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6975 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 178 of 349

  • CPC volume 19 issue 2 Cover and Back matter

  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 2 / March 2010
    Published online by Cambridge University Press:
    05 February 2010, pp. b1-b2
    • Article
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  • WDM and Directed Star Arboricity

  • OMID AMINI, FRÉDÉRIC HAVET, FLORIAN HUC, STÉPHAN THOMASSÉ
  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 2 / March 2010
    Published online by Cambridge University Press:
    05 February 2010, pp. 161-182

  • A digraph is m-labelled if every arc is labelled by an integer in {1, . . ., m}. Motivated by wavelength assignment for multicasts in optical networks, we introduce and study n-fibre colourings of labelled digraphs. These are colourings of the arcs of D such that at each vertex v, and for each colour α, in(v, α) + out(v, α) ≤ n with in(v, α) the number of arcs coloured α entering v and out(v, α) the number of labels l such that there is at least one arc of label l leaving v and coloured with α. The problem is to find the minimum number of colours λn(D) such that the m-labelled digraph D has an n-fibre colouring. In the particular case when D is 1-labelled, λ1(D) is called the directed star arboricity of D, and is denoted by dst(D). We first show that dst(D) ≤ 2Δ−(D)+1, and conjecture that if Δ−(D) ≥ 2, then dst(D) ≤ 2Δ−(D). We also prove that for a subcubic digraph D, then dst(D) ≤ 3, and that if Δ+(D), Δ−(D) ≤ 2, then dst(D) ≤ 4. Finally, we study λn(m, k) = max{λn(D) | D is m-labelled and Δ−(D) ≤ k}. We show that if m ≥ n, then for some constant C. We conjecture that the lower bound should be the correct value of λn(m, k).

  • CPC volume 19 issue 2 Cover and Front matter

  • Journal:
    Combinatorics, Probability and Computing / Volume 19 / Issue 2 / March 2010
    Published online by Cambridge University Press:
    05 February 2010, pp. f1-f2
    • Article
      • You have access
    • PDF
    • Export citation

A First Course in Combinatorial Optimization
  • Jon Lee
  • Published online:
    28 January 2010
    Print publication:
    09 February 2004
  • A First Course in Combinatorial Optimization is a text for a one-semester introductory graduate-level course for students of operations research, mathematics, and computer science. It is a self-contained treatment of the subject, requiring only some mathematical maturity. Topics include: linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and network flows. Central to the exposition is the polyhedral viewpoint, which is the key principle underlying the successful integer-programming approach to combinatorial-optimization problems. Another key unifying topic is matroids. The author does not dwell on data structures and implementation details, preferring to focus on the key mathematical ideas that lead to useful models and algorithms. Problems and exercises are included throughout as well as references for further study.

A Unifying Framework for Structured Analysis and Design Models
  • An Approach Using Initial Algebra Semantics and Category Theory
  • T. H. Tse
  • Published online:
    28 January 2010
    Print publication:
    09 May 1991
  • Structured methodologies are a popular and powerful tool in information systems development. Many different ones exist, each employing a number of models and so a specification must be converted from one form to another during the development process. To solve this problem, Dr Tse proposes in this 1991 book a unifying framework behind popular structured models. He approaches the problem from the viewpoints of algebra and category theory. He not only develops the frameworks but also illustrates their practical and theoretical usefulness. Thus this book will provide insight for software engineers into how methodologies can be formalised and will open up a range of applications and problems for theoretical computer scientists.

Theories of Programming Languages
  • John C. Reynolds
  • Published online:
    28 January 2010
    Print publication:
    13 October 1998
  • First published in 1998, this textbook is a broad but rigourous survey of the theoretical basis for the design, definition and implementation of programming languages and of systems for specifying and proving programme behaviour. Both imperative and functional programming are covered, as well as the ways of integrating these aspects into more general languages. Recognising a unity of technique beneath the diversity of research in programming languages, the author presents an integrated treatment of the basic principles of the subject. He identifies the relatively small number of concepts, such as compositional semantics, binding structure, domains, transition systems and inference rules, that serve as the foundation of the field. Assuming only knowledge of elementary programming and mathematics, this text is perfect for advanced undergraduate and beginning graduate courses in programming language theory and also will appeal to researchers and professionals in designing or implementing computer languages.

