Relation Liftings on Preorders and Posets

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss's coalgebraic over posets

Relation Liftings on Preorders and Posets

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss's coalgebraic over posets

Relation lifting, with an application to the many-valued cover modality

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the "powerset monad" on categories, one is the preservation by T of "exactness" of certain squares. Both characterisations are generalisations of the "classical" results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks. The results presented in this paper enable us to compute predicate liftings of endofunctors of, for example, generalised (ultra)metric spaces. We illustrate this by studying the coalgebraic cover modality in this setting.Comment: 48 pages, accepted for publication in LMC

Moss' logic for ordered coalgebras

We present a finitary coalgebraic logic for $T$-coalgebras, where $T$ is a locally monotone endofunctor of the category of posets and monotone maps that preserves exact squares and finite intersections. The logic uses a single cover modality whose arity is given by the dual of the coalgebra functor $T$, and the semantics of the modality is given by relation lifting. For the finitary setting to work, we need to develop a notion of a base for subobjects of $TX$. This in particular allows us to talk about a finite poset of subformulas for a given formula, and of a finite poset of successors for a given state in a coalgebra. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic and prove its completeness

Coinduction up to in a fibrational setting

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similarity, and a new categorical treatment of weak bisimilarity

Moss' logic for ordered coalgebras

We present a finitary version of Moss' coalgebraic logic for $T$-coalgebras, where $T$ is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor $T_\omega^\partial$, and the semantics of the modality is given by relation lifting. For the semantics to work, $T$ is required to preserve exact squares. For the finitary setting to work, $T_\omega^\partial$ is required to preserve finite intersections. We develop a notion of a base for subobjects of $T_\omega X$. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic, and prove its completeness

A general account of coinduction up-to

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras. In the second part of the paper, we provide a new categorical treatment of weak bisimilarity on labeled transition systems and we prove the soundness of up-to context for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. The weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend the categorical framework developed in the first part to an ordered setting

Extending Set Functors to Generalised Metric Spaces

For a commutative quantale V, the category V-cat can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor T (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor TV on V-cat. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones. Conceptually, this allows us to to solve the same recursive domain equation X ≅ TX in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base. Mathematically, the heart of the matter is to show that, for any commutative quantale V, the “discrete functor Set → V-cat from sets to categories enriched over V is V-cat-dense and has a density presentation that allows us to compute left-Kan extensions along D

Reconciling Shannon and Scott with a Lattice of Computable Information

This paper proposes a reconciliation of two different theories of information. The first, originally proposed in a lesser-known work by Claude Shannon (some five years after the publication of his celebrated quantitative theory of communication), describes how the information content of channels can be described qualitatively, but still abstractly, in terms of information elements, where information elements can be viewed as equivalence relations over the data source domain. Shannon showed that these elements have a partial ordering, expressing when one information element is more informative than another, and that these partially ordered information elements form a complete lattice. In the context of security and information flow this structure has been independently rediscovered several times, and used as a foundation for understanding and reasoning about information flow. The second theory of information is Dana Scott\u27s domain theory, a mathematical framework for giving meaning to programs as continuous functions over a particular topology. Scott\u27s partial ordering also represents when one element is more informative than another, but in the sense of computational progress - i.e. when one element is a more defined or evolved version of another. To give a satisfactory account of information flow in computer programs it is necessary to consider both theories together, in order to understand not only what information is conveyed by a program (viewed as a channel, \ue0 la Shannon) but also how the precision with which that information can be observed is determined by the definedness of its encoding (\ue0 la Scott). To this end we show how these theories can be fruitfully combined, by defining the Lattice of Computable Information (LoCI), a lattice of preorders rather than equivalence relations. LoCI retains the rich lattice structure of Shannon\u27s theory, filters out elements that do not make computational sense, and refines the remaining information elements to reflect how Scott\u27s ordering captures possible varieties in the way that information is presented. We show how the new theory facilitates the first general definition of termination-insensitive information flow properties, a weakened form of information flow property commonly targeted by static program analyses