Syntactic Complexities of Some Classes of Star-Free Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity n of these languages. We study the syntactic complexity of three subclasses of star-free languages. We find tight upper bounds for languages accepted by monotonic, partially monotonic and "nearly monotonic" automata; all three of these classes are star-free. We conjecture that the bound for nearly monotonic languages is also a tight upper bound for star-free languages.

[1]  A. Nerode,et al.  Linear automaton transformations , 1958 .

[2]  Bo Liu,et al.  Quotient Complexity of Star-Free Languages , 2010, Int. J. Found. Comput. Sci..

[3]  Janusz Brzozowski,et al.  Quotient Complexity of Regular Languages , 2009, J. Autom. Lang. Comb..

[4]  Janusz A. Brzozowski,et al.  Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages , 2012, Theor. Comput. Sci..

[5]  A. Cayley.  A theorem on trees , 2009 .

[6]  Robert McNaughton,et al.  Counter-Free Automata (M.I.T. research monograph no. 65) , 1971 .

[7]  Martin Kutrib,et al.  Nondeterministic state complexity of star-free languages , 2012, Theor. Comput. Sci..

[8]  Jeffrey Shallit,et al.  State Complexity and the Monoid of Transformations of a Finite Set , 2004, CIAA.

[9]  Barbara König,et al.  On deterministic finite automata and syntactic monoid size , 2004, Theor. Comput. Sci..

[10]  J. M. Howie,et al.  Products of idempotents in certain semigroups of transformations , 1971, Proceedings of the Edinburgh Mathematical Society.

[11]  Janusz A. Brzozowski,et al.  Syntactic Complexity of Ideal and Closed Languages , 2010, Developments in Language Theory.

[12]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[13]  A. Umar,et al.  Asymptotic Results for Semigroups of Order-Preserving Partial Transformations , 2006 .

[14]  Gracinda M. S. Gomes,et al.  On the ranks of certain semigroups of order-preserving transformations , 1992 .

[15]  Grzegorz Rozenberg,et al.  Developments in Language Theory II , 2002 .

[16]  Peter W. Shor,et al.  A New Proof of Cayley's Formula for Counting Labeled Trees , 1995, J. Comb. Theory, Ser. A.

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

[18]  Sheng Yu,et al.  State Complexity of Regular Languages , 2001, J. Autom. Lang. Comb..

[19]  Mikhail V. Volkov,et al.  Synchronizing generalized monotonic automata , 2005, Theor. Comput. Sci..