An omega-power of a context-free language which is Borel above Delta^0_omega

We use erasers-like basic operations on words to construct a set that is both Borel and above Delta^0_omega, built as a set V^\omega where V is a language of finite words accepted by a pushdown automaton. In particular, this gives a first example of an omega-power of a context free language which is a Borel set of infinite rank.Comment: To appear in the Proceedings of the International Conference Foundations of the Formal Sciences V : Infinite Games, November 26th to 29th, 2004, Bonn, Germany, Stefan Bold, Benedikt L\"owe, Thoralf R\"asch, Johan van Benthem (eds.), College Publications at King's College (Studies in Logic), 200

Observations sur la littérature, à Monsieur ***

XII, 331 p

Transfinite Extension of the Mu-Calculus

In [1] Bradfield found a link between finite differences formed by Sigma(2)(0) sets and the mu-arithmetic introduced by Lubarski [7]. We extend this approach into the transfinite: in allowing countable disjunctions we show that this kind of extended mu-calculus matches neatly to the transfinite difference hierarchy of E-2(0) sets. The difference hierarchy is intimately related to parity games. When passing to infinitely many priorities, it might not longer be true that there is a positional winning strategy. However, if such games are derived from the difference hierarchy, this property still holds true

The Topological Complexity of Models of the Modal μ-Calculus: On The Alternation Free Fragment and Beyond

This is the extended journal version of [C1]. In this paper we define set theoretical operations in terms of µ-formulae. In particular, we introduce the operation given by an action of a µ-definable topological property over a class of models. When considering definability by alternation free formulae, we obtain the $\mu$-calculus counterpart of the Wadge hierarchy for weakly alternating tree automata. It was conjectured that the height of this hierarchy is exactly $\varepsilon_{0}$. We prove that the degree of a tree language definable by an alternation free formula is either below $\omega^\omega$ or above $\omega_1$. However, very little is known about the Wadge hierarchy for the full µ-calculus, the problem being that most of the sets definable by a µ-formula are even not Borel. We make a first step in this direction by introducing the Wadge hierarchy extending the one for the alternating free fragment with an action given by a difference of two $\Pi^1_1$ complete sets

An omega-power of a context-free language which is Borel above Delta^0_omega

To appear in the Proceedings of the International Conference Foundations of the Formal Sciences V : Infinite Games, November 26th to 29th, 2004, Bonn, Germany, Stefan Bold, Benedikt Löwe, Thoralf Räsch, Johan van Benthem (eds.), College Publications at King's College (Studies in Logic), 2007.International audienceWe use erasers-like basic operations on words to construct a set that is both Borel and above Delta^0_omega, built as a set V^\omega where V is a language of finite words accepted by a pushdown automaton. In particular, this gives a first example of an omega-power of a context free language which is a Borel set of infinite rank

The Wadge order on the Scott Domain is not a Well-quasi-order

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{lay}$ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\boldsymbol{\Delta}^0_2$-degrees of the Scott domain. We then show that $\mathbb{P}_{lay}$ both admits infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\boldsymbol{\Delta}^0_2$-degrees of the Scott domain.Comment: 26 pages, 6 figures, submitted to The Journal of Symbolic Logi