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Papers by Dion Coumans
Lecture Notes in Computer Science, 2009
In this paper we describe a language and method for deriving ontologies and ordering databases. T... more In this paper we describe a language and method for deriving ontologies and ordering databases. The ontological structures arrived at are distributive lattices with attribution operations that preserve ∨, ∧ and ^</font >\bot. The preservation of ∧ allows the attributes to model the natural join operation in databases. We start by introducing ontological frameworks and knowledge bases and define the
A Compositional, Diagrammatic Discourse, 2013
Journal of Applied Logic, 2014
Relational semantics, given by Kripke frames, play an essential role in the study of modal and in... more Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [5] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form. In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [4,5] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms. Traditionally, so-called phase semantics are used as models for (provability in) linear logic . These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.
Lecture Notes in Computer Science, 2015
In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (me... more In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implication-formulas. We use these results to derive a new version of subframe formulas for intuitionistic logic and to show that the uniform interpolants of meet-implication-formulas are not necessarily uniform interpolants in the full intuitionistic logic.
Lecture Notes in Computer Science, 2009
In this paper we describe a language and method for deriving ontologies and ordering databases. T... more In this paper we describe a language and method for deriving ontologies and ordering databases. The ontological structures arrived at are distributive lattices with attribution operations that preserve ∨, ∧ and ^</font >\bot. The preservation of ∧ allows the attributes to model the natural join operation in databases. We start by introducing ontological frameworks and knowledge bases and define the
A Compositional, Diagrammatic Discourse, 2013
Journal of Applied Logic, 2014
Relational semantics, given by Kripke frames, play an essential role in the study of modal and in... more Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [5] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form. In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [4,5] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms. Traditionally, so-called phase semantics are used as models for (provability in) linear logic . These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.
Lecture Notes in Computer Science, 2015
In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (me... more In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implication-formulas. We use these results to derive a new version of subframe formulas for intuitionistic logic and to show that the uniform interpolants of meet-implication-formulas are not necessarily uniform interpolants in the full intuitionistic logic.