An automata-theoretic approach to the word problem for ω -terms over R

  • Authors:

  • Jorge Almeida;Marc Zeitoun

  • Affiliations:

  • Centro de Matemática e Dep. Mat. Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal;LIAFA, Université Paris 7 and CNRS UMR 7089, France and LaBRI, Université Bordeaux 1 and CNRS UMR 5800, 351 cours de la Libération, 33405 Talence Cedex, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

This paper studies the pseudovariety R of all finite R-trivial semigroups. We give a representation of pseudowords over R by infinite trees, called R-trees. Then we show that a pseudoword is an @w-term if and only if its associated tree is regular (i.e. it can be folded into a finite graph), or equivalently, if the @w-term has a finite number of tails. We give a linear algorithm to compute a compact representation of the R-tree for @w-terms, which yields a linear solution of the word problem for @w-terms over R. We finally exhibit a basis for the @w-variety generated by R and we show that there is no finite basis. Several results can be compared to recent work of Bloom and Choffrut on long words.