The Quantitative Μ-calculus Diplom-informatikerin

This thesis studies a generalisation of the modal μ-calculus, a modal fixedpoint logic that is an important specification language in formal verification. We define a quantitative generalisation of this logic, meaning that formulae do not just evaluate to true or false anymore but instead to arbitrary real values. First, this logic is evaluated on a quantitative extension of transition systems equipped with quantitative predicates that assign real values to the nodes of the system. Having fixed a quantitative semantics, we investigate which of the classical theorems established for the modal μ-calculus can be lifted to this quantitative setting. The modal μ-calculus is connected to bisimulation, a notion of behavioural equivalence for transition systems. We define a quantitative notion of bisimulation as a distance between systems. First, we show that for systems that have a fixed maximal distance, the evaluation of formulae also differs by at most this distance, thus providing a quantitative version of the classical result that the modal μ-calculus is invariant under bisimulation. The converse direction does not hold for Qμ on arbitrary systems. However, as in the classical case, on finitely-branching systems the converse can be shown for the modal fragment and thus already quantitative modal logic characterises quantitative bisimulation on finitely-branching systems. Next, we consider the model-checking problem which, given a system and a formula, is to decide whether the system is a model of the formula. In the quantitative world, this translates to computing the numerical value of a formula at a given node of the system. The model-checking problem for the modal μ-calculus can be solved by parity games, a class of infinite zero-sum graph games. We introduce a quantitative extension of these games and show that they are the corresponding model-checking games for our logic. After establishing bisimulation invariance and developing model-checking games, we move to systems that are closer to the scenarios arising in practical applications. First, we consider discounted systems, i.e. systems where additionally the edges are labelled with quantities. It is not straightforward to define a negation operator in this setting that allows for the duality properties