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On the topology of concurrent systems

Catarina Faustino, Thomas Kahl, Rodrigo Lopes

TL;DR

The paper proves that the topology of concurrent systems can realize arbitrarily complex spaces: for every nonempty connected polyhedron $|K|$, there exists a shared-variable system whose higher-dimensional automaton (HDA) model has the same homotopy type as $|K|$. The approach builds a cubical barycentric subdivision of a simplicial complex as a precubical set $P$ and then converts it into an HDA $ rak P$, showing $|P| hicksim|K|$. It then ensures accessibility by constructing a bideterministic, accessible HDA ${ rak B}$ with the same homotopy type, and finally encodes ${ rak B}$ as the HDA model of a shared-variable system. This establishes a constructive bridge between directed topology and concrete concurrent systems, allowing any connected polyhedron to be realized as the global state space of a shared-variable program, up to homotopy.

Abstract

Higher-dimensional automata, i.e., pointed labeled precubical sets, are a powerful combinatorial-topological model for concurrent systems. In this paper, we show that for every (nonempty) connected polyhedron there exists a shared-variable system such that the higher-dimensional automaton modeling the state space of the system has the homotopy type of the polyhedron.

On the topology of concurrent systems

TL;DR

The paper proves that the topology of concurrent systems can realize arbitrarily complex spaces: for every nonempty connected polyhedron , there exists a shared-variable system whose higher-dimensional automaton (HDA) model has the same homotopy type as . The approach builds a cubical barycentric subdivision of a simplicial complex as a precubical set and then converts it into an HDA , showing . It then ensures accessibility by constructing a bideterministic, accessible HDA with the same homotopy type, and finally encodes as the HDA model of a shared-variable system. This establishes a constructive bridge between directed topology and concrete concurrent systems, allowing any connected polyhedron to be realized as the global state space of a shared-variable program, up to homotopy.

Abstract

Higher-dimensional automata, i.e., pointed labeled precubical sets, are a powerful combinatorial-topological model for concurrent systems. In this paper, we show that for every (nonempty) connected polyhedron there exists a shared-variable system such that the higher-dimensional automaton modeling the state space of the system has the homotopy type of the polyhedron.
Paper Structure (5 sections, 12 theorems, 66 equations, 5 figures)

This paper contains 5 sections, 12 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The simplicial complex and its cubical barycentric subdivision
  3. The HDA
  4. Accessibility
  5. Shared-variable systems

Key Result

Lemma 2.1

Let $\theta, \psi \in S_n$ and $(t_1, \dots, t_n) \in \Delta_\theta\cap \Delta_\psi$. Then $t_{\theta(i)} = t_{\psi(i)}$ for all $i \in \{1, \dots, n\}$. Moreover, if $1 \leq i_1 < \dots < i_k = n$ are indices such that then $\{\theta(1), \dots , \theta(i_s)\} = \{\psi(1), \dots, \psi(i_s)\}$ for all $s \in \{1, \dots, k\}$.

Figures (5)

  • Figure 1: HDA for Peterson's algorithm. Parallel arrows are supposed to have the same label, and the small arrow indicates the initial state.
  • Figure 2: A 2-cube $(\{u\},\{u,v,w\})$ and its faces
  • Figure 3: The image of the map $f_{\tau, \sigma}$ for $\tau =\{v_1\}$ and $\sigma \setminus \tau = \{w_1<w_2\}$
  • Figure :
  • Figure :

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • ...and 12 more