arXiv subjects
Publications and source records attributed to Paul Gallot.
It is shown that shape preservation is decidable for top-down tree transducers, bottom-up tree transducers, and for compositions of total deterministic macro tree transducers. Moreover, if a transducer is shape preserving, then it can be brought into a particular normal form, where every input node creates exactly one output node.
We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem).
Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in cubic time. As a special case, they apply to the growth of the ambiguity of a nondeterministic tree automaton, i.e. the number of distinct accepting runs over a given input. We deduce analogous decidability results (ignoring complexity) for the growth of the number of results of set queries in Monadic Second-Order logic (MSO) over ranked trees. In the case of polynomial growth of degree $k$, we also prove a reparameterization theorem for such queries: their results can be mapped to $k$-tuples of input nodes in a finite-to-one and MSO-definable fashion. We then apply these tools to study growth rates and subclass membership problems for tree-to-tree functions. Using new proof strategies, we recover and generalize known results concerning polyregular functions, total deterministic macro tree transducers, and partial nondeterministic top-down tree transducers. In particular, we give a procedure to decide polynomial size-to-height increase for both macro tree transducers and MSO set interpretations, and compute the degree. The paper concludes with a survey of a wide range of related work.
We present a novel normal form for (total deterministic) macro tree transducers (mtts), called depth proper normal form. If an mtt is in this normal form, then it is guaranteed that each parameter of each state of the mtt appears at arbitrary depth in the output trees of that state. Intuitively, if some parameter only appears at certain bounded depths in the output trees of a state, then this parameter can be removed by in-lining the corresponding output paths at each call site of that state. We use regular look-ahead in order to determine which of the paths should be in-lined. As a consequence of changing the lookahead, a parameter that was previously appearing at unbounded depths, may be appearing at bounded depths for some new look-ahead; for this reason, our construction has be iterated in order to obtain an mtt in depth-normal form. Using the normal form, we can decide whether the translation of an mtt has linear height increase or has linear size-to-height increase.