This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of many-sorted predicate logic.

We apply mathematical concept analysis to the problem of reengineering configurations. Concept analysis will reconstruct a taxonomy of concepts from a relation between objects and attributes. We use concept analysis to infer configuration structures from existing source code. Our tool NORA/RECS will accept source code, where configuration-specific code pieces are controlled by the preprocessor. The algorithm will compute a so-called concept lattice, which —when visually displayed — offers remarkable insight into the structure and properties of possible configurations. The lattice not only displays tine-grained dependencies between configurations, but also visualizes the overall quality of configuration structures according to software engineering principles. In a second step, interferences between configurations can be analyzed in order to restructure or simplify configurations. Interferences showing up in the lattice indicate high coupling and low cohesion between configuration concepts. Source files can then be simplified according to the lattice structure. Finally, we show how governing expressions can be simplified by utilizing an isomorphism theorem of mathematical concept analysis.

This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...

This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (back-and-forth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations take the form of an adjunction. The BAO associated with a coalgebra can be used for specification, e.g. of classes in object-oriented languages.

A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Two-way similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and two-way similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.

A multivariate normal statistical model defined by the Markov pr er deter-- by an acyclic digric admits ar- efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar - linear rear-- model (# MANOVA model).Her these modelsar extended in anatur way tonor linear rear-- models whose LFs continue to admit suchr- efactor--r frr which maximum likelihoodestimator and likelihoodr (LR) test statistics can beder ed by classical linear methods. The centrdistr -- of the LR test statisticfor testing one such multivariv- norv linear rear model against another isder ed, and there- of theseresesion models to block-r-- enor linear systems is established. It is shown how a collection of nonnested dependentnor linear rear-- models (# seemingly unringly ringly- can be combined into a single multivariv- norvlinear rn grear model by imposing apar-- set of graphical Markov (# conditional independence) restrictions.

Abstract. Static analyses calculate abstract states, and their logics validate properties of the abstract states. We place into perspective the variety of forwards, backwards, functional, and logical completeness used in abstract-interpretation-based static analysis by giving examples and by proving equivalences, implications, and independences. We expose two fundamental Galois connections that underlie the logics for static analyses and reveal a new completeness variant, O-completeness. We also show that the key concept underlying logical completeness is covering, which we use to relate the various forms of completeness. When we use a static analysis, like data-flow analysis or model checking, to validate a program for correctness or code improvement, we must carefully define the domain of properties the analysis can calculate so that it includes both the goal properties we seek to validate as well as intermediate properties that lead to the goals. Say we try to validate {?}y: = −y;x: = y +1{isPositive(x)}; our analysis requires properties like isNegative to calculate a sound precondition:

Abstract. We study the underapproximation of the predicate transformers used to give semantics to the modalities in dynamic and temporal logic. Because predicate transformers operate on state sets, we define appropriate powerdomains for sound approximation. We study four such domains — two are based on “set inclusion ” approximation, and two are based on “quantification ” approximation — and we apply the domains to synthesize the most precise, underapproximating �pre and pre transformers, in the latter case, introducing a focus operation. We also show why the expected abstractions of post and �post are unsound, and we use the powerdomains to guide us to correct, sound underapproximations. 1

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres. 1

Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1