15 research outputs found
Collapsible Pushdown Graphs of Level 2 are Tree-Automatic
We show that graphs generated by collapsible pushdown systems of level 2 are
tree-automatic. Even when we allow -contractions and add a
reachability predicate (with regular constraints) for pairs of configurations,
the structures remain tree-automatic. Hence, their FO theories are decidable,
even when expanded by a reachability predicate. As a corollary, we obtain the
tree-automaticity of the second level of the Caucal-hierarchy.Comment: 12 pages Accepted for STACS 201
Tree-Automatic Well-Founded Trees
We investigate tree-automatic well-founded trees. Using Delhomme's
decomposition technique for tree-automatic structures, we show that the
(ordinal) rank of a tree-automatic well-founded tree is strictly below
omega^omega. Moreover, we make a step towards proving that the ranks of
tree-automatic well-founded partial orders are bounded by omega^omega^omega: we
prove this bound for what we call upwards linear partial orders. As an
application of our result, we show that the isomorphism problem for
tree-automatic well-founded trees is complete for level Delta^0_{omega^omega}
of the hyperarithmetical hierarchy with respect to Turing-reductions.Comment: Will appear in Logical Methods of Computer Scienc
First-Order Model Checking on Generalisations of Pushdown Graphs
We study the first-order model checking problem on two generalisations of
pushdown graphs. The first class is the class of nested pushdown trees. The
other is the class of collapsible pushdown graphs. Our main results are the
following. First-order logic with reachability is uniformly decidable on nested
pushdown trees. Considering first-order logic without reachability, we prove
decidability in doubly exponential alternating time with linearly many
alternations. First-order logic with regular reachability predicates is
uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested
pushdown trees are first-order interpretable in collapsible pushdown graphs of
level 2. This interpretation can be extended to an interpretation of the class
of higher-order nested pushdown trees in the collapsible pushdown graph
hierarchy. We prove that the second level of this new hierarchy of nested trees
has decidable first-order model checking. Our decidability result for
collapsible pushdown graph relies on the fact that level 2 collapsible pushdown
graphs are uniform tree-automatic. Our last result concerns tree-automatic
structures in general. We prove that first-order logic extended by Ramsey
quantifiers is decidable on all tree-automatic structures.Comment: phd thesis, 255 page
Collapsible Pushdown Automata and Recursion Schemes
International audienceWe 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 paper 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 an effective transformations in both directions
A pumping lemma for collapsible pushdown graphs of level 2
We present a pumping lemma for the class of collapsible pushdown graphs of level 2. This pumping lemma even applies to the ε-contractions of level 2 collapsible pushdown graphs. Our pumping lemma also improves the bounds of Hayashi’s pumping lemma for indexed languages