Results 1  10
of
13
The Wadge Hierarchy of Deterministic Tree Languages
 In: Proc. ICALP 2006, Lect. Notes Comput. Sci. 4052
, 2006
"... Vol. 4 (4:15) 2008, pp. 1–44 www.lmcsonline.org ..."
(Show Context)
The NonDeterministic Mostowski Hierarchy and DistanceParity Automata
"... Abstract. Given a Rabin treelanguage and natural numbers i, j, the language is said to be i, jfeasible if it is accepted by a parity automaton using priorities {i, i+1,..., j}. The i, jfeasibility induces a hierarchy over the Rabintree languages called the Mostowski hierarchy. In this paper we p ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Given a Rabin treelanguage and natural numbers i, j, the language is said to be i, jfeasible if it is accepted by a parity automaton using priorities {i, i+1,..., j}. The i, jfeasibility induces a hierarchy over the Rabintree languages called the Mostowski hierarchy. In this paper we prove that the problem of deciding if a language is i, jfeasible is reducible to the uniform universality problem for distanceparity automata. Distanceparity automata form a new model of automata extending both the nested distance desert automata introduced by Kirsten in his proof of decidability of the starheight problem, and parity automata over infinite trees. Distanceparity automata, instead of accepting a language, attach to each tree a cost in ω + 1. The uniform universality problem consists in determining if this cost function is bounded by a finite value. 1
On the topological complexity of weakly recognizable tree languages
 Proc. FCT 2007, LNCS 4639
, 2007
"... Abstract. We show that the family of tree languages recognized by weak alternating automata is closed by three set theoretic operations that correspond to sum, multiplication by ordinals <ω ω, and pseudoexponentiation with the base ω1 of the Wadge degrees. In consequence, the Wadge hierarchy of w ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We show that the family of tree languages recognized by weak alternating automata is closed by three set theoretic operations that correspond to sum, multiplication by ordinals <ω ω, and pseudoexponentiation with the base ω1 of the Wadge degrees. In consequence, the Wadge hierarchy of weakly recognizable tree languages has the height of at least ε0, that is the least fixed point of the exponentiation with the base ω. 1
Deciding the weak definability of Büchi definable tree languages
"... Weakly definable languages of infinite trees are an expressive subclass of regular tree languages definable in terms of weak monadic secondorder logic, or equivalently weak alternating automata. Our main result is that given a Büchi automaton, it is decidable whether the language is weakly definabl ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Weakly definable languages of infinite trees are an expressive subclass of regular tree languages definable in terms of weak monadic secondorder logic, or equivalently weak alternating automata. Our main result is that given a Büchi automaton, it is decidable whether the language is weakly definable. We also show that given a parity automaton, it is decidable whether the language is recognizable by a nondeterministic coBüchi automaton. The decidability proofs build on recent results about cost automata over infinite trees. These automata use counters to define functions from infinite trees to the natural numbers extended with infinity. We reduce to testing whether the functions defined by certain “quasiweak ” cost automata are bounded by a finite value.
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 t ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
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).
www.stacsconf.org WEAK INDEX VERSUS BOREL RANK
"... Abstract. We investigate weak recognizability of deterministic languages of infinite trees. We prove that for deterministic languages the Borel hierarchy and the weak index hierarchy coincide. Furthermore, we propose a procedure computing for a deterministic automaton an equivalent minimal index wea ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We investigate weak recognizability of deterministic languages of infinite trees. We prove that for deterministic languages the Borel hierarchy and the weak index hierarchy coincide. Furthermore, we propose a procedure computing for a deterministic automaton an equivalent minimal index weak automaton with a quadratic number of states. The algorithm works within the time of solving the emptiness problem. 1.
Quasiweak cost automata: A new variant of weakness
 In Supratik Chakraborty and Amit Kumar, editors, FSTTCS, volume 13 of LIPIcs
, 2011
"... Abstract Cost automata have a finite set of counters which can be manipulated on each transition but do not affect control flow. Based on the evolution of the counter values, these automata define functions from a domain like words or trees to N ∪ {∞}, modulo an equivalence relation which ignores e ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract Cost automata have a finite set of counters which can be manipulated on each transition but do not affect control flow. Based on the evolution of the counter values, these automata define functions from a domain like words or trees to N ∪ {∞}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. These automata have been studied by Colcombet et al. as part of a "theory of regular cost functions", an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability like the classical theory. We extend this theory by introducing quasiweak cost automata. Unlike traditional weak automata which have a hardcoded bound on the number of alternations between accepting and rejecting states, quasiweak automata bound the alternations using the counter values (which can vary across runs). We show that these automata are strictly more expressive than weak cost automata over infinite trees. The main result is a Rabinstyle characterization theorem: a function is quasiweak definable if and only if it is definable using two dual forms of nondeterministic Büchi cost automata. This yields a new decidability result for cost functions over infinite trees.
Linear Game Automata: Decidable Hierarchy Problems for StrippedDown Alternating Tree Automata
"... For deterministic tree automata, classical hierarchies, like MostowskiRabin (or index) hierarchy, Borel hierarchy, or Wadge hierarchy, are known to be decidable. However, when it comes to nondeterministic tree automata, none of these hierarchies is even close to be understood. Here we make an att ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
For deterministic tree automata, classical hierarchies, like MostowskiRabin (or index) hierarchy, Borel hierarchy, or Wadge hierarchy, are known to be decidable. However, when it comes to nondeterministic 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—MostowskiRabin, 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.
Automata on Inifinite Trees
, 2011
"... This article presents the basic theory of automata on infinite trees. We introduce the standard model of nondeterministic automata and the generalization to alternating automata. The central part of the theory considers the closure properties of tree languages definable by such automata, the equiva ..."
Abstract
 Add to MetaCart
This article presents the basic theory of automata on infinite trees. We introduce the standard model of nondeterministic automata and the generalization to alternating automata. The central part of the theory considers the closure properties of tree languages definable by such automata, the equivalence of the different models, and their algorithmic properties. We show how these results are obtained and how games of infinite duration can be used as a powerful tool to obtain transparent proofs and algorithms. We also present some applications of tree automata in logic, in particular decidability results and characterizations for monadic secondorder logic over infinite trees, and algorithms for the modal µcalculus.