Automata for unordered unranked trees are relevant for defining schemas and queries for data trees in JSON or XML format. While the existing notions are well-investigated concerning expressiveness, they all lack a proper notion of determinism, which makes it difficult to distinguish subclasses of automata for which problems such as inclusion, equivalence, and minimization can be solved efficiently. In this paper, we propose and investigate different notions of "horizontal determinism", starting from automata for unranked trees in which the horizontal evaluation is performed by finite state automata. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers from coNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending on the choice of the order, we obtain different classes of automata, each of which has the same expressiveness as Counting MSO.
We present a framework for defining automata for unordered data trees that is parametrized by the way in which multisets of children nodes are described. Presburger tree automata and alternating Presburger tree automata are particular instances. We establish the usual equivalence in expressiveness of tree automata and MSO for the automata defined in our framework. We then investigate subclasses of automata for unordered trees for which testing language equivalence is in P-time. For this we start from automata in our framework that describe multisets of children by finite automata, and propose two approaches of how to do this deterministically. We show that a restriction to confluent horizontal evaluation leads to polynomial-time emptiness and universality, but still suffers from coNP-completeness of the emptiness of binary intersections. Finally, efficient algorithms can be obtained by imposing an order of horizontal evaluation globally for all automata in the class. Depending on the choice of the order, we obtain different classes of automata, each of which has the same expressiveness as Counting Mso.
We study a subclass of tree-to-word transducers: linear treeto-word transducers, that cannot use several copies of the input. We aim to study the equivalence problem on this class, by using minimization and normalization techniques. We identify a Myhill-Nerode characterization. It provides a minimal normal form on our class, computable in Exptime. This paper extends an already existing result on tree-to-word transducers without copy or reordering (sequential tree-to-word transducers), by accounting for all the possible reorderings in the output.
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