Advertisement

Applied Categorical Structures

, Volume 15, Issue 4, pp 415–437 | Cite as

Algebra and Geometry of Rewriting

  • Yves LafontEmail author
Article
First Online:

Abstract

We present various results of the last 20 years converging towards a homotopical theory of computation. This new theory is based on two crucial notions: polygraphs (introduced by Albert Burroni) and polygraphic resolutions (introduced by François Métayer). There are two motivations for such a theory:

  • Providing invariants of computational systems to study those systems and prove properties about them;

  • Finding new methods to make computations in algebraic structures coming from geometry or topology.

This means that this theory should be relevant for mathematicians as well as for theoretical computer scientists, since both may find useful tools or concepts for their own domain coming from the other one.

Keywords

Confluence Generators and relations Homology Homotopy Polygraph Resolution Rewriting Termination 

Mathematics Subject Classifications (2000)

16E05 16E40 16S15 18D05 18D10 18G55 20J05 20M05 68Q42 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brown, K.S.: Cohomology of Groups. Springer, Berlin (1982)zbMATHGoogle Scholar
  2. 2.
    Burroni, A.: Higher dimensional word problems with applications to equational logic. Theoret. Comput. Sci. 115, 43–62 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burroni, A.: A new calculation of the Orientals of Street. In: Conference Given in Oxford in April 2000. Available on http://www.math.jussieu.fr/~burroni/
  4. 4.
    Cremanns, R., Otto, F.: Finite derivation type implies the homological finiteness condition FP 3. J. Symbolic Comput. 18, 91–12 (1994)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cremanns, R., Otto, F.: For groups the property of finite derivation type is equivalent to the homological finiteness condition FP 3. J. Symbolic Comput. 22, 155–177 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dehornoy, P., Lafont, Y.: Homology of gaussian groups. Ann. Inst. Fourier 53(2), 489–540 (2003)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Guiraud, Y.: Termination orders for 3-dimensional rewriting. J. Pure Appl. Algebra 207(2), 341–37 (2006)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Katsura, M., Kobayashi, Y.: Constructing finitely presented monoids which have no finite complete presentation. Semigroup Forum 54, 292–302 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kapur, D., Narendran, P.: The Knuth–Bendix completion procedure and Thue systems. SIAM J. Comput. 14(4), 1052–1072 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kapur, D., Narendran, P.: A finite Thue system with decidable word problem and without equivalent finite canonical system. Theoret. Comput. Sci. 35, 337–344 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kobayashi, Y.: Complete rewriting systems and homology of monoid algebras. J. Pure Appl. Algebra 65(3), 263–275 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lafont, Y.: A new finiteness condition for monoids presented by complete rewriting systems (after Craig Squier). J. Pure Appl. Algebra 98(3), 229–244 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lafont, Y.: Towards an algebraic theory of Boolean circuits. J. Pure Appl. Algebra 184(2–3), 257–310 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lafont, Y., Prouté, A.: Church–Rosser property and homology of monoids. Math. Structures Comput. Sci. 1(3), 297–326 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Mac Lane, S.: Homology. Springer, Berlin (1963)Google Scholar
  16. 16.
    Métayer, F.: Resolutions by polygraphs. Theory Appl. Categ. 1(7), 148–184 (2003)Google Scholar
  17. 17.
    Power, J.: An n-categorical pasting theorem. Proceedings of Category Theory, Como 1990. Springer Lecture Notes in Mathematics, 1488, 326–358 (1991)Google Scholar
  18. 18.
    Squier, C., Otto, F., Kobayashi, Y.: A finiteness condition for rewriting systems. Theoret. Comput. Sci. 131, 271–294 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Spanier, C.C.: Algebraic Topology. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  20. 20.
    Squier, C.: Word problems and a homological finiteness condition for monoids. J. Pure Appl. Algebra 49, 201–217 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Street, R.: The algebra of oriented simplexes. J. Pure Appl. Algebra 49, 283–335 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institut de Mathématiques de Luminy (UMR 6206 du CNRS)Université de la Méditerranée (Aix-Marseille 2)Marseille Cedex 9France

Personalised recommendations

Cite article

Buy options