Descriptional Complexity of Unambiguous Nested Word Automata

It is known that converting an n-state nondeterministic nested word automaton (a.k.a. input-driven automaton; a.k.a. visibly pushdown automaton) to a corresponding deterministic automaton requires in the worst case 2Θ(n2) states (R. Alur, P. Madhusudan: Adding nesting structure towords, DLT'06).We show that the same worst case 2Θ(n2) size blowup occurs when converting a nondeterministic nested word automaton to an unambiguous one, and an unambiguous nested word automaton to a deterministic one. In addition, the methods developed in this paper are used to demonstrate that the state complexity of complementation for nondeterministic nested word automata is 2Θ(n2), and that the state complexity of homomorphism for deterministic nested word automata is 2Θ(n2).

[1]  Martin Kutrib,et al.  Nondeterministic Descriptional Complexity Of Regular Languages , 2003, Int. J. Found. Comput. Sci..

[2]  Kai Salomaa,et al.  Limitations of lower bound methods for deterministic nested word automata , 2011, Inf. Comput..

[3]  Kai Salomaa,et al.  Operational state complexity of nested word automata , 2008, Theor. Comput. Sci..

[4]  Oscar H. Ibarra,et al.  Relating the Type of Ambiguity of Finite Automata to the Succinctness of Their Representation , 1989, SIAM J. Comput..

[5]  Jeffrey Shallit,et al.  A Second Course in Formal Languages and Automata Theory , 2008 .

[6]  Marko Vukolic,et al.  SOFSEM 2011: Theory and Practice of Computer Science - 37th Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 22-28, 2011. Proceedings , 2011, SOFSEM.

[7]  Helmut Seidl,et al.  Locating Matches of Tree Patterns in Forests , 1998, FSTTCS.

[8]  Mahesh Viswanathan,et al.  Congruences for Visibly Pushdown Languages , 2005, ICALP.

[9]  Marek Karpinski,et al.  Foundations of Computation Theory , 1983 .

[10]  Joachim Niehren,et al.  Streaming tree automata , 2008, Inf. Process. Lett..

[11]  Gennaro Parlato,et al.  The tree width of auxiliary storage , 2011, POPL '11.

[12]  Juraj Hromkovic,et al.  Communication Complexity and Parallel Computing , 1997, Texts in Theoretical Computer Science An EATCS Series.

[13]  Wojciech Rytter,et al.  On the Maximal Number of Cubic Runs in a String , 2010, LATA.

[14]  Yo-Sub Han,et al.  Nondeterministic state complexity of nested word automata , 2009, Theor. Comput. Sci..

[15]  Burchard von Braunmühl,et al.  Input-Driven Languages are Recognized in log n Space , 1983, FCT.

[16]  Hubert Comon,et al.  Tree automata techniques and applications , 1997 .

[17]  K. Mehlhorn.  Pebbling Moutain Ranges and its Application of DCFL-Recognition , 1980, ICALP.

[18]  Petr Hliněný,et al.  Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings , 2010, MFCS.

[19]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[20]  Haruo Hosoya,et al.  Multi-Return Macro Tree Transducers , 2008, PLAN-X.

[21]  Peter R. J. Asveld.  Review of "V. Geffert, C. Mereghetti & G. Pighizzini, Complementing two-way finite automata, Inform. and Comput. 205 (2007) 1173-1187" , 2008 .

[22]  Marcelo Arenas,et al.  Regular Languages of Nested Words: Fixed Points, Automata, and Synchronization , 2007, ICALP.

[23]  Jonathan Goldstine,et al.  On the Relation between Ambiguity and Nondeterminism in Finite Automata , 1992, Inf. Comput..

[24]  R. Alur,et al.  Adding nesting structure to words , 2006, JACM.

[25]  Alexander Okhotin.  Unambiguous finite automata over a unary alphabet , 2012, Inf. Comput..

[26]  Igor Walukiewicz,et al.  Minimizing Variants of Visibly Pushdown Automata , 2007, MFCS.

[27]  Hing Leung.  Separating Exponentially Ambiguous Finite Automata from Polynomially Ambiguous Finite Automata , 1998, SIAM J. Comput..

[28]  Martin Kutrib,et al.  Descriptional and Computational Complexity of Finite Automata , 2009, LATA.

[29]  Maxime Crochemore,et al.  Finding Patterns In Given Intervals , 2007, Fundam. Informaticae.

[30]  Martin Kutrib,et al.  Nondeterministic Finite Automata-Recent Results on the Descriptional and Computational Complexity , 2008, CIAA.

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

[32]  Yuan Gao,et al.  The State Complexity of Two Combined Operations: Star of Catenation and Star of Reversal , 2008, Fundam. Informaticae.

[33]  Neil Immerman,et al.  First-Order and Temporal Logics for Nested Words , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[34]  Galina Jirásková,et al.  State complexity of some operations on binary regular languages , 2005, Theor. Comput. Sci..

[35]  Alexander Okhotin.  Comparing Linear Conjunctive Languages to Subfamilies of the Context-Free Languages , 2011, SOFSEM.

[36]  Jean-Camille Birget,et al.  Intersection and Union of Regular Languages and State Complexity , 1992, Inf. Process. Lett..

[37]  Hing Leung.  Descriptional complexity of nfa of different ambiguity , 2005, Int. J. Found. Comput. Sci..

[38]  Carlos Martín-Vide,et al.  State complexity of basic language operations combined with reversal , 2008, Inf. Comput..

[39]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[40]  Carlo Mereghetti,et al.  Complementing two-way finite automata , 2007, Inf. Comput..

[41]  E. M. Schmidt.  Succinctness of Descriptions of Context-Free, Regular and Finite Languages , 1977 .