We close affirmatively a question that has been open for long time: decidability of the HOM problem. The HOM problem consists in determining, given a tree homomorphism H and a regular tree language L represented by a tree automaton, whether H ( L ) is regular. In order to decide the HOM problem, we develop new constructions and techniques that are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular languages through tree homomorphisms. Our contributions are based on the following new constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with disequality constraints recognizing the complementary language. We also define a new class of tree automata with arbitrary disequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of a tree automaton with disequality constraints and the image of a regular language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism. We combine the above constructions adequately to provide an algorithm deciding the HOM problem. This is the journal version of a paper presented in the 42nd ACM Symposium on Theory of Computing (STOC 2010). Here, we provide all proofs and examples. Moreover, we obtain better complexity results via the modification of some proofs and a careful complexity analysis. In particular, the obtained time complexity for the decision of HOM is a tower of three exponentials.
Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called Singleton Tree Grammars (STG), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present paper is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and Singleton Tree Grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NPcompleteness is obtained when the terms are represented using the more general formalism of Singleton Tree Grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.
This paper is an investigation of the matching problem for term equations s = t where s contains context variables, and both terms s and t are given using some kind of compressed representation. In this setting, term representation with dags, but also with the more general formalism of singleton tree grammars, are considered. The main result is a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. NP-completeness is obtained when the terms are represented using singleton tree grammars. The special cases of first-order matching and also unification with STGs are shown to be decidable in PTIME.
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