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.