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Volume 10 - April 2000

Contents

Page 2 of 2

Research Article

  • Homotopy invariants of higher dimensional categories and concurrency in computer science

  • PHILIPPE GAUCHER
  • Published online by Cambridge University Press:
    01 August 2000, pp. 481-524

  • The strict globular ω-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ω-category [Cscr ] three homology theories. The first one is called the globular homology. It contains the oriented loops of [Cscr ]. The two other ones are called the negative (respectively, positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (respectively, the merging areas of execution paths or positive corners) of [Cscr ]. Two natural linear maps called the negative (respectively, the positive) Hurewicz morphism from the globular homology to the negative (respectively, positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.

  • Pontrjagin duality and full completeness for multiplicative linear logic (without Mix)

  • MASAHIRO HAMANO
  • Published online by Cambridge University Press:
    01 April 2000, pp. 231-259

  • We prove a full completeness theorem for MLL without the Mix rule. This is done by interpreting a proof as a dinatural transformation in a *-autonomous category of reflexive topological abelian groups first studied by Barr, denoted [Rscr ][Tscr ][Ascr ]. In Section 2, we prove the unique interpretation theorem for a binary provable MLL-sequent. In Section 3, we prove a completeness theorem for binary sequents in MLL without the Mix rule, where we interpret formulas in the category [Rscr ][Tscr ][Ascr ]. The theorem is proved by investigating the concrete structure of [Rscr ][Tscr ][Ascr ], especially that arising from Pontrjagin's work on duality.

  • Higher dimensional automata revisited

  • VAUGHAN PRATT
  • Published online by Cambridge University Press:
    01 August 2000, pp. 525-548

  • The dual of a true concurrency schedule appears to be a false concurrency automaton, a paradox we resolved in a previous paper by extending the latter to higher dimensions. This extension may be formalized via such discrete geometries as n-categories, simplicial complexes, cubical complexes and Chu spaces. We advocate the last as having a clear notion of event, a well-defined process algebra uniformly extending that for event structures, and ease of extension beyond the basic before-during-after analysis.

  • On lists and other abstract data types in the calculus of constructions

  • JONATHAN P. SELDIN
  • Published online by Cambridge University Press:
    01 April 2000, pp. 261-276

  • The representation of the inductively defined abstract data type for lists was left incomplete in Seldin (1997, Section 9). Here that representation is completed, and it is proved that all extra axioms needed are consistent. Among the innovations of this paper is a definition of cdr, whose definition was left for future work in Seldin (1997, Section 9). The results are then extended to other abstract data types – those of Berardi (1993). The method used to define cdr for lists is extended to obtain the definition of an inverse for each argument of each constructor of an abstract data type. These inverses are used to prove the injective property for the constructors. Also, Dedekind's method of defining the natural numbers is used to define a predicate associated with each abstract data type, and the use of this predicate makes it unnecessary to postulate the induction principle. The only axioms left to be proved are those asserting the disjointness of the co-domains of different constructors, and it is shown that those axioms can be proved consistent.

  • Algebraic spans

  • MAURICE HERLIHY, SERGIO RAJSBAUM
  • Published online by Cambridge University Press:
    01 August 2000, pp. 549-573

  • Topological methods have yielded a variety of lower bounds and impossibility results for distributed computing. In this paper, we introduce a new tool for proving impossibility results, which is based on a core theorem of algebraic topology, the acyclic carrier theorem, and unifies, generalizes and extends earlier results.

  • Non-commutative logic II: sequent calculus and phase semantics

  • PAUL RUET
  • Published online by Cambridge University Press:
    01 April 2000, pp. 277-312

  • Non-commutative logic, which is a unification of commutative linear logic and cyclic linear logic, is extended to all linear connectives: additives, exponentials and constants. We give two equivalent versions of the sequent calculus (directly with the structure of order varieties, and with their presentations as partial orders), phase semantics and a cut-elimination theorem. This involves, in particular, the study of the entropy relation between partial orders, and the introduction of a special class of order varieties: the series–parallel order varieties.

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