. The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation. Keywords. Bisimulation, probabilistic transition system, coalgebra, ultrametric space, Borel measure, final coalgebra. 1 Introduction For discrete probabilistic transition systems the notion of probabilistic bisimilarity of Larsen and Skou [LS91] is regarded as the basic process equivalence. The definition was given for reactive systems. However, Van Glabbeek, Smolka and Steffen s...

Abstract. The class of visibly pushdown languages has been recently defined as a subclass of context-free languages with desirable closure properties and tractable decision problems. We study visibly pushdown games, which are games played on visibly pushdown systems where the winning condition is given by a visibly pushdown language. We establish that, unlike pushdown games with pushdown winning conditions, visibly pushdown games are decidable and are 2Exptime-complete. We also show that pushdown games against Ltl specifications and Caret specifications are 3Exptime-complete. Finally, we establish the topological complexity of visibly pushdown languages by showing that they are a subclass of Boolean combinations of Σ3 sets. This leads to an alternative proof that visibly pushdown automata are not determinizable and also shows that visibly pushdown games are determined. 1

An operation M* which constructs from a given structure M a tree-like structure whose domain consists of the finite sequences of elements of M is considered. A notion of automata running on such tree-like structures is defined. It is shown that automata of this kind characterise expressive power of monadic second order logic (MSOL) over tree-like structures. Using this characterisation it is proved that MSOL theory of tree-like structures is effectively reducible to that of the original structures. As another application of the characterisation it is shown that MSOL on trees of arbitrary degree is equivalent to first order logic extended with unary least fixpoint operator.

Abstract. We first consider infinite two-player games on pushdown graphs. In previous work, Cachat, Duparc and Thomas [4] have presented a winning decidable condition that is Σ3-complete in the Borel hierarchy. This was the first example of a decidable winning condition of such Borel complexity. We extend this result by giving a family of decidable winning conditions of arbitrary high finite Borel complexity. From this family, we deduce a family of decidable winning conditions of arbitrary finite Borel complexity for games played on finite graphs. The problem of deciding the winner for these winning conditions is shown to be non-elementary complete. Keywords: Pushdown Automata, Two-player Games, Borel Complexity. 1

Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such that E d ∆ ⊂ E c Γ and d has a as a factor. This generalizes the standard notion of co-induction of actions of groups from actions of subgroups. We then use this construction to show that if Γ is a countable non-amenable group, then Γ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.

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Abstract. There are notions of L2-Betti numbers for discrete groups (Cheeger-

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A new class of languages of infinite words is introduced, called the maxregular languages, extending the class of ω-regular languages. The class has two equivalent descriptions: in terms of automata (a type of deterministic counter automaton), and in terms of logic (weak monadic second-order logic with a bounding quantifier). Effective translations between the logic and automata are given.

Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F. As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007, Forcing Idealized, Preprint) if and only if E is hyperfinite. Using this, we prove that all Σ1 2 sets and Σ1 1 quasi-orders are Borel on Borel reducible to the quasi-order of Borel homomorphism on the class of inhomogeneous Π1 1 on Σ11 σ-ideals.