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Using Membrane Computing for Effective Homology

Abstract

Effective Homology is an algebraic-topological method based on the computational concept of chain homotopy equivalence on a cell complex. Using this algebraic data structure, Effective Homology gives answers to some important computability problems in Algebraic Topology. In a discrete context, Effective Homology can be seen as a combinatorial layer given by a forest graph structure spanning every cell of the complex. In this paper, by taking as input a pixel-based 2D binary object, we present a logarithmic-time uniform solution for describing a chain homotopy operator \(\phi \) for its adjacency graph. This solution is based on Membrane Computing techniques applied to the spanning forest problem and it can be easily extended to higher dimensions.

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Notes

  1. We refer to [21] for basic information in this area, to [22] for a comprehensive presentation and the web site http://ppage.psystems.eu for the up-to-date information.

  2. The objects from \(\varGamma - \mathcal E \) placed in the environment along the computation are not explicitely showed in the configuration since they are not relevant in this approach.

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Acknowledgments

DDP and MAGN acknowledge the support of the projects TIN2008-04487-E and TIN-2009-13192 of the Ministerio de Ciencia e Innovación of Spain and the support of the Project of Excellence with Investigador de Reconocida Valía of the Junta de Andalucía, grant P08-TIC-04200. PR and HAC acknowledge the support of the project MTM2006-03722 of the Ministerio español de Educación y Ciencia and the project PO6-TIC-02268 of Excellence of Junta de Andalucía.

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Correspondence to Daniel Díaz-Pernil.

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Díaz-Pernil, D., Christinal, H.A., Gutiérrez-Naranjo, M.A. et al. Using Membrane Computing for Effective Homology. AAECC 23, 233–249 (2012). https://doi.org/10.1007/s00200-012-0176-6

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  • DOI: https://doi.org/10.1007/s00200-012-0176-6

Keywords

  • Tissue-like P systems
  • Membrane Computing
  • Effective Homology
  • Computational Algebraic Topology
  • Digital Topology