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We introduce the notion of high-order deterministic top-down tree transducers (HODT) whose outputs correspond to single-typed lambda-calculus formulas. These transducers are natural generalizations of known models of top-tree transducers such as: Deterministic Top-Down Tree Transducers, Macro Tree Transducers, Streaming Tree Transducers. . .We focus on the linear restriction of high order tree transducers with look-ahead (HODTRlin), and prove this corresponds to tree to tree functional transformations defined by Monadic Second Order (MSO) logic. We give a specialized procedure for the composition of those transducers that uses a flow analysis based on coherence spaces and allows us to preserve the linearity of transducers. This procedure has a better complexity than classical algorithms for composition of other equivalent tree transducers, but raises the order of transducers. However, we also indicate that the order of a HODTRlin can always be bounded by 3, and give a procedure that reduces the order of a HODTRlin to 3. As those resulting HODTRlin can then be transformed into other equivalent models, this gives an important insight on composition algorithm for other classes of transducers. Finally, we prove that those results partially translate to the case of almost linear HODTR: the class corresponds to the class of tree transformations performed by MSO with unfolding (not closed by composition), and provide a mechanism to reduce the order to 3 in this case. 2012 ACM Subject Classification Theory of computation → Transducers; Theory of computation → Lambda calculus; Theory of computation → Tree languages

Linear High-Order Deterministic Tree Transducers with Regular Look-Ahead