of destination nodes (multimessage multicast), and is achieved again error-free using linear network coding. When each message is to be sent to a different set of destination nodes, however, neither the cutset bound nor linear network coding is optimal in general.

  • 24 - Networking and InformationTheory

  • from Part IV - Extensions
  • Abbas El Gamal, Young-Han Kim, University of California, San Diego
  • Book:
    Network Information Theory
    Published online:
    05 June 2012
    Print publication:
    08 December 2011, pp 600-620

  • Summary

    The source and network models we discussed so far capture many essential ingredients of real-world communication networks, including

  • • noise,

  • • multiple access,

  • • broadcast,

  • • interference,

  • • time variation and uncertainty about channel statistics,

  • • distributed compression and computing,

  • • joint source–channel coding,

  • • multihop relaying,

  • • node cooperation,

  • • interaction and feedback, and

  • • secure communication.

    • Although a general theory for information flow under these models remains elusive, we have seen that there are several coding techniques—some of which are optimal or close to optimal—that promise significant performance improvements over today's practice. Still, the models we discussed do not capture other key aspects of real-world networks.

  • • We assumed that data is always available at the communication nodes. In real-world networks, data is bursty and the nodes have finite buffer sizes.

  • • We assumed that the network has a known and fixed number of users. In real-world networks, users can enter and leave the network at will.

  • • We assumed that the network operation is centralized and communication over the network is synchronous. Many real-world networks are decentralized and communication is asynchronous.

  • • We analyzed performance assuming arbitrarily long delays. In many networking applications, delay is a primary concern.

  • • We ignored the overhead (protocol) needed to set up the communication as well as the cost of feedback and channel state information.

    • While these key aspects of real-world networks have been at the heart of the field of computer networks, they have not been satisfactorily addressed by network information theory, either because of their incompatibility with the basic asymptotic approach of information theory or because the resulting models are messy and intractable. There have been several success stories at the intersection of networking and network information theory, however. In this chapter we discuss three representative examples.

    We first consider the channel coding problem for a DMC with random data arrival. We show that reliable communication is feasible provided that the data arrival rate is less than the channel capacity. Similar results can be established for multiuser channels and multiple data streams. A key new ingredient in this study is the notion of queue stability.

Advanced Topics in Bisimulation and Coinduction
  • Edited by Davide Sangiorgi, Jan Rutten
  • Published online:
    05 November 2011
    Print publication:
    13 October 2011
  • Coinduction is a method for specifying and reasoning about infinite data types and automata with infinite behaviour. In recent years, it has come to play an ever more important role in the theory of computing. It is studied in many disciplines, including process theory and concurrency, modal logic and automata theory. Typically, coinductive proofs demonstrate the equivalence of two objects by constructing a suitable bisimulation relation between them. This collection of surveys is aimed at both researchers and Master's students in computer science and mathematics and deals with various aspects of bisimulation and coinduction, with an emphasis on process theory. Seven chapters cover the following topics: history, algebra and coalgebra, algorithmics, logic, higher-order languages, enhancements of the bisimulation proof method, and probabilities. Exercises are also included to help the reader master new material.

  • 7 - Probabilistic bisimulation

  • Edited by Davide Sangiorgi, Jan Rutten, Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
  • Book:
    Advanced Topics in Bisimulation and Coinduction
    Published online:
    05 November 2011
    Print publication:
    13 October 2011, pp 290-326

  • Summary

    Introduction

    The history of bisimulation is well documented in earlier chapters of this book. In this chapter we will look at a major non-trivial extension of the theory of labelled transition systems: probabilistic transition systems. There are many possible extensions of theoretical and practical interest: real-time, quantitative, independence, spatial and many others. Probability is the best theory we have for handling uncertainty in all of science, not just computer science. It is not an idle extension made for the purpose of exploring what is theoretically possible. Non-determinism is, of course, important, and arises in computer science because sometimes we just cannot do any better or because we lack quantitative data from which to make quantitative predictions. However, one does not find any use of non-determinism in a quantitative science like physics, though it appears in sciences like biology where we have not yet reached a fundamental understanding of the nature of systems.

    When we do have data or quantitative models, it is far preferable to analyse uncertainty probabilistically. A fundamental reason that we want to use probabilistic reasoning is that if we merely reported what is possible and then insisted that no bad things were possible, we would trust very few system designs in real life. For example, we would never trust a communication network, a car, an aeroplane, an investment bank nor would we ever take any medication! In short, only very few idealised systems ever meet purely logical specifications. We need to know the ‘odds’ before we trust any system.

  • 1 - Towards bisimulation

  • Davide Sangiorgi, Università di Bologna
  • Book:
    Introduction to Bisimulation and Coinduction
    Published online:
    05 August 2012
    Print publication:
    13 October 2011, pp 11-27

  • Summary

    We introduce bisimulation and coinduction roughly following the way that led to their discovery in Computer Science. Thus the general topic is the semantics of concurrent languages (or systems), in which several activities, the processes, may run concurrently. Central questions are: what is, mathematically, a process? And what does it mean that two processes are ‘equal’? We seek notions of process and process equality that are both mathematically and practically interesting. For instance, the notions should be amenable to effective techniques for proving equalities, and the equalities themselves should be justifiable, according to the way processes are used.

    We hope that the reader will find this way of proceeding helpful for understanding the meaning of bisimulation and coinduction. The emphasis on processes is also justified by the fact that concurrency remains today the main application area for bisimulation and coinduction.

