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A conjecture on the concatenation product

Published online by Cambridge University Press:  15 July 2002

Jean-Eric Pin  and
Pascal Weil
Jean-Eric Pin
Affiliation:
LIAFA, Université Paris VII et CNRS, Case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France; (Jean-Eric.Pin@liafa.jussieu.fr)
Pascal Weil
Affiliation:
LaBRI, Université Bordeaux I et CNRS, 351 cours de la Libération, 33405 Talence Cedex, France; (Pascal.Weil@labri.fr)
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Abstract

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing–Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 6: A tribute to Aldo de Luca , November 2001 , pp. 597 - 618
Copyright
© EDP Sciences, 2001

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References

J. Almeida, Finite semigroups and universal algebra. Series in Algebra, Vol. 3. WorldScientific, Singapore (1994).
Almeida, J. and Weil, P., Free profinite ${\cal R}$ -trivial monoids. Int. J. Algebra Comput. 7 (1997) 625-671. CrossRef
Arfi, M., Polynomial operations and rational languages, in 4th STACS. Springer, Lecture Notes in Comput. Sci. 247 (1987) 198-206. CrossRef
Arfi, M., Opérations polynomiales et hiérarchies de concaténation. Theoret. Comput. Sci. 91 (1991) 71-84. CrossRef
Blanchet-Sadri, F., On dot-depth two. RAIRO: Theoret. Informatics Appl. 24 (1990) 521-530.
Blanchet-Sadri, F., On a complete set of generators for dot-depth two. Discrete Appl. Math. 50 (1994) 1-25. CrossRef
Brzozowski, J.A., Hierarchies of aperiodic languages. RAIRO Theoret. Informatics. Appl. 10 (1976) 33-49.
Brzozowski, J.A. and Knast, R., The dot-depth hierarchy of star-free languages is infinite. J. Comput. System Sci. 16 (1978) 37-55. CrossRef
Büchi, J.R., Weak second-order arithmetic and finite automata. Z. Math. Logik und Grundl. Math. 6 (1960) 66-92. CrossRef
Champarnaud, J.M. and Hansel, G., AUTOMATE, a computing package for automata and finite semigroups. J. Symb. Comput. 12 (1991) 197-220. CrossRef
Cohen, R.S. and Brzozowski, J.A., Dot-depth of star-free events. J. Comput. System Sci. 5 (1971) 1-15. CrossRef
Cowan, D., Inverse monoids of dot-depth 2. Int. J. Algebra and Comput. 3 (1993) 411-424. CrossRef
S. Eilenberg. Automata, languages and machines, Vol. B. Academic Press, New York (1976).
V. Froidure and J.-E. Pin, Algorithms for computing finite semigroups, in Foundations of Computational Mathematics, edited by F. Cucker and M. Shub. Springer, Berlin (1997) 112-126.
Glaßer, C. and Schmitz, H., Languages of dot-depth 3/2, in Proc. 17th Symposium on Theoretical Aspects of Computer Science. Springer, Lecture Notes in Comput. Sci. 1770 (2000) 555-566. CrossRef
C. Glaßer and H. Schmitz, Decidable Hierarchies of Starfree Languages, in Foundations of Software Technology and Theoretical Computer Science (FSTTCS). Springer, Lecture Notes in Comput. Sci. 1974 (2000) 503-515.
C. Glaßer and H. Schmitz, Concatenation Hierarchies and Forbidden Patterns, Technical Report No. 256. University of Wuerzburg (2000).
C. Glaßer and H. Schmitz, Level 5/2 of the Straubing-Thérien hierarchy for two-letter alphabets (to appear).
Hall, T. and Weil, P., On radical congruence systems. Semigroup Forum 59 (1999) 56-73. CrossRef
Henckell, K., Margolis, S.W., Pin, J.-E. and Rhodes, J., Ash's Type II Theorem, Profinite Topology and Malcev Products. Int. J. Algebra and Comput. 1 (1991) 411-436. CrossRef
Knast, R., A semigroup characterization of dot-depth one languages. RAIRO: Theoret. Informatics Appl. 17 (1983) 321-330.
Knast, R., Some theorems on graph congruences. RAIRO: Theoret. Informatics Appl. 17 (1983) 331-342.
R. McNaughton and S. Pappert, Counter-free Automata. MIT Press (1971).