  • Extending regular expressionswith homomorphic replacement

  • Henning Bordihn, Jürgen Dassow, Markus Holzer
  • Journal:
    RAIRO - Theoretical Informatics and Applications / Volume 44 / Issue 2 / April 2010
    Published online by Cambridge University Press:
    26 January 2010, pp. 229-255
    Print publication:
    April 2010

  • We define H- and EH-expressions as extensions of regular expressions by adding homomorphic and iterated homomorphic replacement as new operations, resp. The definition is analogous to the extension given by Gruska in order to characterize context-free languages. We compare the families of languages obtained by these extensions with the families of regular, linear context-free, context-free, and EDT0Llanguages. Moreover, relations to language families based on patterns, multi-patterns, pattern expressions, H-systems and uniform substitutions are also investigated. Furthermore, we present their closure properties with respect to TRIO operations and discuss the decidability status and complexity of fixed and general membership, emptiness, and the equivalence problem.

  • CHAPTER II - THE PREDICATE CALCULUS AND THE SYSTEM LK

  • Stephen Cook, University of Toronto, Phuong Nguyen, McGill University, Montréal
  • Book:
    Logical Foundations of Proof Complexity
    Published online:
    06 July 2010
    Print publication:
    25 January 2010, pp 9-38

  • Summary

    In this chapter we present the logical foundations for theories of bounded arithmetic. We introduce Gentzen's proof system LK for the predicate calculus, and prove that it is sound, and complete even when proofs have a restricted form called “anchored”. We augment the system LK by adding equality axioms. We prove the Compactness Theorem for predicate calculus, and the Herbrand Theorem.

    In general we distinguish between syntactic notions and semantic notions. Examples of syntactic notions are variables, connectives, formulas, and formal proofs. The semantic notions relate to meaning; for example truth assignments, structures, validity, and logical consequence.

    The first section treats the simple case of propositional calculus.

    Propositional Calculus

    Propositional formulas (called simply formulas in this section) are built from the logical constants ⊥, ⊤ (for False, True), propositional variables (or atoms) P1, P2, …, connectives ¬, ∨, ∧, and parentheses(,). We use P, Q, R, … to stand for propositional variables, A, B, C, … to stand for formulas, and Φ, Ψ, … to stand for sets of formulas. When writing formulas such as (P ∨ (Q ∧ R)), our convention is that P, Q, R, … stand for distinct variables.

  • CHAPTER III - PEANO ARITHMETIC AND ITS SUBSYSTEMS

  • Stephen Cook, University of Toronto, Phuong Nguyen, McGill University, Montréal
  • Book:
    Logical Foundations of Proof Complexity
    Published online:
    06 July 2010
    Print publication:
    25 January 2010, pp 39-72

  • Summary

    Peano Arithmetic is the first order theory of ℕ with simple axioms for +, ·, ≤, and the induction axiom scheme. Here we focus on the subsystem IΔ0 of Peano Arithmetic, in which induction is restricted to bounded formulas. This subsystem plays an essential role in the development of the theories in later chapters: All (two-sorted) theories introduced in this book extend V0, which is a conservative extension of IΔ0. At the end of the chapter we briefly discuss Buss's hierarchy. These single-sorted theories establish a link between bounded arithmetic and the polynomial time hierarchy, and have played a central role in the study of bounded arithmetic. In later chapters we introduce their twosorted versions, including V1, a theory that characterizes P. The theories considered in this chapter are singled-sorted, and the intended domain is ℕ = {0, 1, 2, …}.

    Subsection III.3.3 shows that the relation y = 2x is definable by a bounded formula in the vocabulary of IΔ0, and in Section III.4 this is used to show that bounded formulas represent precisely the relations in the Linear Time Hierarchy (LTH).

    Peano Arithmetic

    See Section II.2 for notions such as vocabulary, formula, and logical consequence.

    Definition III.1.1. A theory over a vocabulary L is a set T of formulas over L which is closed under logical consequence and universal closure.

Page 178 of 349