Abstract
In the seventies, several classification schemes for the rational languages were proposed, based on the alternate use of certain operators (union, complementation, product and star). Some thirty years later, al-though much progress has been done, several of the original problems are still open. Furthermore, their significance has grown considerably over the years, on account of the successive discoveries of surprising links with other fields, like non commutative algebra, finite model theory, structural complexity and topology. In this article, we solve positively a question raised in 1985 about concatenation hierarchies of rational languages, which are constructed by alternating boolean operations and concate-nation products. We establish a simple algebraic connection between the Straubing-Thérien hierarchy, whose basis is the trivial variety, and the group hierarchy, whose basis is the variety of group languages. Thanks to a recent result of Almeida and Steinberg, this reduces the decidability problem for the group hierarchy to a property stronger than decidability for the Straubing-Thérien hierarchy.
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References
J. Almeida and B. Steinberg, On the decidability of iterated semidirect products with applications to complexity, preprint.
M. Arfi, Polynomial operations and rational languages, 4th STACS, Lect. Notes in Comp. Sci. 247, Springer, (1987), 198–206.
M. Arfi, Opérations polynomiales et hiérarchies de concaténation, Theoret. Comput. Sci. 91, (1991), 71–84.
B. Borchert, D. Kuske, F. Stephan, On existentially first-order definable languages and their relation to NP. Proceedings of ICALP 1998, Lect. Notes in Comp. Sci., Springer Verlag, Berlin, Heidelberg, New York, (1998), this volume.
J. A. Brzozowski, Hierarchies of aperiodic languages, RAIRO Inform. Théor. 10, (1976), 33–49.
J.A. Brzozowski and R. Knast, The dot-depth hierarchy of star-free languages is infinite, J. Comput. System Sci. 16, (1978), 37–55.
S. Eilenberg, Automata, languages and machines, Vol. B, Academic Press, New York, 1976.
K. Henckell and J. Rhodes, The theorem of Knast, the PG = BG and Type II Conjectures, in J. Rhodes (ed.) Monoids and Semigroups with Applications, Word Scientific, (1991), 453–463.
R. Knast, A semigroup characterization of dot-depth one languages, RAIRO Inform. Théor. 17, (1983), 321–330.
R. Knast, Some theorems on graph congruences, RAIRO Inform. Théor. 17, (1983), 331–342.
S. W. Margolis and J.E. Pin, Product of group languages, FCT Conference, Lect. Notes in Comp. Sci. 199, (1985), 285–299.
J.-E. Pin, Hiérarchies de concaténation, RAIRO Informatique Théorique 18, (1984), 23–46.
J.-E. Pin, Variétés de langages formels, Masson, Paris, 1984; English translation: Varieties of formal languages, Plenum, New-York, 1986.
J.-E. Pin, Logic on words, Bulletin of the European Association of Theoretical Computer Science 54, (1994), 145–165.
J.-E. Pin, Finite semigroups and recognizable languages: an introduction, in NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (ed.), Kluwer academic publishers, (1995), 1–32.
J.-E. Pin, BG = PG, a success story, in NATO Advanced Study Institute Semi-groups, Formal Languages and Groups, J. Fountain (ed.), Kluwer academic publishers, (1995), 33–47.
J.-E. Pin, A variety theorem without complementation, Izvestiya VUZ Matematika 39 (1995) 80–90. English version, Russian Mathem. (Iz. VUZ) 39 (1995) 74–83.
J.-E. Pin, Logic, semigroups and automata on words, Annals of Mathematics and Artificial Intelligence, (1996), 16, 343–384.
J.-E. Pin, Polynomial closure of group languages and open sets of the Hall topology, Theoretical Computer Science 169, (1996), 185–200.
J.-E. Pin, Syntactic semigroups, in Handbook of language theory, G. Rozenberg et A. Salomaa eds., vol. 1, ch. 10, pp. 679–746, Springer (1997).
J.-E. Pin and H. Straubing, 1981, Monoids of upper triangular matrices, Colloquia Mathematica Societatis Janos Bolyai 39, Semigroups, Szeged, 259–272.
J.-E. Pin and P. Weil, Polynomial closure and unambiguous product, Theory Comput. Systems 30, (1997), 1–39.
Ch. Reutenauer, Sur les variétés de langages et de monoÏdes, Lect. Notes in Comp. Sci. 67, (1979) 260–265.
M.P. Schützenberger, On finite monoids having only trivial subgroups, Information and Control 8, (1965), 190–194.
I. Simon, Piecewise testable events, Proc. 2nd GI Conf., Lect. Notes in Comp. Sci. 33, Springer Verlag, Berlin, Heidelberg, New York, (1975), 214–222.
I. Simon, The product of rational languages, Proceedings of ICALP 1993, Lect. Notes in Comp. Sci. 700, Springer Verlag, Berlin, Heidelberg, New York, (1993), 430–444.
H. Straubing, Families of recognizable sets corresponding to certain varieties of finite monoids, J. Pure Appl. Algebra, 15, (1979), 305–318.
H. Straubing, A generalization of the Schützenberger product of finite monoids, Theoret. Comp. Sci. 13, (1981), 137–150.
H. Straubing, Finite semigroups varieties of the form U * D, J. Pure Appl. Algebra 36, (1985), 53–94.
H. Straubing, Semigroups and languages of dot-depth two, Theoret. Comp. Sci. 58, (1988), 361–378.
H. Straubing, The wreath product and its application, in Formal properties of finite automata and applications, J.-E. Pin (ed.), Lect. Notes in Comp. Sci. 386, Springer Verlag, Berlin, Heidelberg, New York, (1989), 15–24.
H. Straubing and P. Weil, On a conjecture concerning dot-depth two languages, Theoret. Comp. Sci. 104, (1992), 161–183.
D. Thérien, Classification of finite monoids: the language approach, Theoret. Comp. Sci. 14, (1981), 195–208.
W. Thomas, Classifying regular events in symbolic logic, J. Comput. Syst. Sci 25, (1982), 360–375.
P. Weil, Closure of varieties of languages under products with counter, J. Comput. System Sci. 45, (1992), 316–339.
P. Weil, Some results on the dot-depth hierarchy, Semigroup Forum 46, (1993), 352–370.
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Pin, Jé. (1998). Bridges for concatenation hierarchies. In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055073
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DOI: https://doi.org/10.1007/BFb0055073
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