Analysis of Boolean Equation Systems through Structure Graphs

We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph

Unprovability of the Logical Characterization of Bisimulation

We quickly review labelled Markov processes (LMP) and provide a counterexample showing that in general measurable spaces, event bisimilarity and state bisimilarity differ in LMP. This shows that the logic in Desharnais [*] does not characterize state bisimulation in non-analytic measurable spaces. Furthermore we show that, under current foundations of Mathematics, such logical characterization is unprovable for spaces that are projections of a coanalytic set. Underlying this construction there is a proof that stationary Markov processes over general measurable spaces do not have semi-pullbacks. ([*] J. Desharnais, Labelled Markov Processes. School of Computer Science. McGill University, Montr\'eal (1999))Comment: Extended introduction and comments; extra section on semi-pullbacks; 11 pages Some background details added; extra example on the non-locality of state bisimilarity; 14 page

A Definition Scheme for Quantitative Bisimulation

FuTS, state-to-function transition systems are generalizations of labeled transition systems and of familiar notions of quantitative semantical models as continuous-time Markov chains, interactive Markov chains, and Markov automata. A general scheme for the definition of a notion of strong bisimulation associated with a FuTS is proposed. It is shown that this notion of bisimulation for a FuTS coincides with the coalgebraic notion of behavioral equivalence associated to the functor on Set given by the type of the FuTS. For a series of concrete quantitative semantical models the notion of bisimulation as reported in the literature is proven to coincide with the notion of quantitative bisimulation obtained from the scheme. The comparison includes models with orthogonal behaviour, like interactive Markov chains, and with multiple levels of behavior, like Markov automata. As a consequence of the general result relating FuTS bisimulation and behavioral equivalence we obtain, in a systematic way, a coalgebraic underpinning of all quantitative bisimulations discussed.Comment: In Proceedings QAPL 2015, arXiv:1509.0816

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

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)