A note on partially ordered tree automata

A recent paper by Bouajjani, Muscholl and Touili shows that the class of languages accepted by partially ordered word automata (or equivalently accepted by @S"2-formulae) is closed under semi-commutation and it suggested the following open question: can we extend this result to tree languages? This problem can be addressed by proving (1) that the class of tree regular languages accepted by @S"2 formulae is strictly included in the class of languages accepted by partially ordered automata, and (2) that Bouajjani and the others results cannot be extended to tree.

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