On a Conjecture Concerning Dot-Depth Two Languages

Abstract In this paper, we study the second level of the dot-depth hierarchy for star-free regular languages. We investigate a necessary condition stated by Straubing for a language to have dot-depth two, and prove that it is sufficient for languages whose syntactic monoid is inverse with three inverse generators. Also we disprove a conjecture according to which Straubing's condition would be equivalent to both dot-depth two and another condition expressed in terms of two-sided semidirect product.

[1]  Howard Straubing,et al.  FINITE SEMIGROUP VARIETIES OF THE FORM V,D , 1985 .

[2]  David F. Cowan,et al.  Inverse Monoids of dot-Depth Two , 1993, Int. J. Algebra Comput..

[3]  J. Gerhard,et al.  The lattice of equational classes of idempotent semigroups , 1970 .

[4]  Pascal Weil,et al.  Inverse Monoids of Dot-Depth Two , 1989, Theor. Comput. Sci..

[5]  Janusz A. Brzozowski,et al.  Characterizations of locally testable events , 1973, Discret. Math..

[6]  J. Green,et al.  On semi-groups in which xr = x , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  Janusz A. Brzozowski,et al.  The Dot-Depth Hierarchy of Star-Free Languages is Infinite , 1978, J. Comput. Syst. Sci..

[8]  Gilbert Helmberg,et al.  A THEOREM ON EQUIDISTRIBUTION ON COMPACT GROUPS , 1958 .

[9]  J. B. Stephen.  Presentations of inverse monoids , 1990 .

[10]  John Rhodes,et al.  The kernel of monoid morphisms , 1989 .

[11]  Pascal Weil.  Inverse monoids and the dot -depth hierarchy , 1988 .

[12]  Howard Straubing,et al.  Semigroups and Languages of Dot-Depth Two , 1988, Theor. Comput. Sci..

[13]  Faith Ellen,et al.  A Characterization of a Dot-Depth Two Analogue of Generalized Definite Languages , 1979, ICALP.

[14]  Wolfgang Thomas,et al.  Classifying Regular Events in Symbolic Logic , 1982, J. Comput. Syst. Sci..

[15]  Imre Simon,et al.  Piecewise testable events , 1975, Automata Theory and Formal Languages.

[16]  Howard Straubing.  Semigroups and Languages of Dot-Depth 2 , 1986, ICALP.

[17]  Faith Ellen,et al.  On generalized locally testable languages , 1984, Discret. Math..

[18]  J. Howie.  An introduction to semigroup theory , 1976 .

[19]  Janusz A. Brzozowski,et al.  Dot-Depth of Star-Free Events , 1971, Journal of computer and system sciences (Print).

[20]  Joseph Buchanan Stephen.  Applications of automata theory to presentations of monoids and inverse monoids , 1987 .

[21]  Robert Knast,et al.  A Semigroup Characterization of Dot-Depth one Languages , 1983, RAIRO Theor. Informatics Appl..

[22]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[23]  Shelly L. Wismath,et al.  The lattices of varieties and pseudovarieties of band monoids , 1986 .

[24]  Christophe Reutenauer,et al.  Une Topologie du Monoïde Libre , 1979 .

[25]  Samuel Eilenberg.  Automata, Languages and Machines, Vol. B , 1976 .

[26]  Bret Tilson,et al.  Categories as algebra: An essential ingredient in the theory of monoids , 1987 .

[27]  Naoki Kimura,et al.  THE STRUCTURE OF IDEMPOTENT SEMIGROUPS. I , 1958 .