7,148 research outputs found
Rigid Tree Automata and Applications
International audienceWe introduce the class of Rigid Tree Automata (RTA), an extension of standard bottom-up automata on ranked trees with distinguished states called rigid. Rigid states define a restriction on the computation of RTA on trees: RTA can test for equality in subtrees reaching the same rigid state. RTA are able to perform local and global tests of equality between subtrees, non-linear tree pattern matching, and some inequality and disequality tests as well. Properties like determinism, pumping lemma, Boolean closure, and several decision problems are studied in detail. In particular, the emptiness problem is shown decidable in linear time for RTA whereas membership of a given tree to the language of a given RTA is NP-complete. Our main result is the decidability of whether a given tree belongs to the rewrite closure of an RTA language under a restricted family of term rewriting systems, whereas this closure is not an RTA language. This result, one of the first on rewrite closure of languages of tree automata with constraints, is enabling the extension of model checking procedures based on finite tree automata techniques, in particular for the verification of communicating processes with several local non rewritable memories, like security protocols. Finally, a comparison of RTA with several classes of tree automata with local and global equality tests, with dag automata and Horn clause formalisms is also provided
Isomorphisms of scattered automatic linear orders
We prove that the isomorphism of scattered tree automatic linear orders as
well as the existence of automorphisms of scattered word automatic linear
orders are undecidable. For the existence of automatic automorphisms of word
automatic linear orders, we determine the exact level of undecidability in the
arithmetical hierarchy
On the logical definability of certain graph and poset languages
We show that it is equivalent, for certain sets of finite graphs, to be
definable in CMS (counting monadic second-order logic, a natural extension of
monadic second-order logic), and to be recognizable in an algebraic framework
induced by the notion of modular decomposition of a finite graph. More
precisely, we consider the set of composition operations on graphs
which occur in the modular decomposition of finite graphs. If is a subset
of , we say that a graph is an \calF-graph if it can be
decomposed using only operations in . A set of -graphs is recognizable if
it is a union of classes in a finite-index equivalence relation which is
preserved by the operations in . We show that if is finite and its
elements enjoy only a limited amount of commutativity -- a property which we
call weak rigidity, then recognizability is equivalent to CMS-definability.
This requirement is weak enough to be satisfied whenever all -graphs are
posets, that is, transitive dags. In particular, our result generalizes Kuske's
recent result on series-parallel poset languages
Non-elementary amenable subgroups of automata groups
We consider groups of automorphisms of locally finite trees, and give
conditions on its subgroups that imply that they are not elementary amenable.
This covers all known examples of groups that are not elementary amenable and
act on the trees: groups of intermediate growths and Basilica group, by giving
a more straightforward proof. Moreover, we deduce that all finitely generated
branch groups are not elementary amenable, which was conjectured by Grigorchuk
- …