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Rewriting Higher-order Stack Trees

### Vincent Penelle

Higher-order pushdown systems and ground tree rewriting systems can
be seen as extensions of suffix word rewriting systems. Both classes
generate infinite graphs with interesting logical
properties. Indeed, the satisfaction of any formula written in
monadic second order logic (respectively first order logic with
reachability predicates) can be decided on such a graph.

The purpose of this talk is to propose a common extension to both
higher-order stack operations and ground tree rewriting. We introduce a model
of higher-order ground tree rewriting over trees labelled by
higher-order stacks (henceforth called stack trees), which syntactically
coincides with ordinary ground tree rewriting at order 1 and with the dynamics
of higher-order pushdown automata over unary trees. The infinite
graphs generated by this class have a decidable first order logic with
reachability.

Formally, an order n stack tree is a tree labelled by order n-1 stacks.
Operations of ground stack tree rewriting are represented by a certain class
of connected DAGs labelled by a set of basic operations over stack trees
describing of the relative application positions of the basic
operations appearing on it. Applying a DAG to a stack tree t intuitively
amounts to paste its input vertices to some leaves of t and to simplify
the obtained structure, applying the basic operations labelling the edges of the
DAG to the leaves they are appended to, until either a new stack tree is
obtained or the process fails, in which case the application of the DAG to t
at the chosen position is deemed impossible. This model is a common extension
to those of higher-order stack operations presented by Carayol and of ground tree
transducers presented by Dauchet and Tison.

As further results we can define a notion of recognisable sets of
operations through a generalisation. The proof that the graphs generated
by a ground stack tree rewriting system have a decidable first order theory
with reachability is inspired by the technique of finite set interpretations
presented by Colcombet and Loding.