We construct witnesses that can be used to derive strategies in fixpoint games and provide proof that the least fixpoint of a function is either above or not below some given bound. We rely on a lattice-theoretical approach, including a Galois connection that connects a lattice representing the "logic universe", where the witness lives, with another lattice representing the "behaviour universe", over which the function is defined. In fact we consider two types of games - primal and dual games - and in both cases show how to derive winning strategies in the game from witnesses and construct witnesses from strategies. The two games differ wrt. their rules and the choice of basis of the lattice. The theory can be instantiated to well-known examples: in particular we compare with the construction of distinguishing formulas in standard bisimilarity and behavioural metrics for probabilistic systems. As a new case study we consider witnesses for certifying lower bounds for the termination probability for Markov chains.

Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the linear-time/ branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics vary in granularity, and are often characterized by fragments of suitable modal logics. In the latter respect, the quantitative case is, however, more involved than the two-valued one; in fact, we show that probabilistic metric trace distance cannot be characterized by any compositionally defined modal logic with unary modalities. We go on to provide a unifying treatment of spectra of behavioural metrics in the emerging framework of graded monads, working in coalgebraic generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we introduce algebraic presentations of graded monads on the category of metric spaces. Moreover, we provide a general criterion for a given real-valued modal logic to characterize a given behavioural distance. As a case study, we apply this criterion to obtain a new characteristic modal logic for trace distance in fuzzy metric transition systems.

We address the task of deriving fixpoint equations from modal logics characterizing behavioural equivalences and metrics (summarized under the term conformances). We rely on earlier work that obtains Hennessy-Milner theorems as corollaries to a fixpoint preservation property along Galois connections between suitable lattices. We instantiate this to the setting of coalgebras, in which we spell out the compatibility property ensuring that we can derive a behaviour function whose greatest fixpoint coincides with the logical conformance. We then concentrate on the linear-time case, for which we study coalgebras based on the machine functor living in Eilenberg-Moore categories, a scenario for which we obtain a particularly simple logic and fixpoint equation. The theory is instantiated to concrete examples, both in the branching-time case (bisimilarity and behavioural metrics) and in the linear-time case (trace equivalences and trace distances).