Common estimation algorithms, such as least squares estimation or the Kalman filter, operate on a state in a state space S that is represented as a real-valued vector. However, for many quantities, most notably orientations in 3D, S is not a vector space, but a so-called manifold, i.e. it behaves like a vector space locally but has a more complex global topological structure. For integrating these quantities, several ad-hoc approaches have been proposed.Here, we present a principled solution to this problem where the structure of the manifold S is encapsulated by two operators, state displacement : S × R n → S and its inverse : S × S → R n . These operators provide a local vector-space view δ → x δ around a given state x. Generic estimation algorithms can then work on the manifold S mainly by replacing +/− with / where appropriate. We analyze these operators axiomatically, and demonstrate their use in least-squares estimation and the Unscented Kalman Filter. Moreover, we exploit the idea of encapsulation from a software engineering perspective in the Manifold Toolkit, where the / operators mediate between a "flat-vector" view for the generic algorithm and a "named-members" view for the problem specific functions.

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy-Milner logic, graded modal logic and probabilistic modal logic.

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modelled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioural equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviourally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.Coalgebra has in recent years emerged as an appropriate framework for the treatment of reactive systems in a very general sense [30]; in particular, coalgebra provides a unifying perspective on notions such as coinduction, corecursion, and bisimulation. It has turned out that modal logic is a good candidate for being the basic logic of coalgebra in the same sense as equational logic is the basic logic of algebra. For example, classes of coalgebras defined by modal axioms can be regarded as the dual of varieties [18,20]. Moreover, coalgebraic modal logic as considered in [13,19,[24][25][26]28] is invariant under behavioural equivalence. Conversely, in [24][25][26], sufficient conditions are given for coalgebraic modal logics to be expressive in the sense that logically indistinguishable states are behaviourally equivalent; this is a generalisation of the classical result for Hennessy-Milner logic [12]. These results depend on conditions imposed on the signature functor, i.e. the data type in which collections of successor states are organised.Indeed, coalgebraic logic as introduced by Moss [22], which may be regarded as a form of modal logic, is expressive for the (very large) class of so-called set-based functors; however, from the point of view of practical E-mail address: Lutz.Schroeder@dfki.de. 1 Financial support by the DFG project HasCASL2 (KR 1191/7-2) is gratefully acknowledged. 0304-3975/$ -see front matter c

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE . This leads to a unified derivation of tight PSPACE -bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.