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 ${\bf \Delta^0_2}$ of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure

Definable Operations On Weakly Recognizable Sets of Trees

Alternating automata on infinite trees induce operations on languages which do not preserve natural equivalence relations, like having the same Mostowski--Rabin index, the same Borel rank, or being continuously reducible to each other (Wadge equivalence). In order to prevent this, alternation needs to be restricted to the choice of direction in the tree. For weak alternating automata with restricted alternation a small set of computable operations generates all definable operations, which implies that the Wadge degree of a given automaton is computable. The weak index and the Borel rank coincide, and are computable. An equivalent automaton of minimal index can be computed in polynomial time (if the productive states of the automaton are given)

Contracting a chordal graph to a split graph or a tree

The problems Contractibility and Induced Minor are to test whether a graph G contains a graph H as a contraction or as an induced minor, respectively. We show that these two problems can be solved in |VG|f(|VH|)VGf(VH) time if G is a chordal input graph and H is a split graph or a tree. In contrast, we show that containment relations extending Subgraph Isomorphism can be solved in linear time if G is a chordal input graph and H is an arbitrary graph not part of the input

An Algebraic Theory of Complexity for Valued Constraints: Establishing a Galois Connection.

The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages. © 2011 Springer-Verlag GmbH

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 above—Mostowski-Rabin, Borel, and Wadge—are 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: (ω^ω)^ω

The Wadge Hierarchy of Deterministic Tree Languages

Vol. 4 (4:15) 2008, pp. 1–44 www.lmcs-online.or