S.W. Margolis and J.-E. Pin, Product of group languages, in FCT Conference. Springer, Lecture Notes in Comput. Sci. 199 (1985) 285-299.
S.W. Margolis and B. Steinberg, Power semigroups and polynomial closure, in Proc. of the Third International Colloquium on Words, Languages and Combinatorics (to appear).
Pin, J.-E., Hiérarchies de concaténation. RAIRO: Theoret. Informatics Appl. 18 (1984) 23-46.
J.-E. Pin, Variétés de langages formels. Masson, Paris (1984); English translation: Varieties of formal languages. Plenum, New-York (1986).
Pin, J.-E., A property of the Schützenberger product. Semigroup Forum 35 (1987) 53-62. CrossRef
Pin, J.-E., A variety theorem without complementation. Izvestiya VUZ Matematika 39 (1995) 80-90.
J.-E. Pin, Syntactic semigroups, Chap. 10, in Handbook of formal languages, edited byG. Rozenberg and A. Salomaa. Springer (1997).
Pin, J.-E., Bridges for concatenation hierarchies, in 25th ICALP. Springer, Berlin, Lecture Notes in Comput. Sci. 1443 (1998) 431-442. CrossRef
J.-E. Pin, Algebraic tools for the concatenation product. Theoret. Comput. Sci. (to appear).
Pin, J.-E. and Straubing, H., Monoids of upper triangular matrices. Colloq. Math. Soc. János Bolyai 39 (1981) 259-272.
Pin, J.-E. and Weil, P., Reiterman, A theorem for pseudovarieties of finite first-order structures. Algebra Universalis 35 (1996) 577-595. CrossRef
Pinand, J.-E. Weil, P., Profinite semigroups, Mal'cev products and identities. J. Algebra 182 (1996) 604-626.
Pin, J.-E. and Weil, P., Polynomial closure and unambiguous product. Theory Comput. Systems 30 (1997) 1-39. CrossRef
J.-E. Pin and P. Weil, The wreath product principle for ordered semigroups. Comm. in Algebra (to appear).
Schützenberger, M.P., On finite monoids having only trivial subgroups. Inform. and Control 8 (1965) 190-194. CrossRef
Selivanov, V., A logical approach to decidability of hierarchies of regular star-free languages, in Proc. STACS 2001. Springer, Lecture Notes in Comput. Sci. 2010 (2001) 539-550. CrossRef
Simon, I., Piecewise testable events, in Proc. 2nd GI Conf. Springer, Lecture Notes in Comput. Sci. 33 (1975) 214-222. CrossRef
Steinberg, B., Polynomial closure and topology. Int. J. Algebra and Comput. 10 (2000) 603-624. CrossRef
Straubing, H., Aperiodic homomorphisms and the concatenation product of recognizable sets. J. Pure Appl. Algebra 15 (1979) 319-327. CrossRef
Straubing, H., A generalization of the Schützenberger product of finite monoids. Theory Comput. Systems 13 (1981) 137-150. CrossRef
Straubing, H., Relational morphisms and operations on recognizable sets. RAIRO: Theoret. Informatics Appl. 15 (1981) 149-159.
Straubing, H., Finite semigroups varieties of the form V*D. J. Pure Appl. Algebra 36 (1985) 53-94. CrossRef
H. Straubing. Semigroups and languages of dot-depth two. Theoret. Comput. Sci. 58 (1988) 361-378.
Straubing, H. and Weil, P., On a conjecture concerning dot-depth two languages. Theoret. Comput. Sci. 104 (1992) 161-183. CrossRef
Thérien, D., Classification of finite monoids: The language approach. Theoret. Comput. Sci. 14 (1981) 195-208. CrossRef
Thomas, W., Classifying regular events in symbolic logic. J. Comput. Syst. Sci. 25 (1982) 360-375. CrossRef
Thomas, W., An application of the Ehrenfeucht-Fraïssé game in formal language theory. Bull. Soc. Math. France 16 (1984) 11-21.
Thomas, W., A concatenation game and the dot-depth hierarchy, in Computation Theory and Logic, edited by E. Böger. Springer, Lecture Notes in Comput. Sci. 270 (1987) 415-426. CrossRef
Trotterand, P. Weil, P., The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue. Algebra Universalis 37 (1997) 491-526. CrossRef
Weil, P., Inverse monoids of dot-depth two. Theoret. Comput. Sci. 66 (1989) 233-245. CrossRef
Weil, P., Some results on the dot-depth hierarchy. Semigroup Forum 46 (1993) 352-370. CrossRef