Monotonic Distributive Semilattices

In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the {→, ∧, ⊤}-fragment of intuitionistic logic is the variety of implicative meet-semilattices (Chellas 1980; Hansen 2003). In this paper we introduce and study the class of distributive meet-semilattices endowed with a monotonic modal operator m. We study the representation theory of these algebras using the theory of canonical extensions and we give a topological duality for them. Also, we show how our new duality extends to some particular subclasses.Fil: Celani, Sergio Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; ArgentinaFil: Menchón, María Paula. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Synthesis of Strategies and the Hoare Logic of Angelic Nondeterminism

Abstract. We study a propositional variant of Hoare logic that can be used for reasoning about programs that exhibit both angelic and demonic nondeterminism. We work in an uninterpreted setting, where the mean-ing of the atomic actions is specified axiomatically using hypotheses of a certain form. Our logical formalism is entirely compositional and it sub-sumes the non-compositional formalism of safety games on finite graphs. We present sound and complete Hoare-style (partial-correctness) calculi that are useful for establishing Hoare assertions, as well as for synthesiz-ing implementations. The computational complexity of the Hoare theory of dual nondeterminism is investigated using operational models, and it is shown that the theory is complete for exponential time.

Involutions on Relational Program Calculi

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Relational semantics through duality

Abstract. In this paper we show how the classical duality results extended to a Duality via Truth contribute to development of a relational semantics for various modal-like logics. In particular, we present a Duality via Truth for some classes of information algebras and frames. We also show that the full categorical formulation of classical duality extends to a full Duality via Truth.

Monotone predicate transformers as Up-closed multirelations

In the study of semantic models for computations two independent views predominate: relational models and predicate transformer semantics. Recently the traditional relational view of computations as binary relations between states has been generalised to multirelations between states and properties allowing the simultaneous treatment of angelic and demonic nondeterminism. In this paper the two-level nature of multirelations is exploited to provide a factorisation of up-closed multirelations which clarifies exactly how multirelations model nondeterminism. Moreover, monotone predicate transformers are, in the precise sense of duality, up-closed multirelations. As such they are shown to provide a notion of effectivity of a specification for achieving a given postcondition. © Springer-Verlag Berlin Heidelberg 2006.Conference Pape

Discrete duality for relation algebras and cylindric algebras

Following the representation theorems for relation algebras and cylindric algebras presented in [5] and [7] we develop discrete duality for relation algebras and relation frames, and for cylindric algebras and cylindric frames. © 2009 Springer-Verlag Berlin Heidelberg.Conference Pape

Context algebras, context frames, and their discrete duality

The data structures dealt with in formal concept analysis are referred to as contexts. In this paper we study contexts within the framework of discrete duality. We show that contexts can be adequately represented by a class of sufficiency algebras called context algebras. On the logical side we define a class of context frames which are the semantic structures for context logic, a lattice-based logic associated with the class of context algebras. We prove a discrete duality between context algebras and context frames, and we develop a Hilbert style axiomatization of context logic and prove its completeness with respect to context frames. Then we prove a duality via truth theorem showing that both context algebras and context frames provide the adequate semantic structures for context logic. We discuss applications of context algebras and context logic to the specification and verification of various problems concerning contexts such as implications (attribute dependencies) in contexts, and derivation of implications from finite sets of implications. © 2008 Springer Berlin Heidelberg.Conference Pape

Discrete Duality and Its Applications to Reasoning with Incomplete Information

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Three dual ontologies

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Modelling the algebra of weakest preconditions

Dijkstra's weakest precondition semantics, as presented in textbook form by Gries, may be viewed as an equational algebra. The problem then is to find a reasonable (set-theoric) model of this algebra. This paper provides one