Liveness Properties in Geometric Logic for Domain-Theoretic Streams

We devise a version of Linear Temporal Logic (LTL) on a denotational domain of streams. We investigate this logic in terms of domain theory, (point-free) topology and geometric logic. This yields the first steps toward an extension of the "Domain Theory in Logical Form" paradigm to temporal liveness properties. We show that the negation-free formulae of LTL induce sober subspaces of streams, but that this is in general not the case in presence of negation. We propose a direct, inductive, translation of negation-free LTL to geometric logic. This translation reflects the approximations used to compute the usual fixpoint representations of LTL modalities. As a motivating example, we handle a natural input-output specification for the usual filter function on streams

Free constructions and coproducts of d-frames

A general theory of presentations for d-frames does not yet exist. We review the difficulties and give sufficient conditions for when they can be overcome. As an application we prove that the category of d-frames is closed under coproducts

Duality and canonical extensions for stably compact spaces

We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.Comment: 29 pages, 1 figur

Strongly Complete Logics for Coalgebras

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts. Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor. Part II investigates algebras for a functor over ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical extensions of Boolean algebras with operators to this setting. Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T. Based on Part II we prove the logic to be strongly complete under a reasonable condition on T

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page

Formal Concept Analysis and Resolution in Algebraic Domains

We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic cpos. We show how to identify the notion of formal concept in the domain theoretic setting. In particular, we show that a special instance of the resolution rule from the domain logic coincides with the concept closure operator from formal concept analysis. The results shed light on the use of contexts and domains for knowledge representation and reasoning purposes.Comment: 14 pages. We have rewritten the old version according to the suggestions of some referees. The results are the same. The presentation is completely differen