## TOPOLOGICAL DEFORMATION OF HIGHER DIMENSIONAL AUTOMATA (2003)

Venue: | HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.39–82 |

Citations: | 40 - 18 self |

### BibTeX

@MISC{Gaucher03topologicaldeformation,

author = {Philippe Gaucher and Eric Goubault},

title = { TOPOLOGICAL DEFORMATION OF HIGHER DIMENSIONAL AUTOMATA},

year = {2003}

}

### OpenURL

### Abstract

A local po-space is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local po-spaces. We introduce here a particular kind of local po-space, the “globular CW-complexes”, for which we formalize these notions of deformations and which are sufficient to formalize

### Citations

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Citation Context ...t adjoint that will be denoted by (−) ⊥ . Proof. The category Top has a generator: the one-point space; it is cocomplete and wellcopowered. The result follows from the Special Adjoint Functor theorem =-=[25]-=-. If X and Y are two topological spaces, the topological space Cop(X, Y ) will be by definition the set Top(X, Y ) of continuous maps from X to Y endowed with the compact-open topology: a basis for th... |

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Citation Context .... Such a X is called a globular CW-complex and X0 and the collection of −→ e n α and its attaching maps φα : −→ S n−1 −→ Xn−1 is called the cellular decomposition of X. As for usual CW-complexes (see =-=[20]-=- Proposition A.2.), a globular cellular decomposition of a given globular CW-complex X yields characteristic maps φα : −→ D nα → X satisfying: 4 This condition will appear to be necessary in the seque... |

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Citation Context ...es Q1, . . . , Qn, abstracted as a sequence of locks and unlocks on (semaphores associated with) shared objects, Qi = R1a1 i .R2a2 i · · · Rnia ni i (Rk being 1 as E. W. Dijkstra originally put it in =-=[4]-=-, now more usually called deadlock. 2 Of course this operation must be done “atomically”, meaning that the semaphore itself must be handled in a mutually exclusive manner: this is done at the hardware... |

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Citation Context ...sed with the realization of precubical sets as local po-spaces of [6] in some sense) transforming a precubical set into a globular CW-complex. We first need a few (classical) remarks. Definition 3.6. =-=[2]-=- [22] A precubical set (or HDA) consists of a family of sets (Mn)n�0 and of ∂ α i a family of face maps Mn ��Mn−1 for α ∈ {0, 1} and 1 � i � n which satisfies the following axiom (called sometime the ... |

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Citation Context ...on of points (states), edges (transitions), squares, cubes and hypercubes (higher-dimensional transitions representing the truly-concurrent execution of some number of actions). This is introduced in =-=[27]-=- as well as possible formalizations using n-categories, and a notion of homotopy. These precubical sets are called Higher-Dimensional Automata (HDA) following [27] because it really makes sense to con... |

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Citation Context ...m is co-cartesian (for n ∈ N), ⊔ ∂σx ∐ x∈Mn+1 ✲ Tn(M) ∐ x∈Mn+1 ∂D [n+1] ⊆ ❄ x∈Mn+1 D [n+1] where ∂D [n+1] = Tn(D [n+1]) and ∂σx = σx|∂D [n+1] . ⊔ x∈M n+1 ⊆ σx ❄ ✲ Tn+1(M) Proof. We mimic the proof of =-=[8]-=-: it suffices to prove that the diagram below (in the category of sets) is cocartesian for all p � n + 1, ∐ x∈Mn+1 (∂D ⊔ (∂σx)p x∈Mn+1 ✲ [n+1])p (Tn(M))p ∐ ⊆ ❄ x∈Mn+1 (D [n+1])p ⊔ x∈M n+1 ⊆ (σx)p ❄ ✲ ... |

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Citation Context ...al spaces, we endow the disjoint sum X ⊔Y with the final topology induced by both inclusion maps X ⊂ X ⊔Y and Y ⊂ X ⊔Y . Both following lemmas summarize well-known facts about topological spaces: see =-=[28]-=- exercises 8.12 and 8.13. Lemma 3.1. Let φ be a closed continuous map from X to Y and let Z ⊂ Y . Let U be an open subset of X containing φ −1 (Z). Then there exists an open subset V of Y such that Z ... |

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Citation Context ...reasons to do that which will be clearer in the future developments. This section focuses on a very striking one. The fundamental algebraic structure which has emerged from the ω-categorical approach =-=[12, 10, 14]-=- is the diagram of Figure 14 where C is an ω-category. The analogue in the globular CW-complex framework is the diagram of Figure 15 where PX is the space of dipaths between two elements of the 0-skel... |

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Citation Context ... spaces with a closed (global) partial order. More general topological models are needed in general, in which the partial order is only defined locally, and have been introduced and motivated in [7], =-=[5]-=- and [6]. The precise definitions and properties are given in Section 3.1. The natural combinatorial notion which discretizes this topological framework is that of a precubical set, which is a collect... |

46 |
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Citation Context ... 000 111 000 111 000 111 000 111 Pb Vb (1,1) T1 Figure 3: The progress graph corresponding to Pb.V b.Pa.V a | Pa.V a.Pb.V b There are other formulations of the same problems using homological methods =-=[15]-=-, strict globular ω-categories [12]. An important motivation in these pieces of work is that of “reducing the complexity” of the semantics (given by a local po-space for instance) by considering defor... |

