The HOM Problem is EXPTIME-Complete

The HOM problem questions whether the image of a given regular tree language through a given tree homomorphism is also regular. Decidability of HOM is an important theoretical question which was open for a long time. Recently, HOM has been proved decidable with a triple exponential time algorithm. In this paper we obtain an exponential time algorithm for this problem, and conclude that it is EXPTIME-complete. The proof builds upon previous results and techniques on tree automata with constraints.

[1]  Sophie Tison,et al.  Regular Tree Languages and Rewrite Systems , 1995, Fundam. Informaticae.

[2]  Guillem Godoy,et al.  Decidable Classes of Tree Automata Mixing Local and Global Constraints Modulo Flat Theories , 2013, Log. Methods Comput. Sci..

[3]  Sophie Tison,et al.  The Recognizability Problem for Tree Automata with Comparisons between Brothers , 1999, FoSSaCS.

[4]  Joost Engelfriet,et al.  Bottom-up and top-down tree transformations— a comparison , 1975, Mathematical systems theory.

[5]  Sebastian Maneth,et al.  Deciding Regularity of the Set of Instances of a Set of Terms with Regular Constraints is EXPTIME-Complete , 2009, SIAM J. Comput..

[6]  Gregory Kucherov,et al.  Decidability of Regularity and Related Properties of Ground Normal Form Languages , 1992, CTRS.

[7]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[8]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[9]  Sophie Tison,et al.  Reduction de la non-linearite des morphismes d'arbres Recognizable tree-languages and non-linear morphisms , 2002, Theor. Comput. Sci..

[10]  Murali Mani,et al.  Taxonomy of XML schema languages using formal language theory , 2005, TOIT.

[11]  Jesse B. Wright,et al.  Algebraic Automata and Context-Free Sets , 1967, Inf. Control..

[12]  Giuseppe Castagna,et al.  XML Typechecking , 2009, Encyclopedia of Database Systems.

[13]  John Doner,et al.  Tree Acceptors and Some of Their Applications , 1970, J. Comput. Syst. Sci..

[14]  Zoltán Fülöp.  Undecidable Properties of Deterministic Top-Down Tree Transducers , 1994, Theor. Comput. Sci..

[15]  Dieter Hofbauer,et al.  Computing Linearizations Using Test Sets , 1992, CTRS.

[16]  Sebastian Maneth,et al.  Classes of Tree Homomorphisms with Decidable Preservation of Regularity , 2008, FoSSaCS.

[17]  Florent Jacquemard,et al.  Ground reducibility is EXPTIME-complete , 2003, Inf. Comput..

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

[19]  Jean-Marc Talbot,et al.  Tree Automata with Global Constraints , 2008, Int. J. Found. Comput. Sci..

[20]  Sophie Tison,et al.  Deciding the Satisfiability of Quantifier free Formulae on One-Step Rewriting , 1999, RTA.

[21]  James W. Thatcher,et al.  Transformations and translations from the point of view of generalized finite automata theory , 1969, STOC.

[22]  Guillem Godoy,et al.  The HOM problem is decidable , 2013, J. ACM.

[23]  Ferenc Gécseg,et al.  Tree Languages , 1997, Handbook of Formal Languages.

[24]  Sándor Vágvölgyi,et al.  For a rewrite system it is decidable whether the set of irreducible, ground terms is regognizable , 1992, Bull. EATCS.

[25]  Max Dauchet,et al.  Automata for Reduction Properties Solving , 1995, J. Symb. Comput..

[26]  Peter Lammich,et al.  Tree Automata , 2009, Arch. Formal Proofs.

[27]  Sophie Tison,et al.  Set Constraints and Automata , 1999, Inf. Comput..