Collapsible Pushdown Automata and Recursion Schemes

We consider recursion schemes (not assumed to be homogeneously typed , and hence not necessarily safe ) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed deterministic term rewriting system that generates, when one applies the rewriting rules ad infinitum , an infinite tree, called its value tree . A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this article, we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it, and contracts its silent transitions, which leads to an infinite tree, which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving effective transformations in both directions.