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It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game (with n nodes and m distinct values) is proven to be in the class of fixed parameter tractable (FPT) problems (when parameterised over m). Both results improve known bounds, from runtime n O( p n/ log(n)) to O(n log(m)+6 ) (note that m  n) and from an XP-algorithm with runtime O(n ⇥(m) ) for fixed parameter m to an FPT-algorithm with runtime O(n 5 ) + g(m), for some function g depending on m only. As an application it is proven that coloured Muller games with n nodes and m colours can be decided in time O((m m • n) 5 ); this bound cannot be improved to O((2 m • n) c ), for any c, unless FPT = W[1].

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Abstract. A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay 23 and Chaitin 10 we s a y that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's 23 -like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability o f a universal self-delimiting Turing machine Chaitin's number, 9 is also a random r.e. real. Solovay showed that any Chaitin numberis -like. In this paper we show that the converse implication is true as well: any -like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these group are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the Word Problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2nilpotent groups and Baumslag-Solitar groups B(1, n) are graph automatic, as well as many other metabelian groups.