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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...
Stone duality for markov processes
 In Proceedings of the 28th Annual IEEE Symposium on Logic in Computer Science: LICS
"... We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove a Stonetype duality theorem between countable Aumann algebras and countablygenerated continuousspace Markov proc ..."
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We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove a Stonetype duality theorem between countable Aumann algebras and countablygenerated continuousspace Markov processes. Our results subsume existing results on completeness of probabilistic modal logics for Markov processes. 1.
Experimental Testing of Feature Structures and Unification
"... This paper presents two experiments where feature structures and unification provide an explanatory framework for what has been called illusory conjunctions in visual perception. Feature Structures and Unification has been successfully applied to computational analyses of natural languages. However, ..."
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This paper presents two experiments where feature structures and unification provide an explanatory framework for what has been called illusory conjunctions in visual perception. Feature Structures and Unification has been successfully applied to computational analyses of natural languages. However, this efficient computational technique has not been experimentally tested among human subjects. This is an attempt to show some psychological validity for the notion of feature structures and unification.
Strong Completeness for Markovian Logics
"... Abstract. In this paper we present Hilbertstyle axiomatizations for three logics for reasoning about continuousspace Markov processes (MPs): (i) a logic for MPs defined for probability distributions on measurable state spaces, (ii) a logic for MPs defined for subprobability distributions and (iii ..."
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Abstract. In this paper we present Hilbertstyle axiomatizations for three logics for reasoning about continuousspace Markov processes (MPs): (i) a logic for MPs defined for probability distributions on measurable state spaces, (ii) a logic for MPs defined for subprobability distributions and (iii) a logic defined for arbitrary distributions. These logics are not compact so one needs infinitary rules in order to obtain strong completeness results. We propose a new infinitary rule that replaces the socalled Countable Additivity Rule (CAR) currently used in the literature to address the problem of proving strong completeness for these and similar logics. Unlike the CAR, our rule has a countable set of instances; consequently it allows us to apply the RasiowaSikorski lemma for establishing strong completeness. Our proof method is novel and it can be used for other logics as well. 1
École Doctorale IAEM Lorraine Département de Formation Doctorale en informatique
"... appliquée à l’analyse et la vérification de systèmes infinis Habilitation présentée et soutenue publiquement le 14/11/2011 pour l’obtention d’une Habilitation à Diriger des Recherches de l’Université Nancy 2 (spécialité informatique) par ..."
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appliquée à l’analyse et la vérification de systèmes infinis Habilitation présentée et soutenue publiquement le 14/11/2011 pour l’obtention d’une Habilitation à Diriger des Recherches de l’Université Nancy 2 (spécialité informatique) par