    We compare processes and functions in Section 1.1. We will see that processes do not fit the input/output schema of functions. A process has an interactive behaviour, and it is essential to take this into account. We formalise the idea of behaviour in Section 1.2 via labelled transition systems (LTSs), together with notations and terminology for them. We discuss the issue of equality between behaviours in Section 1.3. We first try to re-use notions of equality from Graph Theory and Automata Theory. The failure of these attempts leads us to proposing bisimilarity, in Section 1.4. We introduce the reader to the bisimulation proof method through a number of examples.

  • Preface

  • Edited by Davide Sangiorgi, Jan Rutten, Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
  • Book:
    Advanced Topics in Bisimulation and Coinduction
    Published online:
    05 November 2011
    Print publication:
    13 October 2011, pp xi-xiv

  • Summary

    This book is about bisimulation and coinduction. It is the companion book of the volume An Introduction to Bisimulation and Coinduction, by Davide Sangiorgi (Cambridge University Press, 2011), which deals with the basics of bisimulation and coinduction, with an emphasis on labelled transition systems, processes, and other notions from the theory of concurrency.

    In the present volume, we have collected a number of chapters, by different authors, on several advanced topics in bisimulation and coinduction. These chapters either treat specific aspects of bisimulation and coinduction in great detail, including their history, algorithmics, enhanced proof methods and logic. Or they generalise the basic notions of bisimulation and coinduction to different or more general settings, such as coalgebra, higher-order languages and probabilistic systems. Below we briefly summarise the chapters in this volume.

    • The origins of bisimulation and coinduction, by Davide Sangiorgi

    In this chapter, the origins of the notions of bisimulation and coinduction are traced back to different fields, notably computer science, modal logic, and set theory.

    • An introduction to (co)algebra and (co)induction, by Bart Jacobs and Jan Rutten

    Here the notions of bisimulation and coinduction are explained in terms of coalgebras. These mathematical structures generalise all kinds of infinitedata structures and automata, including streams (infinite lists), deterministic and probabilistic automata, and labelled transition systems. Coalgebras are formally dual to algebras and it is this duality that is used to put both induction and coinduction into a common perspective.

  • Preface

  • Davide Sangiorgi, Università di Bologna
  • Book:
    Introduction to Bisimulation and Coinduction
    Published online:
    05 August 2012
    Print publication:
    13 October 2011, pp xi-xii

  • Summary

    This book is an introduction to bisimulation and coinduction and a precursor to the companion book on more advanced topics. Between them, the books analyse the most fundamental aspects of bisimulation and coinduction, exploring concepts and techniques that can be transported to many areas. Bisimulation is a special case of coinduction, by far the most studied coinductive concept. Bisimulation was discovered in Concurrency Theory and processes remain the main application area. This explains the special emphasis on bisimulation and processes that one finds throughout the two volumes.

    This volume treats basic topics. It explains coinduction, and its duality with induction, from various angles, starting from some simple results of fixed-point theory. It then goes on to bisimulation, as a tool for defining behavioural equality among processes (bisimilarity), and for proving such equalities. It compares bisimulation with other notions of behavioural equivalence. It also presents a simple process calculus, both to show algebraic techniques for bisimulation and to illustrate the combination of inductive and coinductive reasoning.

    The companion volume, Advanced Topics in Bisimulation and Coinduction, edited by Davide Sangiorgi and Jan Rutten, deals with more specialised topics. A chapter recalls the history of the discovery of bisimulation and coinduction. Another chapter unravels the duality between induction and coinduction, both as defining principles and as proof techniques, in terms of the duality between the mathematical notions of algebra and coalgebra and properties such as initiality and finality.

  • 6 - Refinements of simulation

  • Davide Sangiorgi, Università di Bologna
  • Book:
    Introduction to Bisimulation and Coinduction
    Published online:
    05 August 2012
    Print publication:
    13 October 2011, pp 168-181

  • Summary

    The simulation equivalence of Exercise 1.4.17 drops the symmetry of the bisimulation game: the challenge transitions may only be launched by one of the processes in the pairs. We have seen that simulation equivalence is strictly coarser than bisimilarity. Unfortunately, it does not respect deadlock. In this section we discuss a few refinements of simulation equivalence without this drawback. They are coinductively defined, much like bisimilarity, while being coarser than bisimilarity. Thus they can allow us to use coinduction in situations where bisimilarity may be over-discriminating. Another possible advantage of a simulationlike relation is that it naturally yields a preorder (with all the advantages mentioned in Section 5.5). With respect to simulation-based relations, however, bisimilarity remains mathematically more robust and natural. The most interesting refinements we examine are represented by ready similarity and coupled similarity.

    We begin in Section 6.1 with complete simulation, and continue in Section 6.2 with ready simulation. They are to simulation what complete trace equivalence and failure (or ready) equivalence are to trace equivalence. In Section 6.3 we discuss two-nested simulation equivalence. In Section 6.4 we consider the weak versions of the relations in the previous sections. In Section 6.5 we present coupled similarity, which aims at relaxing the bisimulation requirements on the internal actions of processes. Finally, in Section 6.6, we summarise the various equivalences discussed in this and previous chapters.

    Complete simulation

    The arguments about the deadlock-insensitivity of trace equivalence in Section 1.3.2, such as the equality between the processes in Figure 1.4, apply to simulation equivalence too.

Page 46 of 99