22 research outputs found
Deciding the Borel complexity of regular tree languages
We show that it is decidable whether a given a regular tree language belongs
to the class of the Borel hierarchy, or equivalently whether
the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure
A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic
We define a infinitary labelled sequent calculus for PDL, G3PDL^{\infty}. A
finitarily representable cyclic system, G3PDL^{\omega}, is then given. We show
that both are sound and complete with respect to standard models of PDL and,
further, that G3PDL^{\infty} is cut-free complete. We additionally investigate
proof-search strategies in the cyclic system for the fragment of PDL without
tests
Linear Game Automata: Decidable Hierarchy Problems for Stripped-Down Alternating Tree Automata
For deterministic tree automata, classical hierarchies, like Mostowski-Rabin (or index) hierarchy, Borel hierarchy, or Wadge hierarchy, are known to be decidable. However, when it comes to non-deterministic tree automata, none of these hierarchies is even close to be understood. Here we make an attempt in paving the way towards a clear understanding of tree automata. We concentrate on the class of linear game automata (LGA), and prove within this new context, that all corresponding hierarchies mentioned aboveâMostowski-Rabin, Borel, and Wadgeâare decidable. The class LGA is obtained by taking linear tree automata with alternation restricted to the choice of path in the input tree. Despite their simplicity, LGA recognize sets of arbitrary high Borel rank. The actual richness of LGA is revealed by the height of their Wadge hierarchy: (Ï^Ï)^Ï
The Wadge Hierarchy of Deterministic Tree Languages
Vol. 4 (4:15) 2008, pp. 1â44 www.lmcs-online.or
Cardinality quantifiers in MLO over trees
We study an extension of monadic second-order logic of order with the uncountability quantifier âthere exist uncountably many setsâ. We prove that, over the class of finitely branching trees, this extension is equally expressive to plain monadic second-order logic of order. Additionally we find that the continuum hypothesis holds for classes of sets definable in monadic second-order logic over finitely branching trees, which is notable for not all of these classes are analytic. Our approach is based on Shelahâs composition method and uses basic results from descriptive set theory. The elimination result is constructive, yielding a decision procedure for the extended logic. Furthermore, by the well-known correspondence between monadic second-order logic and tree automata, our findings translate to analogous results on the extension of first-order logic by cardinality quantifiers over injectively presentable Rabin-automatic structures, generalizing the work of Kuske and Lohrey
On deciding topological classes of deterministic tree languages
Abstract. It has been proved by NiwiĆski and Walukiewicz that a deterministic tree language is either Î 1 1-complete or it is on the level Î 0 3 of the Borel hierarchy, and that it can be decided effectively which of the two takes place. In this paper we show how to decide if the language recognized by a given deterministic tree automaton is on the Î 0 2, the ÎŁ 0 2, or the ÎŁ 0 3 level. Together with the previous results it gives a procedure calculating the exact position of a deterministic tree language in the topological hierarchy