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23
Local Possibilistic Logic
 Journal of Applied NonClassical Logic
, 1997
"... Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natur ..."
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Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natural interpretation of logical connectives as operations on fuzzy sets. Due to the quantal structure of information states, we obtain a system which shares several features with (exponentialfree) intuitionistic linear logic. Soundness and completeness are proved, parametrically on the choice of the tnorm operation.
A Pointfree approach to Constructive Analysis in Type Theory
, 1997
"... The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from ..."
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The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the HeineBorel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the HahnBanach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional MartinLöf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...
On some peculiar aspects of the constructive theory of pointfree spaces
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Formal Topologies on the Set of FirstOrder Formulae
 Journal of Symbolic Logic
, 1998
"... this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for ..."
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this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for instance, in Fourman and Grayson [6], where the analogy between points of a locale and models of a theory is emphasised; the identification of formal points with Henkin sets, gives a precise form to this analogy. We replace the use of locales by formal topology, which can be expressed in a predicative framework such as MartinLof's type theory. Prooftheoretic issues are also considered by Dragalin [4], who presents a topological completeness proof using only finitary inductive definitions. Palmgren and Moerdijk [10] is also concerned with constructions of models: using sheaf semantics, they obtain a stronger conservativity result than the one in [3]. We will first investigate the difference between the DedekindMacNeille cover and the inductive cover. It easy to see that \Delta DM is stronger than \Delta I , that is, OE \Delta I U implies OE \Delta DM U , but the converse does not hold in general. The notion of point is not primitive in formal topology and therefore it is natural to require that a formal topology has some notion of positivity defined on the basic neighbourhoods; that a neighbourhood is positive then corresponds to, in ordinary point based topology, that it is inhabited by some point. We will show several negative results on positivity, both for the inductive topology and the DedekindMacNeille topology. The points of an inductive topology correspond to Henkin sets, but the DedekindMacNeille topology has, in general, no points. Our reasoning is constructi...
An Application of Constructive Completeness.
 In Proceedings of the Workshop TYPES '95
, 1995
"... this paper, we explore one possible effective version of this theorem, that uses topological models in a pointfree setting, following Sambin [11]. The truthvalues, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this m ..."
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this paper, we explore one possible effective version of this theorem, that uses topological models in a pointfree setting, following Sambin [11]. The truthvalues, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this more abstract notion of model. The first is that, by using formal topology, we get a remarkably simple completeness proof; it seems indeed simpler than the usual classical completeness proof. The second is that this completeness proof is now constructive and elementary. In particular, it does not use any impredicativity and can be formalized in intuitionistic type theory; this is of importance for us, since we want to develop model theory in a computer system for type theory. Formal topology has been developed in the type theory implementation ALF [1] by Cederquist [2] and the completeness proof we use has been checked in ALF by Persson [9]. In view of the extreme simplicity of this proof, it might be feared that it has no interesting applications. We show that this is not the case by analysing a conservativity theorem due to Dragalin [4] concerning a nonstandard extension of Heyting arithmetic. We can transpose directly the usual model theoretic conservativity argument, that we sketched above, in this framework. It seems likely that a direct syntactical proof of this result would have to be more involved. The first part of this paper presents a definition of topological models, Sambin's completeness proof, and an alternative completeness proof; we also discuss how Beth models relate to our approach. The second part shows how to use this in order to give a proof of Dragalin's conservativity result; our proof is different from his and, we believe, simpler. In [8] a stronger ...
Minimal Invariant Spaces in Formal Topology
 The Journal of Symbolic Logic
, 1996
"... this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a ..."
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this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace
Priestley Duality for Strong Proximity Lattices
"... In 1937 Marshall Stone extended his celebrated representation theorem for Boolean algebras to distributive lattices. In modern terminology, the representing topological spaces are zerodimensional stably compact, but typically not Hausdorff. In 1970, Hilary Priestley realised that Stone’s topology ..."
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In 1937 Marshall Stone extended his celebrated representation theorem for Boolean algebras to distributive lattices. In modern terminology, the representing topological spaces are zerodimensional stably compact, but typically not Hausdorff. In 1970, Hilary Priestley realised that Stone’s topology could be enriched to yield orderdisconnected compact ordered spaces. In the present paper, we generalise Priestley duality to a representation theorem for strong proximity lattices. For these a “Stonetype ” duality was given in 1995 in joint work between Philipp Sünderhauf and the second author, which established a close link between these algebraic structures and the class of all stably compact spaces. The feature which distinguishes the present work from this duality is that the proximity relation of strong proximity lattices is “preserved ” in the dual, where it manifests itself as a form of “apartness. ” This suggests a link with constructive mathematics which in this paper we can only hint at. Apartness seems particularly attractive in view of potential applications of the theory in areas of semantics where continuous phenomena play a role; there, it is the distinctness between different states which is observable, not equality. The idea of separating states is also taken up in our discussion of possible morphisms for which the representation theorem extends to an equivalence of categories.
Sequents, Frames, and Completeness
"... . Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the ..."
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. Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the unifying principle of the spatiality of coherence logic. In particular, the domain of disjunctive states, derived from the hyperresolution rule as used in disjunctive logic programs, can be seen as the frame freely generated from the opposite of a sequent structure. At the categorical level, we present equivalences among the categories of sequent structures, distributive lattices, and spectral locales using appropriate morphisms. Key words: sequent structures, lattices, frames, domain theory, resolution, category. Introduction Entailment relations were introduced by Scott as an abstract description of Gentzen's sequent calculus [1315]. It can be seen as a generalisation of the ear...
Fusion of Symbolic Knowledge and Uncertain Information in Robotics
 International Journal of Intelligent Systems
, 2001
"... The interpretation of data coming from the real world may require different and often complementary uncertainty models: some are better described by possibility theory, others are intrinsically probabilistic. A logic for belief functions is introduced to axiomatize both semantics as special cases. A ..."
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The interpretation of data coming from the real world may require different and often complementary uncertainty models: some are better described by possibility theory, others are intrinsically probabilistic. A logic for belief functions is introduced to axiomatize both semantics as special cases. As it properly extends classical logic, it also allows the fusion of data with different semantics and symbolic knowledge. The approach has been applied to the problem of mobile robot localization. For each place in the environment, a set of logical propositions allows the system to calculate the belief of the robot's presence as a function of the partial evidences provided by the individual sensors.