On the Height of Towers of Subsequences and Prefixes

A tower is a sequence of words alternating between two languages in such a way that every word is a subsequence of the following word. The height of the tower is the number of words in the sequence. If there is no infinite tower (a tower of infinite height), then the height of all towers between the languages is bounded. We study upper and lower bounds on the height of maximal finite towers between two regular languages with respect to the size of the NFA (respectively the DFA) representation. Our motivation to study the bounds on maximal finite towers comes from a method to compute a piecewise testable separator of two regular languages. We show that the upper bound is polynomial in the number of states and exponential in the size of the alphabet, and that it is asymptotically tight if the size of the alphabet is fixed. If the alphabet may grow, then, using an alphabet of size approximately the number of states of the automata, the lower bound on the height of towers is exponential with respect to that number. In this case, there is a gap between the lower and upper bound, and the asymptotically optimal bound remains an open problem. Since, in many cases, the constructed towers are sequences of prefixes, we also study towers of prefixes.

[1]  Thomas Place,et al.  Separation and the Successor Relation , 2015, STACS.

[2]  Michaël Thomazo,et al.  Alternating Towers and Piecewise Testable Separators , 2014, ArXiv.

[3]  Wim Martens,et al.  A Characterization for Decidable Separability by Piecewise Testable Languages , 2014, Discret. Math. Theor. Comput. Sci..

[4]  Tomás Masopust,et al.  On Upper and Lower Bounds on the Length of Alternating Towers , 2014, MFCS.

[5]  Wim Martens,et al.  Separability by Short Subsequences and Subwords , 2015, ICDT.

[6]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

[7]  Philippe Schnoebelen,et al.  On the index of Simon's congruence for piecewise testability , 2015, Inf. Process. Lett..

[8]  Wim Martens,et al.  Efficient Separability of Regular Languages by Subsequences and Suffixes , 2013, ICALP.

[9]  Howard Straubing,et al.  A Generalization of the Schützenberger Product of Finite Monoids , 1981, Theor. Comput. Sci..

[10]  Thomas Place,et al.  Separating Regular Languages by Piecewise Testable and Unambiguous Languages , 2013, MFCS.

[11]  Tomás Masopust,et al.  Separability by Piecewise Testable Languages is PTime-Complete , 2017, Theor. Comput. Sci..

[12]  Markus Krötzsch,et al.  Deciding Universality of ptNFAs is PSpace-Complete , 2018, SOFSEM.

[13]  Denis Thérien,et al.  Classification of Finite Monoids: The Language Approach , 1981, Theor. Comput. Sci..

[14]  Thomas Schwentick,et al.  BonXai , 2017, ACM Trans. Database Syst..

[15]  Jorge Almeida,et al.  Implicit operations on finite J-trivial semigroups and a conjecture of I. Simon , 1991 .

[16]  Jacques Stern,et al.  Characterizations of Some Classes of Regular Events , 1985, Theor. Comput. Sci..

[17]  Jorge Almeida,et al.  Finite Semigroups and Universal Algebra , 1995 .

[18]  Thomas Place,et al.  Going Higher in the First-Order Quantifier Alternation Hierarchy on Words , 2014, ICALP.