On P-transitive graphs and applications

We introduce a new class of graphs which we call P-transitive graphs, lying between transitive and 3-transitive graphs. First we show that the analogue of de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we show that the mu-calculus fixpoint hierarchy is infinite for P-transitive graphs. Both results contrast with the case of transitive graphs. We give also an undecidability result for an enriched mu-calculus on P-transitive graphs. Finally, we consider a polynomial time reduction from the model checking problem on arbitrary graphs to the model checking problem on P-transitive graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081

The Modal μ-Calculus Hierarchy on Restricted Classes of Transition Systems

We discuss the strictness of the modal µ-calculus hierarchy over some restricted classes of transition systems. First, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for reflexive systems the same results holds for finite models. Second, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the finite model theorem for transitive transition systems is also proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment

Game semantics for the constructive μ\mu-calculus

We define game semantics for the constructive $\mu$-calculus and prove its correctness. We use these game semantics to prove that the $\mu$-calculus collapses to modal logic over $\mathsf{CS5}$ frames. Finally, we prove the completeness of $\mathsf{\mu CS5}$ over $\mathsf{CS5}$ frames

The Alternation Hierarchy for the Theory of mu-lattices

The alternation hierarchy problem asks whether every mu-term,that is a term built up using also a least fixed point constructoras well as a greatest fixed point constructor, is equivalent to amu-term where the number of nested fixed point of a different typeis bounded by a fixed number.In this paper we give a proof that the alternation hierarchyfor the theory of mu-lattices is strict, meaning that such numberdoes not exist if mu-terms are built up from the basic lattice operations and are interpreted as expected. The proof relies on theexplicit characterization of free mu-lattices by means of games andstrategies

μ計算と算術の階層構造の探索---ゲール・スチュワートゲームの視点より

Tohoku University横山啓太課

On the separation question for tree languages

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game

Unambiguous Languages Exhaust the Index Hierarchy

This work is a study of the expressive power of unambiguity in the case of automata over infinite trees. An automaton is called unambiguous if it has at most one accepting run on every input, the language of such an automaton is called an unambiguous language. It is known that not every regular language of infinite trees is unambiguous. Except that, very little is known about which regular tree languages are unambiguous. This paper answers the question whether unambiguous languages are of bounded complexity among all regular tree languages. The notion of complexity is the canonical one, called the (parity or Rabin/Mostowski) index hierarchy. The answer is negative, as exhibited by a family of examples of unambiguous languages the cannot be recognised by any alternating parity tree automata of bounded range of priorities. Hardness of the examples is based on the theory of signatures, previously studied by Walukiewicz. The technical core of the article is a definition of the canonical signatures together with a parity game that compares signatures of a given pair of parity games (of the same index)