39 | Algebraic Topology and Concurrency
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(Show Context)
Citation Context ...space is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance =-=[6]-=-) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant co... |

34 |
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Citation Context ...with the realization of precubical sets as local po-spaces of [6] in some sense) transforming a precubical set into a globular CW-complex. We first need a few (classical) remarks. Definition 3.6. [2] =-=[22]-=- A precubical set (or HDA) consists of a family of sets (Mn)n�0 and of ∂ α i a family of face maps Mn ��Mn−1 for α ∈ {0, 1} and 1 � i � n which satisfies the following axiom (called sometime the cube ... |

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Citation Context ... on whether one chooses the cubical approach or the globular approach to model the execution paths, the higher dimensional homotopies between them and the compositions between them (see Section 1 and =-=[16]-=- for more explanations). Loosely speaking, directed cells such as Glob(e n ) are “equivalent” modulo directed deformations to a n-cube with the usual cartesian partial ordering defined by (x1, . . .,x... |

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Citation Context ...nical rewriting system. In fact, it tells us a lot more: if M can be presented by a finite canonical rewriting system, then the third homology group of M is of finite type. In fact, we can even prove =-=[23, 29]-=- that all homology groups are of finite type. One of the resolutions [19] constructs a cubical set on which the monoïd acts freely on the left. This means that all orbits of points under the action of... |

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Citation Context ...sed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known =-=[11]-=- that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant computer scientific properties like presence or absence of deadlocks. Un... |

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Citation Context ...reasons to do that which will be clearer in the future developments. This section focuses on a very striking one. The fundamental algebraic structure which has emerged from the ω-categorical approach =-=[12, 10, 14]-=- is the diagram of Figure 14 where C is an ω-category. The analogue in the globular CW-complex framework is the diagram of Figure 15 where PX is the space of dipaths between two elements of the 0-skel... |

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Citation Context ... a as shown in [6]. We see that the realization of this precubical set as defined above is exactly the globular CW-complex of Example 2.9. 3.4 Globular CW-complex and d-space Marco Grandis defines in =-=[17, 18]-=- a notion of d-space to study the geometry of directed spaces up to a form of dihomotopy. There is a functor which associates to each globular CW-complex X a d-space dX as follows: • the underlying to... |

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Citation Context ...ïdal operation. It is also probably due to the fact that the real criterion would have to be homotopical and not just homological. These ideas are already reflected in [29] and in the more recent one =-=[24]-=-. In fact, in the case of concurrency theory, we only deal with “partially commutative” monoïds i.e. free monoïds modulo the commutativity of certains pair of actions, or of certains n-uples of action... |

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Citation Context ...s of resources a and b) whereas in Figure 3, there are essentially three dipaths up to dihomotopy. Progress graphs have actually a nice topological model; they are compact order-Hausdorff spaces (see =-=[26]-=-, [21]), i.e. are compact Hausdorff spaces with a closed (global) partial order. More general topological models are needed in general, in which the partial order is only defined locally, and have bee... |

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Citation Context ...d by a finite canonical rewriting system, then the third homology group of M is of finite type. In fact, we can even prove [23, 29] that all homology groups are of finite type. One of the resolutions =-=[19]-=- constructs a cubical set on which the monoïd acts freely on the left. This means that all orbits of points under the action of M are trajectories on a cubical set; then, the homology of the monoïd is... |

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Citation Context ... a as shown in [6]. We see that the realization of this precubical set as defined above is exactly the globular CW-complex of Example 2.9. 3.4 Globular CW-complex and d-space Marco Grandis defines in =-=[17, 18]-=- a notion of d-space to study the geometry of directed spaces up to a form of dihomotopy. There is a functor which associates to each globular CW-complex X a d-space dX as follows: • the underlying to... |

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Citation Context ...nical rewriting system. In fact, it tells us a lot more: if M can be presented by a finite canonical rewriting system, then the third homology group of M is of finite type. In fact, we can even prove =-=[23, 29]-=- that all homology groups are of finite type. One of the resolutions [19] constructs a cubical set on which the monoïd acts freely on the left. This means that all orbits of points under the action of... |

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Citation Context ...mbedding of the category of homotopy types into the new category of dihomotopy types (Theorem 5.9). This embedding has a lot of important consequences that are sketched in the perspectives section of =-=[13]-=-. Once the right notion has been given, we make explicit the link between the globular CWcomplexes and some geometric notions above mentioned, that is the local po-spaces and the precubical sets in Se... |

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Citation Context ...ses, the partial order of a global po-space X will be simply denoted by �. Here are two fundamental examples of global po-spaces for the sequel: 1. The achronal segment I is defined to be the segment =-=[0, 1]-=- endowed with the closed partial ordering x �I y if and only if x = y. 2. The directed segment −→ I is defined to be the segment [0, 1] endowed with the closed partial ordering x �−→ I y if and only i... |

1 |
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Citation Context ...W-complexes g from Glob(Y ) to Glob(X) such that g ◦ f is S-homotopic to the identity of Glob(X) and f ◦ g S-homotopic to the identity of Glob(Y ). 8 This conjecture has been actually solved later on =-=[9]-=-. 37� �� N gl− N gl (C) h − ���������� h + ���� ���� � (C) N gl+ (C) Figure 14: The fundamental diagram Proof. The composite x ◦ P(f) ◦ i X �� PGlob(X) P(f) �� PGlob(Y ) �� Y is a homotopy equivalenc... |