Jacques Duparc

A hierarchy of deterministic context-free ω-languages

Twenty years ago, Klaus. W. Wagner came up with a hierarchy of ?-regular sets that actually bears his name. It turned out to be exactly the Wadge hierarchy of the sets of ?-words recognized by deterministic finite automata. We describe the Wadge hierarchy of context-free ?-languages, which stands as an extension of Wagners work from automata to pushdown automata.

Solving Pushdown Games with a Sigma3 Winning Condition

We study infinite two-player games over pushdown graphs with a winning condition that refers explicitly to the infinity of the game graph: A play is won by player 0 if some vertex is visited infinity often during the play. We show that the set of winning plays is a proper ?3-set in the Borel hierarchy, thus transcending the Boolean closure of ?2-sets which arises with the standard automata theoretic winning conditions (such as the Muller, Rabin, or parity condition). We also show that this ?3-game over pushdown graphs can be solved effectively (by a computation of the winning region of player 0 and his memoryless winning strategy). This seems to be a first example of an effectively solvable game beyondt he second level of the Borel hierarchy.

Computer science and the fine structure of Borel sets

(I) Wadge defined a natural refinement of the Borel hierarchy, now called the Wadge hierarchy WH. The fundamental properties of WH follow from the results of Kuratowski, Martin, Wadge and Louveau. We give a transparent restatement and proof of Wadges main theorem. Our method is new for it yields a wide and unexpected extension: from Borel sets of reals to a class of natural but non Borel sets of infinite sequences. Wadges theorem is quite ineffective and our generalization clearly worsens in this respect. Yet paradoxically our method is appropriate to effectivize this whole theory in the context discussed below. (II) Wagner defined on Buchi automata (accepting words of length ω) a hierarchy and proved for it an effective analog of Wadges results. We extend Wagners results to more general kinds of automata: counters, push-down automata and Buchi automata reading transfinite words. The notions and methods developed in (I) are quite useful for this extension, and we start to use them in order to look for extensions of the fundamental effective determinacy results of Buchi–Landweber, Rabin; and of Courcelle–Walukiewicz.

THE MISSING LINK FOR ω-RATIONAL SETS, AUTOMATA, AND SEMIGROUPS

In 1997, following the works of Klaus W. Wagner on ω-regular sets, Olivier Carton and Dominique Perrin introduced the notions of chains and superchains for ω-semigroups. There is a clear correspondence between the algebraic representation of each of these operations and the automata-theoretical one. Unfortunately, chains and superchains do not suffice to describe the whole Wagner hierarchy. We introduce a third notion that completes the task undertaken by these two authors.

On the topological complexity of weakly recognizable tree languages

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 < ωω, 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 e0, that is the least fixed point of the exponentiation with the base ω.

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).

The Wadge Hierarchy of Petri Nets ω-Languages

We describe the Wadge hierarchy of the ω-languages recognized by deterministic Petri nets. This is an extension of the celebrated Wagner hierarchy which turned out to be the Wadge hierarchy of the ω-regular languages. Petri nets are an improvement of automata. They may be defined as partially blind multi-counter automata. We show that the whole hierarchy has height \(\omega^{\omega^2}\), and give a description of the restrictions of this hierarchy to every fixed number of partially blind counters.

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: (ωω)ω.

Transfinite extension of the mu-calculus

In [1] Bradfield found a link between finite differences formed by Σ

Expressive Power of Non-deterministic Evolving Recurrent Neural Networks in Terms of Their Attractor Dynamics

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