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

The Arity Hierarchy in the Polyadic μ\mu-Calculus

The polyadic mu-calculus is a modal fixpoint logic whose formulas define relations of nodes rather than just sets in labelled transition systems. It can express exactly the polynomial-time computable and bisimulation-invariant queries on finite graphs. In this paper we show a hierarchy result with respect to expressive power inside the polyadic mu-calculus: for every level of fixpoint alternation, greater arity of relations gives rise to higher expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282

The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of structures. A prominent instance is the class of infinite words over which the alternation-free fragment is already as expressive as the full mu-calculus. Our current understanding of when and why the mu-calculus alternation hierarchy is not strict is limited. This paper makes progress in answering these questions by showing that the alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure's graph decomposes it into graphs in which all paths are of finite length. Our collapse results are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202

On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components

For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1). This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1)

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

Disjunctive form and the modal μ\mu alternation hierarchy

This paper studies the relationship between disjunctive form, a syntactic normal form for the modal mu calculus, and the alternation hierarchy. First it shows that all disjunctive formulas which have equivalent tableau have the same syntactic alternation depth. However, tableau equivalence only preserves alternation depth for the disjunctive fragment: there are disjunctive formulas with arbitrarily high alternation depth that are tableau equivalent to alternation-free non-disjunctive formulas. Conversely, there are non-disjunctive formulas of arbitrarily high alternation depth that are tableau equivalent to disjunctive formulas without alternations. This answers negatively the so far open question of whether disjunctive form preserves alternation depth. The classes of formulas studied here illustrate a previously undocumented type of avoidable syntactic complexity which may contribute to our understanding of why deciding the alternation hierarchy is still an open problem.Comment: In Proceedings FICS 2015, arXiv:1509.0282

The Variable Hierarchy for the Games mu-Calculus

Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables